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+{"text":"\n\n\n\\section{Introduction}\nPairwise sentence scoring tasks have wide applications in NLP. They can be used in information retrieval, question answering, duplicate question detection, or clustering. \nAn approach that sets new state-of-the-art performance for many tasks including pairwise sentence scoring is BERT \\cite{devlin2018bert}. \nBoth sentences are passed to the network and attention is applied across all tokens of the inputs. \nThis approach, where both sentences are simultaneously passed to the network, is called \\textit{cross-encoder} \\cite{Humeau2020Poly-encoders}. \n\n\\begin{figure}[t]\n\\centering\n\\begin{center}\n    \\includegraphics[trim=70 0 50 0,clip,width=0.45\\textwidth]{data\/English-STS.pdf}\n    \\captionof{figure}{Spearman rank correlation ($\\rho$) test scores for different STS Benchmark (English) training sizes.}\n    \\label{fig:english-sts}\n\\end{center}\n\\end{figure}\n\nA downside of cross-encoders is the extreme computational overhead for many tasks. For example, clustering of 10,000 sentences has a quadratic complexity with a cross-encoder and would require about 65 hours with BERT \\cite{reimers-2019-sentence-bert}. End-to-end information retrieval is also not possible with cross-encoders, as they do not yield independent representations for the inputs that could be indexed. \nIn contrast, \\textit{bi-encoders} such as Sentence BERT (SBERT) \\cite{reimers-2019-sentence-bert}  encode each sentence independently and map them to a dense vector space. \nThis allows efficient indexing and comparison. For example, the complexity of clustering 10,000 sentences is reduced from 65 hours to about 5 seconds \\cite{reimers-2019-sentence-bert}. Many real-world applications hence depend on the quality of bi-encoders.\n\nA drawback of the SBERT bi-encoder is usually a lower performance in comparison with the BERT cross-encoder. \nWe depict this in \\autoref{fig:english-sts}, where we compare a fine-tuned cross-encoder (BERT) and a fine-tuned  bi-encoder (SBERT) over the popular English STS Benchmark dataset\\footnote{\\href{http:\/\/ixa2.si.ehu.es\/stswiki\/index.php\/STSbenchmark}{http:\/\/ixa2.si.ehu.es\/stswiki\/index.php\/STSbenchmark}} \\cite{cer-etal-2017-semeval} for different training sizes and spearman rank correlation ($\\rho$) on the test split.\n\nThis performance gap is the largest when little training data is available. \nThe BERT cross-encoder can compare both inputs simultaneously, while the SBERT bi-encoder has to solve the much more challenging task of mapping inputs independently to a meaningful vector space which requires a sufficient amount of training examples for fine-tuning.\n \nIn this work, we present a data augmentation method, which we call \\textit{Augmented SBERT} (AugSBERT), that uses a BERT cross-encoder to improve the performance for the SBERT bi-encoder. We use the cross-encoder to label new input pairs, which are added to the training set for the bi-encoder.  The SBERT bi-encoder is then fine-tuned on this larger augmented training set, which yields a significant performance increase. As we show, selecting the input pairs for soft-labeling with the cross-encoder is non-trivial and crucial for improving performance. \nOur method is easy to apply to many pair classification and regression problems, as we show in the exhaustive evaluation of our approach.\n\nFirst, we evaluate the proposed AugSBERT method on four diverse tasks: Argument similarity, semantic textual similarity, duplicate question detection, and news paraphrase identification. We observe consistent performance increases of 1 to 6 percentage points over the state of the art SBERT bi-encoder's performance.\nNext, we demonstrate the strength of AugSBERT in a domain adaptation scenario.\nSince the bi-encoder is not able to map the new domain to a sensible vector space, the performance drop on the target domain for SBERT bi-encoders is much higher than for BERT cross-encoders. \nIn this scenario, AugSBERT achieves a performance increase of up to 37 percentage points. \n\n\n\n\\section{Related Work}\nSentence embeddings are a well studied area in recent literature. Earlier techniques included unsupervised methods such as Skip-thought vectors \\cite{10.5555\/2969442.2969607} and supervised methods such as InferSent \\cite{conneau-etal-2017-supervised} or USE \\cite{cer-etal-2018-universal}. \nFor pairwise scoring tasks, more recent sentence embedding techniques are also able to encode a pair of sentences jointly.\nAmong these, BERT \\cite{devlin2018bert} can be used as a cross-encoder.\nBoth inputs are separated by a special \\texttt{SEP} token and multi-head attention is applied over all input tokens. While the BERT cross-encoder achieves high performances for many sentence pair-tasks, a drawback is that no independent sentence representations are generated. This drawback was addressed by SBERT \\cite{reimers-2019-sentence-bert}, which applies BERT independently on the inputs followed by mean pooling on the output to create fixed-sized sentence embeddings.\n\n\\newcite{Humeau2020Poly-encoders} showed that cross-encoders typically outperform bi-encoders on sentence scoring tasks. They proposed a third strategy (poly-encoders), that is in-between cross- and bi-encoders. Poly-encoders utilize two separate transformers, one for the candidate and one for the context. A given candidate is represented by one vector, while the context is jointly encoded with the candidates (similar to cross-encoders). Unlike cross-encoder's full self attention technique, poly-encoders apply attention between two inputs only at the top layer. Poly-encoders have the drawback that they are only practical for certain applications: The score function is not symmetric, i.e., they cannot be applied for tasks with a symmetric similarity relation. Further, poly-encoder representations cannot be efficiently indexed, causing issues for retrieval tasks with large corpora sizes. \n\n\\newcite{chen-etal-2020-dipair} propose the DiPair architecture which, similar to our work, also uses a cross-encoder model to annotate unlabeled pairs for fine-tuning a bi-encoder model. \nDiPair focuses on inference speed and provides a detailed ablation for optimal bi-encoder architectures for performance versus speed trade-offs. The focus of our work are sampling techniques, which we find crucial for performance boosts in the bi-encoder model while keeping its architecture constant.\n\n\nOur proposed data augmentation approach is based on semi-supervision \\cite{10.1145\/279943.279962} for in-domain tasks, which has been applied successfully for a wide range of tasks. \\newcite{uva-etal-2018-injecting} train a SVM model with few gold samples and apply semi-supervision with pre-training neural networks. Another common strategy is to generate paraphrases of existent sentences, for example, by replacing words with synonyms \\cite{wei-zou-2019-eda}, by using round-trip translation \\cite{wei2018fast,xie2020unsupervised}, or with seq2seq-models \\cite{kumar-etal-2019-submodular}. Other approaches generate synthetic data by using generative adversarial networks \\cite{tanaka2019data}, by using a language model to replace certain words \\cite{cbert-aug} or to generate complete sentences \\cite{anaby2019not}. These data augmentation approaches have in common that they were applied to single sentence classification tasks. In our work, we focus on \\textit{sentence pair tasks}, for which we need to generate suitable sentence pairs. As we show, randomly combining sentences is insufficient. Sampling appropriate pairs has a decisive impact on performance \nwhich corresponds to recent findings on similar datasets \\cite{peinelt-etal-2019-aiming}.\n\n\n\\section{Methods}\nIn this section we present Augmented SBERT for diverse sentence pair in-domain tasks. We also evaluate our method for domain adaptation tasks.\n\n\\subsection{Augmented SBERT}\n\n    \n     Given a pre-trained, well-performing cross-encoder, we sample sentence pairs according to a certain \\textit{sampling strategy} (discussed later) and label these using the cross-encoder. We call these weakly labeled examples the \\textit{silver dataset} and they will be merged with the gold training dataset. We then train the bi-encoder on this extended training dataset. We refer to this model as Augmented SBERT (AugSBERT). The process is illustrated in \\autoref{fig:augsbert}.\n    \n    \\begin{figure}[h]\n    \\centering\n    \\begin{center}\n        \\includegraphics[trim=75 5 60 0,clip,width=0.4\\textwidth]{data\/flowchart.pdf}\n        \\captionof{figure}{Augmented SBERT In-domain approach}\n        \\label{fig:augsbert}\n    \\end{center}\n    \\end{figure}\n\n    \\paragraph{Pair Sampling Strategies} The novel sentence pairs, that are to be labeled with the cross-encoder, can either be new data or we can re-use individual sentences from the gold training set and re-combine pairs. In our in-domain experiments, we re-use the sentences from the gold training set. This is of course only possible if not all combinations have been annotated. However, this is seldom the case as there are $n\\times(n-1)\/2$ possible combinations for $n$ sentences. Weakly labeling all possible combinations would create an extreme computational overhead, and, as our experiments show, would likely not lead to a performance improvement. Instead, using the right sampling strategy is crucial to achieve a performance improvement.\n    \n    \\textbf{\\textit{Random Sampling (RS):}} We randomly sample a sentence pair and weakly label it with the cross-encoder. Randomly selecting two sentences usually leads to a dissimilar (negative) pair; positive pairs are extremely rare. This skews the label distribution of the silver dataset heavily towards negative pairs.\n    \n    \\textbf{\\textit{Kernel Density Estimation (KDE):}} We aim to get a similar label distribution for the silver dataset as for the gold training set. To do so, we weakly label a large set of randomly sampled pairs and then keep only certain pairs. For classification tasks, we keep all the positive pairs. Subsequently we randomly sample out negative pairs from the remaining dominant negative silver-pairs, in a ratio identical to the gold dataset training distribution (positives\/negatives). For regression tasks, we use kernel density estimation (KDE) to estimate the continuous density functions $F_{gold}(s)$ and $F_{silver}(s)$ for scores $s$. \n    We try to minimize KL Divergence \\cite{kullback1951} between distributions\n \n    using a sampling function which retains a sample with score $s$ with probability $Q(s)$: \n \n    \n    \\begin{small}\n    \\begin{equation*}\n      Q(s) =\n        \\begin{cases}\n            \\hfil 1 & \\text{if } F_{gold}(s) \\geq F_{silver}(s) \\\\ \n          \\\\\n          \\dfrac{F_{gold}(s)}{F_{silver}(s)} & \\text{if } F_{gold}(s) < F_{silver}(s) \\\\\n        \\end{cases}   \n    \\end{equation*}\n    \\end{small}\n    \n    Note, that the KDE sampling strategy is computationally inefficient as it requires labeling many, randomly drawn samples, which are later discarded.\n    \n    \\textbf{\\textit{BM25 Sampling (BM25):}} In information retrieval, the Okapi BM25 \\cite{Amati2009} algorithm is based on lexical overlap and is commonly used as a scoring function by many search engines. We utilize ElasticSearch\\footnote{\\href{https:\/\/www.elastic.co\/}{https:\/\/www.elastic.co\/}} for the creation of indices which helps in fast retrieval of search query results. For our experiments, we index every unique sentence, query for each sentence and retrieve the top $k$ similar sentences. These pairs are then weakly labeled using the cross-encoder. Indexing and retrieving similar sentences is efficient and all weakly labeled pairs will be used in the silver dataset.\n    \n     \\textbf{\\textit{Semantic Search Sampling (SS):}} A drawback of BM25 is that only sentences with lexical overlap can be found. Synonymous sentences with no or little lexical overlap will not be returned, and hence, not be part of the silver dataset. We train a bi-encoder (SBERT) on the gold training set as described in \\autoref{sec:experimental_setup} and use it to sample further, similar sentence pairs. We use cosine-similarity and retrieve for every sentence the top $k$ most similar sentences in our collection. For large collections, approximate nearest neighbour search like Faiss\\footnote{\\href{https:\/\/github.com\/facebookresearch\/faiss}{https:\/\/github.com\/facebookresearch\/faiss}} could be used to quickly retrieve the $k$ most similar sentences. \n     \n    \\textbf{\\textit{BM25~+~Semantic Search Sampling (BM25-S.S.):}} We apply both BM25 and Semantic Search (S.S.) sampling techniques simultaneously. Aggregating the strategies helps capture the lexical and semantically similar sentences but skews the label distribution towards negative pairs.\n     \n     \n    \\paragraph{Seed Optimization} \\newcite{dodge2020fine} show a high dependence on the random seed for transformer based models like BERT, as it converges to different minima that generalize differently to unseen data \\cite{LeCun1998, Erhan2010, reimers-gurevych-2017-reporting}. This is especially the case for small training datasets. In our experiments, we apply \\textit{seed optimization}: We train with 5 random seeds and select the model that performs best on the development set. In order to speed this up, we apply \\textit{early stopping} at 20\\% of the training steps and only continue training the best performing model until the end. We empirically found that we can predict the final score with high confidence at 20\\% of the training steps (\\autoref{sec:seed-opt}). \n   \n\n\\begin{figure}[htb]\n    \\centering\n    \\begin{center}\n        \\includegraphics[trim=0 10 0  0,clip,width=0.5\\textwidth]{data\/flowchart-domain-transfer.pdf}\n        \\captionof{figure}{Domain adaptation with AugSBERT.}\n        \\label{fig:augsbert-cross-domain}\n    \\end{center}\n\\end{figure}\n\n\\subsection{Domain Adaptation with AugSBERT}\n\\label{sec:DomainAdaptationwithAugSBERT}\n\nUntil now we discussed Augmented SBERT for in-domain setups, i.e., when the training and test data are from the same domain. \nHowever, we expect an even higher performance gap of SBERT on out-of-domain data.\nThis is because SBERT fails to map sentences with unseen terminology to a sensible vector space. Unfortunately, annotated data for new domains is rarely available. \n\nHence, we evaluate the proposed data augmentation strategy for domain adaptation: We first fine-tune a cross-encoder (BERT) over the source domain containing pairwise annotations. After fine-tuning, we use this fine-tuned cross-encoder to label the target domain. Once labeling is complete, we train the bi-encoder (SBERT) over the labeled target domain sentence pairs (\\autoref{fig:augsbert-cross-domain}). \n\n\n\n\n\\section{Datasets}\n\n    \\begin{table*}[t!]\n        \\centering\n        \\small\n        \\begin{tabular}{ c | c c c c c }\n            \\toprule\n               \\textbf{Dataset} & \\textbf{Spanish-STS} & \\textbf{BWS (cross-topic)}  & \\textbf{BWS (in-topic)} & \\textbf{Quora-QP} & \\textbf{MRPC} \\\\\n             \\midrule\n             \\# training-samples & 1,400 & 2125 & 2471 & 10,000 & 4,340 \\\\\n             \\# development-samples & 220 & 425 & 478 & 3,000 & 731 \\\\\n             \\# testing-samples & 250 & 850 & 451 & 3,000 & 730 \\\\\n             \\midrule\n             \\# total-samples & \\textbf{1,870} & \\textbf{3,400} & \\textbf{3,400} & \\textbf{16,000} & \\textbf{5,801} \\\\\n             \\bottomrule\n        \\end{tabular}\n        \\centering\n        \\caption{Summary of all datasets being used for diverse in-domain sentence pair tasks in this paper.}\n        \\label{tab:my_label_1}\n    \\end{table*}\n\n\\begin{table*}[t]\n    \\centering\n    \\small\n        \\begin{tabular}{l|p{5cm}p{7cm}|c}\n        \\toprule\n        \\textbf{Dataset} & \\textbf{Sentence 1} & \\textbf{Sentence 2} & \\textbf{Score}\\\\ \\midrule\n        BWS & Cloning treats children as objects.\n        & It encourages parents to regard their children as property. & 0.89 \\\\ \\midrule\n        \n        Quora-QP & How does one cook broccoli? & What are the best ways to cook broccoli? & 1 \\\\ \\midrule\n        \n        Spanish-STS & Dos hombres en trajes rojos practicando artes marciales.\n        & Dos hombre en uniformes de artes marciales entrenando.\n        & 0.80 \\\\ \\midrule\n        MRPC & The DVD-CCA then appealed to the state Supreme Court. \n        & DVD CCA appealed that decision to the U.S. Supreme Court. & 1 \\\\ \\bottomrule\n        \\end{tabular}\n        \\caption{Dataset examples for our in-domain tasks. We report the normalized similarity score $[0,1]$ for regression tasks and the binary label $\\{0,1\\}$ for classification tasks.}\n        \\label{tab:dataset-examples}\n    \\end{table*}\n\n\nSentence pair scoring can be differentiated in regression and classification tasks. Regression tasks assign a score to indicate the similarity between the inputs. For classification tasks, we have distinct labels, for example, \\textit{paraphrase} vs. \\textit{non-paraphrase}. \n\n\\subsection{Single-Domain Datasets}\n\nIn our single-domain (i.e. in-domain) experiments, we use two sentence pair regression tasks: semantic textual similarity and argument similarity. \nFurthermore, we use two binary sentence pair classification tasks: Duplicate question detection and news paraphrase identification. \nExamples for all datasets are given in \\autoref{tab:dataset-examples}.\n\n\n\n    \\textbf{SemEval Spanish STS:} Semantic Textual Similarity (STS)\\footnote{\\href{https:\/\/ixa2.si.ehu.es\/stswiki}{https:\/\/ixa2.si.ehu.es\/stswiki}} is the task of assessing the degree of similarity between two sentences over a scale ranging from $[0,5]$ with $0$ indicating no semantic overlap and $5$ indicating identical content \\cite{agirre2016semeval}. We choose Spanish STS data to test our methods for a different language than English. For our training and development dataset, we use the datasets provided by SemEval STS 2014 \\cite{agirre-etal-2014-semeval} and SemEval STS 2015 \\cite{agirre2015semeval}. These consist of  annotated sentence pairs from news articles and from Wikipedia. As test set, we use SemEval STS 2017 \\cite{cer-etal-2017-semeval}, which annotated image caption pairs from SNLI \\cite{snli:emnlp2015}. For all our experiments, we normalise the original similarity scores to $[0,1]$ by dividing the score by 5.\n    \n    \\textbf{BWS Argument Similarity Dataset (BWS):} Existing similarity datasets have the disadvantage that the sentence pair selection\/sampling process is not always comprehensible.\n    To overcome this limitation, we create and publicly release a novel dataset\\footnote{Public Data Release (BWS Argument Similarity Corpus): \\href{https:\/\/tudatalib.ulb.tu-darmstadt.de\/handle\/tudatalib\/2496.2}{https:\/\/tudatalib.ulb.tu-darmstadt.de\/handle\/tudatalib\/2496}} \n    for argument similarity.\n    \n    We annotate sentential arguments on controversial topics on a continuous scale. We use the dataset by \\newcite{Stab2018b}, which contains pro and con stance arguments for eight controversial topics ($T_1$ - $T_8$) (``cloning'', ``abortion'', ``minimum wage'', ``marijuana legalization'', ``nuclear energy'', ``death penalty'', ``gun control'', ``school uniforms'') retrieved from heterogeneous web sources.\n \n    Previous work addressing argument similarity \\cite{MisraEW16,Reimers:2019:ACL} used discrete scales. However, expressing an inherently continuous property in this way is counter-intuitive and potentially unreliable due to different assumptions made when binning a range of values into a discrete class \\cite{RePEc:uwp:landec:v:86:y:2010:iii:1:p530-544}.\n   \n    Collecting continuous annotations is complex due to selection bias and due to a lack of consistency for a single annotator \\cite{kendallcorrelation}. \n    To solve the consistency problem, we apply a comparative approach, which converts the annotation into a preference problem: the annotators stated their preference on pairs of sentential arguments. We utilized the Best-Worst Scaling (\\textit{BWS}) method \\cite{kiritchenko-mohammad-2016-capturing} to reduce the number of required annotations.\n   \n    For each topic regardless of stance, all arguments were randomly paired and for ensuring a certain proportion of similar arguments within the pairings, a distant supervision filtering strategy was implemented by labeling pairs with scores between 0 and 1 using the system proposed by \\newcite{MisraEW16}. \n    Next, all argument pairs were sampled with a desired similarity distribution, by creating argument pair bins across three categories: top 1\\%, top 2-50\\% and remaining pairs. \n    As the final step, we randomly drew pairs from the top 1\\% with 50\\% probability, and with each 25\\% from the two other bins.\n    \n    The resulting argument pairs were annotated using crowdsourcing via the Amazon Mechanical Turk Platform.\n    For each annotation task, workers were shown four argument pairs and had to select the most and least similar pair amongst them. Each of these tasks was assigned to four different workers. \n    To assess the quality of the resulting annotations, we used split-half reliability measure \\cite{10.2307\/1434452}.\n   \n   \n    Workers' votes were split by half and used to independently rank all argument pairs with the BWS method for each half on each task.\n    Finally, the Spearman's rank correlation between the resulting rankings is calculated as a proxy for consistency.\n    The resulting average correlation across all topics in our dataset is 0.66 (random splits are repeated 25 times and final scores averaged), which, given the small number of votes per half (two), is in an acceptable range and reflects the difficulty of this task \\cite{kiritchenko-mohammad-2016-capturing}.\n    \\autoref{tab:bws_topic} lists the mean split-half reliability estimates for all topics (averaged over 25 random splits) in the dataset.\n    \n    \\begin{table}[ht]\n    \\centering\n    \\small\n    \\begin{tabular}{lc|lc}\n    \\toprule\n         \\textbf{Topic~$T$} & \\textbf{Score} & \\textbf{Topic~$T$} & \\textbf{Score} \\\\ \\midrule\n         Cloning & 0.84 & Nuclear energy & 0.64\\\\\n         Abortion & 0.79 & Death penalty  & 0.58 \\\\\n         Minimum wage & 0.50 & Gun control  & 0.59 \\\\\n         Marijuana legal. & 0.57 & School uniforms  & 0.64 \\\\\n         \\midrule\n         \\multicolumn{4}{c}{Whole dataset = 0.66} \\\\\n        \\bottomrule\n    \\end{tabular}\n    \\caption{Mean split-half reliability estimate is calculated using Spearman's rank correlation  $\\rho$ per topic $T$ and over the whole \\emph{BWS Argument Similarity} dataset.}\n    \\label{tab:bws_topic}\n\\end{table}\n\n    \n    \n    We use the resulting \\textit{BWS Argument Similarity Dataset} with different splitting strategies in our paper.\n    In \\emph{cross-topic} tasks, we fix topics ($T_1$ - $T_5$) for training, $T_6$ for development and ($T_7$ and $T_8$) for test sets. This is a difficult task, as models are evaluated on completely unseen topics. \n    \n    \n    Note that the cross-topic experiments on this dataset are quite different from cross-domain tasks (\\autoref{sec:DomainAdaptationwithAugSBERT}): \n    the model fine-tunes in-domain on fixed topics ($T_1$ - $T_5$ in our case) and is evaluated on unseen topics, whereas in the domain adaptation experiments we fine-tune on target domain data.\n    For \\emph{in-topic}, we randomly sample fixed and disjoint pairs from each and every topic ($T_1$ - $T_8$) and create our train, development and test splits with approximately equal number of pairs from each topic.\n\n    \\textbf{Quora Question Pairs (Quora-QP):} Duplicate question classification identifies whether two questions are duplicates. Quora released a dataset\\footnote{\\href{{https:\/\/www.quora.com\/q\/quoradata\/First-Quora-Dataset-Release-Question-Pairs}}{https:\/\/www.quora.com\/q\/quoradata\/First-Quora-Dataset-Release-Question-Pairs}} containing 404,290 question pairs. We start with the same dataset partitions from \\newcite{ijcai2017-579}\\footnote{ \\href{https:\/\/drive.google.com\/file\/d\/0B0PlTAo--BnaQWlsZl9FZ3l1c28}{https:\/\/drive.google.com\/file\/d\/0B0PlTAo--BnaQWlsZl9FZ3l1c28 }}. We remove all overlaps and ensure that a question in one split of the dataset does not appear in any other split to mitigate the transductive classification problem  \\cite{10.1007\/978-3-642-15880-3_42}. As we observe performance differences between cross- and bi-encoders mainly for small datasets, we randomly downsample the training set to 10,000 pairs while preserving the original balance of non-duplicate to duplicate question pairs.\n    \n    \\textbf{Microsoft Research Paraphrase Corpus (MRPC):} \\newcite{dolan-etal-2004-unsupervised} presented a paraphrase identification dataset consisting of sentence pairs automatically extracted from online news sources. Each pair was manually annotated by two human judges whether they describe the same news event. \n    We use the originally provided train-test splits\\footnote{\\href{https:\/\/github.com\/wasiahmad\/paraphrase_identification}{https:\/\/github.com\/wasiahmad\/paraphrase\\_identification}}. We ensured that all splits have disjoint sentences.\n\n\\subsection{Multi-Domain Datasets}\n\nOne of the most prominent sentence pair classification tasks with datasets from multiple domains is \\textit{duplicate question detection}.\nSince our focus is on pairwise sentence scoring, we model this task as a question vs. question (title\/headline) binary classification task.\n    \n    \\textbf{AskUbuntu,  Quora,  Sprint, and SuperUser:}\n    We replicate the setup of \\newcite{shah-etal-2018-adversarial} for domain adaptation experiments. The AskUbuntu and SuperUser data comes from Stack Exchange, which is a family of technical community support forums. Sprint FAQ is a crawled dataset from the Sprint technical forum website. We exclude Apple and Android datasets due to unavailability of labeled question pairs. The Quora dataset (originally derived from the Quora website) is artificially balanced by removing negative question pairs. The statistics for the datasets can be found in \\autoref{tab:multi-domain}. \n    Since negative question pairs are not explicitly labeled, \\newcite{shah-etal-2018-adversarial} add 100 randomly sampled (presumably) negative question pairs per duplicate question for all datasets except Quora, which has explicit negatives. \n    \n\\begin{table}[t]\n    \\centering\n    \\small\n    \\resizebox{.49\\textwidth}{!}{%\n    \\begin{tabular}{l c  c  c  c}\n    \\toprule\n    \\textbf{Dataset $k$} &  \\textbf{Train \/ Dev \/ Test} & \\textbf{Train} & \\textbf{Dev \/ Test} \\\\ \n    \\textbf{} &  (Total Pairs) & (Ratio) & (Ratio) \\\\\n    \\midrule\n    \\textbf{AskUbuntu} & 919706 \/ 101k \/ 101k & 1 : 100 & 1 : 100 \\\\ \n    \\textbf{Quora} & 254142 \/ 10k \/ 10k & 3.71 : 100 & 1 : 1\\\\ \n    \\textbf{Sprint} & 919100 \/ 101k \/ 101k & 1 : 100 & 1 : 100\\\\  \n    \\textbf{SuperUser} & 919706 \/ 101k \/ 101k & 1 : 100 & 1 : 100\\\\ \\bottomrule\n    \\end{tabular}\n    }\n    \\centering\n    \\caption{Summary of multi-domain datasets originally proposed by \\newcite{shah-etal-2018-adversarial} and used for our domain adaptation experiments. Ratio denotes the duplicate pairs (positives) vs. not duplicate pairs (negatives).}\n    \\label{tab:multi-domain}\n\\end{table}\n  \n\\begin{table*}[t]\n    \\centering\n    \\resizebox{\\textwidth}{!}{%\n    \\small\n    \\begin{tabular}{l | c | c | c | c  c | c }\n         \\toprule\n         \\multicolumn{1}{l}{Task} & \\multicolumn{1}{l}{} &\n        \\multicolumn{3}{c}{Regression ($\\rho \\times 100$)}    &\n        \\multicolumn{2}{c}{Classification ($F_1$)}    \\\\ \n        \\cmidrule(lr){3-5}\n        \\cmidrule(lr){6-7}\n        \\multicolumn{1}{l}{Model \/ Dataset} &\n        \\multicolumn{1}{c}{(Seed Opt.)} & \n        \\multicolumn{1}{c}{\\textbf{Spanish-STS}} &\n        \\multicolumn{1}{c}{\\textbf{BWS (cross-topic)}} &\n        \\multicolumn{1}{c}{\\textbf{BWS (in-topic)}} &\n        \\multicolumn{1}{c}{\\textbf{Quora-QP}} &\n        \\multicolumn{1}{c}{\\textbf{MRPC}} \\\\\n        \\midrule\n            Baseline & - & 30.27 & 5.53 & 6.98 & 66.67 & 80.80 \\\\\n            USE \\cite{yang2019multilingual} & - & 86.86 & 53.43 & 57.23 & 74.16 & 81.51 \\\\\n        \\midrule\n            BERT & \\xmark & 77.50 $\\pm$ 1.49 & 65.06 $\\pm$ 1.06  &  65.91 $\\pm$ 1.20 & 80.40 $\\pm$ 1.05 & 88.95 $\\pm$ 0.67  \\\\\n            SBERT & \\xmark & 68.36 $\\pm$ 5.28 & 58.04 $\\pm$ 1.46 & 61.20 $\\pm$ 1.66 & 73.44 $\\pm$ 0.65 & 84.44 $\\pm$ 0.68 \\\\ \n        \\midrule\n            BERT (\\emph{Upper-bound}) & \\cmark &  77.74 $\\pm$ 1.24 & 65.78 $\\pm$ 0.78 & 66.54 $\\pm$ 0.94 & 81.23 $\\pm$ 0.93 &  89.00 $\\pm$ 0.56 \\\\\n            SBERT (\\emph{Lower-bound}) & \\cmark & 72.07 $\\pm$ 2.05 & 60.54 $\\pm$ 0.99 & 63.77 $\\pm$ 2.29 & 74.66\t $\\pm$ 0.31 & 84.39 $\\pm$ 0.51 \\\\\n            SBERT-NLPAug & \\cmark & 74.11 $\\pm$ 2.58 & 58.15 $\\pm$ 1.66 & 61.15 $\\pm$ 0.86 & 73.08 $\\pm$ 0.42 & 84.47 $\\pm$ 0.79 \\\\            \n            \n        \\midrule\n\n            AugSBERT-R.S. & \\cmark & 62.05 $\\pm$ 2.53 & 59.95 $\\pm$ 0.70 & 64.54 $\\pm$ 1.90 & 73.42 $\\pm$ 0.74 & 82.28 $\\pm$ 0.38 \\\\\n            AugSBERT-KDE & \\cmark & 74.67 $\\pm$ 1.01 & \\textbf{61.49 $\\pm$ 0.71} & \\textbf{69.76 $\\pm$ 0.50} & \\textbf{79.31 $\\pm$ 0.46} & 84.33 $\\pm$ 0.27 \\\\\n            AugSBERT-BM25 & \\cmark & 75.08 $\\pm$ 1.94 & 61.48 $\\pm$ 0.73 & 68.63 $\\pm$ 0.79 & 79.01 $\\pm$ 0.45 & \\textbf{85.46 $\\pm$ 0.52} \\\\\n            AugSBERT-S.S. & \\cmark & 74.99 $\\pm$ 2.30 & 61.05 $\\pm$ 1.02 & 68.06 $\\pm$ 0.93 & 77.20 $\\pm$ 0.41 & 82.42 $\\pm$ 0.32 \\\\\n            AugSBERT-BM25+S.S. & \\cmark & \\textbf{76.24 $\\pm$ 1.42} & 59.41 $\\pm$ 0.98 & 63.30 $\\pm$ 1.34 & 72.45 $\\pm$ 0.77 & 82.68 $\\pm$ 0.33 \\\\\n        \\bottomrule\n    \\end{tabular}\n    }\n    \\centering\n    \\caption{Summary of all the datasets being used for the in-domain tasks in this paper. STS and BWS are regression tasks, where we report Spearman's rank correlation $\\rho \\times 100$. Quora-QP and MRPC are classification tasks, where we report $F_1$ score of the positive class. Scores with the best AugSBERT strategy are highlighted. Corresponding development set performances can be found in \\autoref{sec:dev-performances}, \\autoref{tab:dev-results}.}\n    \\label{tab:results}\n\\end{table*}\n\n\n\n\\section{Experimental Setup}\\label{sec:experimental_setup}\nWe conduct our experiments using PyTorch Huggingface's transformers \\cite{wolf2019huggingfaces} and the sentence-transformers framework\\footnote{\\href{https:\/\/github.com\/UKPLab\/sentence-transformers}{https:\/\/github.com\/UKPLab\/sentence-transformers}} \\cite{reimers-2019-sentence-bert}. The latter showed that BERT outperforms other transformer-like networks when used as bi-encoder. For English datasets, we use \\textit{bert-base-uncased}  and for the Spanish dataset we use \\textit{bert-base-multilingual-cased}. Every AugSBERT model exhibits computational speeds identical to the SBERT model \\cite{reimers-2019-sentence-bert}.\n    \n    \\textbf{Cross-encoders} \\quad We fine-tune the BERT-uncased model by optimizing a variety of hyperparameters: hidden-layer sizes, learning-rates and batch-sizes. We add a linear layer with sigmoid activation on top of the \\textit{[CLS]} token to output scores 0 to 1. We achieve optimal results with the combination: learning rate of $1\\times10^{-5}$, hidden-layer sizes in $\\{200,400\\}$ and a batch-size of $16$. Refer to  \\autoref{tab:BERT-hyper-opt} in \\autoref{sec:hyperparameter-tuning}.\n  \n    \\textbf{Bi-encoders} \\quad We fine-tune SBERT with a batch-size of $16$, a fixed learning rate of $2\\times10^{-5}$, and AdamW optimizer. \\autoref{tab:SBERT-hyper-opt} in \\autoref{sec:hyperparameter-tuning} lists hyper-parameters we initially evaluated.\n    \n    \\textbf{BM25 and Semantic Search} \\quad We evaluate for various top $k$ in $\\{3, ..., 18\\}$. We conclude the impact of $k$ is not big and overall accomplish best results with $k=3$ or $k=5$ for our experiments. More details in \\autoref{sec:top-k}.\n    \n    \\textbf{Evaluation} \\quad If not otherwise stated, we repeat our in-domain experiments with 10 different random seeds and report mean scores along with standard deviation. For in-domain regression tasks (STS and BWS), we report the Spearman's rank correlation ($\\rho \\times 100$) between predicted and gold similarity scores and for in-domain classification tasks (Quora-QP, MRPC), we determine the optimal threshold from the development set and use it for the test set. We report the $F_1$ score of the positive label. For all domain adaptation tasks, we weakly-label the target domain training dataset and measure AUC(0.05) as the metric since it is more robust against false negatives \\cite{shah-etal-2018-adversarial}. AUC(0.05) is the area under the curve of the true positive rate as function of the false positive rate (\\emph{fpr}), from \\emph{fpr} = 0 to \\emph{fpr} = 0.05.\n    \n    \\textbf{Baselines} \\quad For the in-domain regression tasks, we use  Jaccard similarity to measure the word overlap of the two input sentences. For the in-domain classification tasks, we use a majority label baseline. Further, we compare our results against Universal Sentence Encoder (USE) \\cite{yang2019multilingual}, which is a popular state-of-the-art sentence embedding model trained on a wide rang of training data. We utilise the multilingual model\\footnote{\\href{https:\/\/tfhub.dev\/google\/universal-sentence-encoder-multilingual-large\/3}{https:\/\/tfhub.dev\/google\/universal-sentence-encoder-multilingual-large\/3}}. Fine-tuning code for USE is not available, hence, we utilise USE as a comparison to a large scale, pre-trained sentence embedding method. Further, we compare our data augmentation strategy AugSBERT against a straightforward data augmentation strategy provided by NLPAug, which implements 15 methods for text data augmentation.\\footnote{\\href{https:\/\/github.com\/makcedward\/nlpaug}{https:\/\/github.com\/makcedward\/nlpaug}} We include synonym replacement replacing words in sentences with synonyms utilizing a BERT language model. We empirically found synonym-replacement to work best from the rest of the methods provided in NLPAug.\n    \n\\section{Results and Discussion}  \n\n\n\\begin{table*}[t]\n    \\centering\n   \n    \\small\n    \\begin{tabular}{l | l |c  c | c | c | c  }\n        \\toprule\n        \\multicolumn{2}{l}{} &\n        \\multicolumn{1}{c}{In-Domain}    &\n        \\multicolumn{4}{c}{Cross-Domain}    \\\\ \n        \\cmidrule(lr){3-3}\n        \\cmidrule(lr){4-7}\n        \\multicolumn{1}{l}{\\textbf{Source}} &\n        \\multicolumn{1}{l}{\\textbf{Target}} &\n        \\multicolumn{1}{c}{\\textbf{SBERT}} &\n        \\multicolumn{1}{c}{\\textbf{AugSBERT}} &\n        \\multicolumn{1}{c}{\\textbf{SBERT}} &\n        \\multicolumn{1}{c}{\\textbf{Bi-LSTM}} &\n        \\multicolumn{1}{c}{\\textbf{Bi-LSTM}} \\\\\n        \\multicolumn{1}{l}{(Train)} &\n        \\multicolumn{1}{l}{(Evaluate)} &\n        \\multicolumn{1}{c}{(\\emph{Upper-bound})} &\n        \\multicolumn{1}{c}{} &\n        \\multicolumn{1}{c}{(\\emph{Lower-bound})} &\n        \\multicolumn{1}{c}{(Direct)} &\n        \\multicolumn{1}{c}{(Adversarial)} \\\\\n        \\midrule\n             & Quora & 0.504 &\\textbf{0.496} & \\textbf{0.496} & 0.059 & 0.066 \\\\\n            AskUbuntu & Sprint & 0.869 & 0.852 & 0.747 & \\textbf{0.93} & 0.923  \\\\\n             & SuperUser & 0.802 & 0.779 &  0.738 & \\textbf{0.806} & 0.798   \\\\\n        \\midrule\n             & AskUbuntu & 0.715  & \\textbf{0.602} & 0.501 & 0.351 & 0.328   \\\\\n            Quora & Sprint & 0.869 & \\textbf{0.875} & 0.505 & \\textbf{0.875} & 0.867    \\\\\n             & SuperUser & 0.802 & \\textbf{0.645} & 0.504 & 0.523 & 0.485   \\\\\n        \\midrule\n             & AskUbuntu & 0.715 & \\textbf{0.709} & 0.637 & 0.629 & 0.627  \\\\\n            SuperUser & Quora & 0.504 &\\textbf{ 0.495} &  0.495 & 0.058 & 0.067   \\\\\n             & Sprint & 0.869 & 0.876 &  0.785 & 0.936 & \\textbf{0.937}   \\\\\n        \\midrule\n             & AskUbuntu & 0.715 & \\textbf{0.663} & 0.613 & 0.519 & 0.543   \\\\\n            Sprint & Quora & 0.504 & 0.495 & \\textbf{0.496} & 0.048 & 0.063    \\\\\n             & SuperUser & 0.802 & \\textbf{0.769} & 0.660 & 0.658 & 0.636   \\\\\n        \\bottomrule\n    \\end{tabular}\n   \n    \\centering\n    \\caption{AUC(0.05) scores for domain adaptation experiments. All except SBERT (in-domain) are evaluated in cross-domain setup with the best transfer strategy highlighted. We adapt \\cite{shah-etal-2018-adversarial} Bi-LSTM models. Corresponding development set performances can be found in \\autoref{sec:dev-performances}, \\autoref{tab:dev-cross-domain}.}\n    \\label{tab:results-cross-domain}\n\\end{table*}\n\n\n\\subsection{In-Domain Experiments for AugSBERT}\n\n\\autoref{tab:results} summarizes all results for all in-domain datasets. The plain bi-encoder (SBERT w\/o Seed Opt.) consistently underperforms (4.5 - 9.1 points) the cross-encoder across all in-domain tasks. Optimizing the seed helps SBERT more than BERT, however, the performance gap remains open (2.8 - 8.2 points). Training with multiple random seeds and selecting the best performing model on the development set can significantly improve the performance. For the smallest dataset (STS), we observe large performance differences between different random seeds. The best and worst seed for SBERT have a performance difference of more than 21 points. For larger datasets, the dependence on the random seed decreases. \nWe observe bad training runs can often be identified and stopped early using the early stopping algorithm \\cite{dodge2020fine}. \nDetailed results with seed optimization can be found in \\autoref{sec:seed-opt}.\n\nOur proposed AugSBERT approach improves the performance for all tasks by 1 up to 6 points, significantly outperforming the existing bi-encoder SBERT and reducing the performance difference to the cross-encoder BERT. It outperforms the synonym replacement data augmentation technique (\\textit{NLPAug}) for all tasks. Simple word replacement strategies as shown are not helpful for data augmentation in sentence-pair tasks, even leading to worse performances compared to models without augmentation for BWS and Quora-QP. Compared to the off the shelf USE model, we see a significant improvement with AugSBERT for all tasks except Spanish-STS. This is presumably due to the fact that USE was trained on the SNLI corpus \\cite{snli:emnlp2015}, which was used as basis for the Spanish STS test set, i.e., USE has seen the test sentence pairs during training.\n\n\n\nFor the novel BWS argument similarity dataset, we observe AugSBERT only gives a minor improvement for cross-topic split. We assume this is due to cross-topic setting being a challenging task, mapping sentences of an unseen topic to a vector space such that similar arguments are close. However, on known topics (in-topic),  AugSBERT shows its full capabilities and even outperforms the cross-encoder. \nWe think this is due a better generalization of SBERT bi-enconder compared to the BERT cross-encoder. Sentences from known topics (in-topic) are mapped well within a vector space by a bi-encoder. \n\n\\begin{figure}[htb!]\n    \\begin{center}\n        \\includegraphics[width=1.0\\linewidth]{data\/STS-distributions.pdf}\n        \\captionof{figure}{Comparison of the density distributions of gold standard with silver standard for various sampling techniques on Spanish-STS (in-domain) dataset.}\n        \\label{tab:distributions}\n    \\end{center}\n\\end{figure}\n    \n\\textbf{Pairwise Sampling} We observe that the sampling strategy is critical to achieve an improvement using AugSBERT. Random sampling (R.S.) decreases performance compared to training SBERT without any additional silver data in most cases. \nBM25 sampling and KDE produces the best AugSBERT results, followed by Semantic Search (S.S.). \n\\autoref{tab:distributions}, which shows the score distribution for the gold and silver dataset for Spanish-STS, visualizes the reason for this. \nWith random sampling, we observe an extremely high number of low similarity pairs. This is expected, as randomly sampling two sentences yields in nearly all cases a dissimilar pair.  \nIn contrast, \\textit{BM25} generates a silver dataset with similar score distribution to the gold training set. \nIt is still skewed towards low similarity pairs, but has the highest percentage of high similarity pairs. \n\\textit{BM25+S.S.}, apart on Spanish-STS, overall performs worse in this combination than the individual methods. It even underperforms random sampling on the BWS and Quora-QP datasets. We believe this is due to the aggregation of a high number of dissimilar pairs from the sampling strategies combined.\n\\textit{KDE} shows the highest performance in three tasks, but only marginally outperforms BM25 in two of these. \nGiven that BM25 is the most computationally efficient sampling strategy and also creates smaller silver datasets (numbers are given in \\autoref{sec:silver-sizes}, \\autoref{tab:silver}), it is likely the best choice for practical applications.\n\n\n\\subsection{Domain Adaptation with AugSBERT}\nWe evaluate the suitability of AugSBERT for the task of domain adaptation. We use duplicate question detection data from different (specialized) online communities. Results are shown in\n\\autoref{tab:results-cross-domain}. We can see in almost all combinations that AugSBERT outperforms SBERT trained on out-of-domain data (cross-domain). \nOn the Sprint dataset (target), the improvement can be as large as 37 points. In few cases, AugSBERT even outperforms SBERT trained on gold in-domain target data.\n\nWe observe that AugSBERT benefits a lot when the source domain is rather generic (e.g.\\ Quora) and the target domain is rather specific (e.g.\\ Sprint).\nWe assume this is due to Quora forum covering many different topics including both technical and non-technical questions, transferred well by a cross-encoder to label the specific target domain (thus benefiting AugSBERT). Vice-versa, when we go from a specific domain (Sprint) to a generic target domain (Quora), only a slight performance increase is noted.\n\nFor comparison, \\autoref{tab:results-cross-domain} also shows the state-of-the-art results from \\newcite{shah-etal-2018-adversarial}, who applied direct and adversarial domain adaptation with a Bi-LSTM bi-encoder. With the exception of the Sprint dataset (target), we outperform that system with substantial improvement for many combinations.\n\n\n\n\\section{Conclusion}\n\nWe presented a simple, yet effective data augmentation approach called AugSBERT to improve bi-encoders for pairwise sentence scoring tasks. The idea is based on using a more powerful cross-encoder to soft-label new sentence pairs and to include these into the training set. \n\nWe saw a performance improvement of up to 6 points for in-domain experiments. However, selecting the right sentence pairs for soft-labeling is crucial and the naive approach of randomly selecting pairs fails to achieve a performance gain. We compared several sampling strategies and found that BM25 sampling provides the best trade-off between performance gain and computational complexity. \n\nThe presented AugSBERT approach can also be used for domain adaptation, by soft-labeling data on the target domain. In that case, we observe an improvement of up to 37 points compared to an SBERT model purely trained on the source domain.\n\n\\section*{Acknowledgements}\n\nThis work has been supported by the German Federal Ministry of Education and Research (BMBF) under the promotional reference 03VP02540 (ArgumenText), by the German Research Foundation through the German-Israeli Project Cooperation (DIP, grant DA 1600\/1-1 and\ngrant GU 798\/17-1) and has been funded by the German Federal Ministry of Education and Research and the Hessian Ministry of Higher Education, Research, Science and the Arts within their joint support of the National Research Center for Applied Cybersecurity ATHENE. We would like to thank Andreas R\u00fcckl\u00e9, Jan-Christoph Klie, Mohsen Mesgar, Kevin Stowe and the anonymous reviewers for their feedback.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{s1}\nThe Higgs field in the Standard Model (SM) is fundamental to ensure its renormalizability and unitarity. \nHowever, as pointed out by Susskind \\cite{Susskind}, there are self-energy corrections to the\nHiggs propagator which would require an incredible fine-tuning of $10^{-34}$ parts in $1$,\nif we assume that the limit of validity of the SM is at the Planck scale. This problem does not\nappear in other theories which present {\\it Naturalness}. The concept of Naturalness was defined by Susskind as when \n{\\it the behavior of the world at ordinary energies is not exceedingly sensitive to the values of the fundamental parameters} \\cite{Susskind2}.\nTheories with scalar fields present unnaturalness because the mass of the scalar, a phenomenological parameter, exceedingly depends on the \ncutoff of the theory. In the context of the Standard Model, we can see this,\nafter a regularization and a renormalization scheme, via the corrected mass of the Higgs boson \\cite{{Veltman},{Castro}}\n\\begin{align}\n  m ^ 2_H =& m^2+\\frac{3 \\Lambda^2}{8\\pi ^2 \\upsilon^2}[m_{Z}^2+2m_{W}^2+m^2-4m_{t}^2]+ \\nonumber\\\\\n &+ O(\\ln\\frac{\\Lambda}{m})\n  \\label{1}\n\\end{align}\nwhere $m_{Z}$, $m_{W}$, $m$ and $m_{t}$ stand for $Z^{0}$, $W^{\\pm}$, Higgs and top quark masses, respectively, and $\\Lambda$ is the energy scale of \nthe theory. We see that large corrections, unnaturalness and the consequent fine-tuning problem are all caused by $\\delta m^2=m^2_H-m^2$ being \nproportional to the energy scale squared. \n\nTherefore, proposals to eliminate this quadratic energy scale have been put forth \\cite {Veltman,Bardeen,Grange,Aoki}. One of them consisted in choosing the sum in square brackets in eq. (\\ref {1}) to be zero \\cite {Veltman}. This gives a constraint among the masses. However, its validity at any energy scale is not guaranteed \\cite{{Bardeen},{Chaichian}}. Such constraint overestimates the Higgs boson mass \\cite {Castro} as compared with the expected value \\cite{Higgs}. \n\nIn ref. \\cite{Bardeen} is argued that the $\\Lambda^2$ dependence of equation (\\ref{1}) is not in consonance with scale symmetry breaking. In order to see this consider the classical energy-momentum tensor which is broken by the mass term\n\\begin{equation}\n \\Theta_{\\mu}^{\\mu}\\mid_{classical} = m^2 \\bar{H} H.\n\\label{e1}\n\\end{equation}\nwhere $H$ denotes the Higgs field.\n\nWe have a symmetry in equation (\\ref{e1}) in the limit $m\\rightarrow 0$. The quantum corrections also break that conservation law:\n\\begin {eqnarray}\n\\centering\n  &\\Theta_{\\mu}^{\\mu}\\mid_{1-loop} = \\delta m^2 \\bar{H} H + \\sum_i \\frac{\\partial \\mathcal{L}}{\\partial \\lambda_i} \\beta_{\\lambda_i}\n \\nonumber\\\\\n &\\delta m^2= \\frac{3 \\Lambda^2}{8\\pi ^2 \\upsilon^2}[m_{Z}^2+2m_{W}^2+m^2-4m_{t}^2] + \\nonumber\\\\\n&+ O(\\ln\\frac{\\Lambda}{m})\n  \\label{e2}\n\\end{eqnarray}\nwhere $\\beta_{\\lambda_i}$ is the beta function associated with the coupling $\\lambda_i$. \n\nIf we try to restore the classical limit, taking the limits $m\\rightarrow 0$ and $\\beta_{\\lambda_i}\\rightarrow 0$ in equation (\\ref{e2}), the result is\n\\begin {eqnarray}\n\\centering\n  &\\Theta_{\\mu}^{\\mu}\\mid_{1-loop} = [\\frac{3 \\Lambda^2}{8\\pi ^2 \\upsilon^2}(m_{Z}^2+2m_{W}^2-4m_{t}^2)]\\bar{H} H\n  \\label{e3}\n\\end{eqnarray}\n\nEquation (\\ref{e3}) shows that we do not restore the classical limit ($\\Theta_{\\mu}^{\\mu}=0$) when we take the classical \nlimit ($m\\rightarrow 0$ and $\\beta_{\\lambda_i}\\rightarrow 0$). In the limit where $m$ goes to zero, in order to preserve the \nstructure of the anomalous divergence of the scale current, $\\delta m^2$ must scale as $m^2$ rather than $\\Lambda^2$. The \nconclusion is that this $\\Lambda^2$ dependence  is incompatible with the consistency of scale symmetry breaking,  which leads \nus to analyze the possibility and implications of it being an artifact of regularization.  In fact in eq. (\\ref{e2}) the \n$\\Lambda^2$ dependence results from the use of a sharp cutoff regularization, which breaks scale symmetry\nbut not in a physical way, i.e. not related with the beta functions. The scheme causes spurious divergences.\n\nA similar conclusion is drawn in ref. \\cite{Grange}, using the Taylor-Lagrange renormalization. Furthermore in the Wilsonian renormalization group approach it was argued in ref. \\cite{Aoki} that the consistency of scale invariance breaking of the SM is an alternative solution to the Naturalness problem.\n\nIn this work we show in a regularization independent way that the coefficient of the quadratic divergence which gives rise to the Naturalness problem is intrinsically arbitrary and should be set to zero on the grounds of the symmetry arguments discussed above. This is in consonance with the approach proposed in ref. \\cite{Jackiw}, in which arbitrary parameters stemming from perturbation theory calculations should be fixed by the underlying symmetries of the model. \n\nAs we will explicitly show in the next section, a quadratic divergence can be generally parametrized as $I_{quad}(\\mu ^2)=\\frac {i}{(4\\pi )^2}[c_{2}\\Lambda^ 2 +(1+c_{1})\\mu ^2+\\mu^ 2 \\ln \\frac{\\Lambda ^2}{\\mu ^2}]$, where $c_2$ is an arbitrary regularization\ndependent constant.\n\nMoreover, after these considerations, we can reestablish perturbation theory choosing the ratio $\\delta m^2\/m^2_H$ in a range between $0$ to $1$.\nWe obtain bounds for the Higgs boson mass in a $m_H$ {\\it versus} $\\Lambda$ diagram. We compare our bounds with those existent in the literature \\cite {{Regiao2},{Regiao1},{Regiao3}}.\n\n\\section{Implicit Regularization and arbitrariness in divergent amplitudes}\\label {s2}\n\nAn ultraviolet (UV) divergent amplitude will be regularized implicitly \\cite{IR}. In other words, we assume the existence of a regulator function\nfor the integral and we separate the regularization dependent content from the finite one by using the following identity\n\\begin{align}\n \\frac{1}{[(k-l)^2-\\mu ^2]}&=\\sum_{j=0}^{n-1}\\frac{(-1)^j(l^2-2 l.k)^j}{(k^2-\\mu ^2)^{j+1}}+ \\nonumber\\\\\n &+ \\frac{(-1)^n(l^2-2 l.k)^n}{(k^2-\\mu ^2)^n[(k-l)^2-\\mu ^2]}\n \\label{2}\n\\end{align} \nwhere $k$ is the internal momentum, $\\mu$ is the mass of the internal particle and $n$ is chosen such that the denominator of an UV divergent integral does not contain an external momentum $l$. We get basic divergent integrals which are defined as\n\\begin{equation}\n I_{quad}(\\mu ^2)\\equiv \\int^{\\Lambda} \\frac{d^4 k}{(2\\pi)^4} \\frac{1}{(k^2-\\mu ^2)}\n \\label{3}\n\\end{equation}\n\\begin{equation}\n I_{log}(\\mu ^2)\\equiv \\int^{\\Lambda} \\frac{d^4 k}{(2\\pi)^4} \\frac{1}{(k^2-\\mu ^2)^2}\n \\label{4}\n\\end{equation}\nwhere the index $\\Lambda$ means that we assume the existence of a regulator just in order to perform mathematical manipulations with the \nintegrand.\n\nFor example, one of the corrections to the Higgs propagator (see figure \\ref {fig1}) is rewritten using eq. (\\ref {2}) for $n=1$\n \n\\begin{align}\n &\\int^{\\Lambda}\\frac{d^4 k}{(2\\pi)^4} \\frac{1}{(k^2-m ^2)[(k-p)^2-m^2]}= I_{log}(m^2) +\\nonumber \\\\\n &+\\int^{\\Lambda}\\frac{d^4 k}{(2\\pi)^4} \\frac{p^2-2 p.k}{(k^2-m^2)^2[(k-p)^2-m ^2]}\n  \\label{5}\n\\end{align}\nThe second term in (\\ref {5}) is finite in the UV limit by power counting of the internal momentum.\n\nThe most general parametrization of the basic divergent integral $ I_{log}(\\mu ^2)$ can be constructed as follows. By noting that\n\\begin{equation}\n \\frac{\\partial I_{log}(\\mu ^2)}{\\partial \\mu ^2}=\\frac {-i}{(4\\pi )^2 \\mu^2}\n \\label{6}\n\\end{equation}\nand\n\\begin{equation}\n \\frac{\\partial I_{quad}(\\mu ^2)}{\\partial \\mu ^2}=I_{log}(\\mu ^2),\n \\label{7}\n\\end{equation}\nshould be obeyed by any regularization scheme, a general parametrization with explicit scale dependence that satisfies (\\ref{6}) and (\\ref{7}) is given by\n\\begin{equation}\n I_{log}(\\mu ^2)=\\frac {i}{(4\\pi )^2}[\\ln \\frac{\\Lambda ^2}{\\mu ^2}+c_{1}]\n \\label{8}\n\\end{equation}\n\\begin{equation}\n I_{quad}(\\mu ^2)=\\frac {i}{(4\\pi )^2}[c_{2}\\Lambda^ 2 +(1+c_{1})\\mu ^2+\\mu^ 2 \\ln \\frac{\\Lambda ^2}{\\mu ^2}],\n \\label{9}\n\\end{equation}\nwhere $c_1$ and $c_2$ are dimensionless and arbitrary constants.\nSuch arbitrary and regularization dependent constants are inherent to perturbation theory and can be fixed on symmetry grounds \\cite{Jackiw}. For example, gauge \nand momentum routing invariance in QED demands that quadratic surface terms, which are related to arbitrary constants like $c_1$ and $c_2$, should vanish \\cite {Adriano}.There are also some examples of extended QED where gauge invariance is used to fix Lorentz-violating terms \\cite {Gustavo}.\n\nA plethora of regularization schemes have been constructed to be used where gauge invariant Dimensional Regularization may fail, namely in the so called dimensional specific theories among which super-symmetric, chiral and topological quantum field theories figure in. A natural question would be which basic properties should a method that does not resort to analytical continuation in the space-time dimension should retain in order to be invariant. We start by illustrating with  simple examples following \\cite{Varin:2006de}. Let $\\Delta$ be the superficial degree of divergence of a $1$-loop integral where the momentum $k$ runs. Consider the following $\\Delta=2$ integrals,\n\\begin{equation}\nA = \\int_k \\frac{k^2}{(k^2-\\mu ^2)^2},\n\\end{equation}\nand\n\\begin{equation}\nB = I_{quad} (\\mu ^2) + \\mu ^2 I_{log} (\\mu ^2),\n\\end{equation}\nwhere $\\int_k \\equiv \\int d^4k\/(2 \\pi)^4$ and we recover the standard notation of eqs. \\ref{3} and \\ref{4}.\n\nWe expect  $A=B$ be guaranteed by any regularization procedure. However this is not the case. Proper-time regularization \\cite{ZinnJustin:1993wc}, for instance, introduces a cut-off $\\Lambda$ after Wick rotation via the following identity at the level of propagators\n\\begin{align}\n\\frac{\\Gamma(n)}{(k^2+\\mu ^2)^n} = \\int_0^{\\infty} d\\tau \\tau^{n-1} e^{- \\tau (k^2+\\mu ^2)} \\rightarrow \\nonumber\\\\\n\\rightarrow \\int_{1\/\\Lambda^2}^{\\infty}  d\\tau \\tau^{n-1} e^{- \\tau (k^2+\\mu ^2)}.\n\\end{align}\nThus it is trivial to obtain within the proper-time method that $A \\neq B$ since\n\\begin{equation}\nA_{\\Lambda} = \\frac{-2 i}{(4 \\pi)^2} (\\Lambda^2 - \\mu ^2 \\ln \\Lambda^2\/\\mu ^2),\n\\end{equation}\nwhereas\n\\begin{equation}\nB_{\\Lambda} = \\frac{- i}{(4 \\pi)^2} (\\Lambda^2 - 2 \\mu ^2 \\ln \\Lambda^2\/\\mu ^2).\n\\end{equation}\n\n\n\nAs we can see above, an inadequate regularization scheme could fix the constants $c_{1}$ and $c_{2}$ in a way that would lead us to inappropriate conclusions. In the context of this work, the value $c_{2}=-1$, obtained with a sharp cutoff regularization, would lead to the fine-tuning problem. In the context of scalar QED coupled to gravitation, a non-vanishing value of $c_{2}$ would make this theory asymptotically free \\cite{Jean}.\n\n\\begin{figure}[!htb]\n \\centering\n \\includegraphics[scale=0.55]{figure1.pdf}\n \\caption{One of the corrections to the Higgs propagator.}\n \\label{fig1}\n\\end{figure}\n\n\\section{1-loop corrections to the Higgs propagator and theoretical bounds on the Higgs mass}\\label {s3}\n\nAfter spontaneous symmetry breaking, the contributions to the Higgs boson propagator include massive and Goldstone fields in the Landau gauge ($\\xi =0$).\n\n\\begin{figure}[!htb]\n \\centering\n \\includegraphics[scale=0.8]{figure2.pdf}\n \\caption{1-Loop correction to the Higgs propagator in the Landau gauge. S, V and F stands for scalar, vector boson and fermionic field, respectively.}\n \\label{fig2}\n\\end{figure}\n\nFigure \\ref {fig2} shows all 1-loop contributions. We also consider only the heaviest fermion ( top quark) for the \nfermionic 1-loop diagrams. \n\nIf one regularizes with a sharp covariant Euclidean cutoff \nand an on-mass-shell renormalization, for example, one has the Higgs mass correction given by equation (\\ref {1}).  The quadratic \ncutoff of that equation means that we have an upper bound for this theory around 2 TeV \\cite{{Chaichian},{Espinosa}}. This implies that the Standard Model, \nas an effective theory, should be valid up to energy scales that have already been reached by LHC \\cite\n{Espinosa}. Furthermore, one could expect the appearance of new physics around this energy, which so far did not emerge. So one can question whether the scale for the onset of new physics is really of that order. In addition, as the Higgs\nmass is expected to be 125 GeV \\cite{Higgs} we also would expect the cutoff to be much larger than that mass to preserve\nthe hierarchy of the theory.\n\nIn terms of the basic divergent integrals defined in eqs. \\ref{8} and \\ref{9}, the 1-loop contributions in figure \\ref{fig2} can be written in an on-mass-shell renormalization, for example, as\n\\begin{align}\n&\\delta m ^ 2 =\\frac{6i}{\\upsilon^2}[m_{Z}^2+2m_{W}^2+m^2-4m_{t}^2]I_{quad}(m ^2)+ \\nonumber\\\\\n&\\frac{-3im^2}{\\upsilon^2}[3m^2+6m^2_W+m^2_Z-6m^2_t]I_{log}(m ^2)\n\\label{10}\n\\end{align}\nwhere we neglect finite terms in the UV limit because they are small compared to terms proportional to $\\Lambda$ and\n$\\ln \\Lambda$ when $\\Lambda$ is large. For example, the finite integral in the UV limit of eq. (\\ref {5}) is much smaller than one when $p^2= m^2_H$.\n\nNow, considering eq. (\\ref {9}), $\\delta m^2$ correction reads\n\\begin{equation}\n  \\delta m ^ 2 = \\frac{-3 c_{2}}{8 \\pi^2 \\upsilon^2}[m_{Z}^2+2m_{W}^2+m^2-4m_{t}^2]\\Lambda ^2+... \n  \\label{11}\n\\end{equation}\nwhere the ellipses stand for contributions from $I_{log}(m ^2)$ and other terms from eq. (\\ref {9}).\n\nBecause $c_{2}$ is arbitrary as discussed in section \\ref{s2} it is consistent with a vanishing quadratic divergence. This assumption is justified by compatibility with scale\nsymmetry breaking which at one loop can be broken only by terms proportional to the beta function and terms which vanish in the limit $m\\rightarrow0$ as argued in \nsection \\ref{s1}. Thus it consists of a theoretical symmetry argument that fix an arbitrariness in this case.\n\nThe constant $c_1$ from eqs. (\\ref {8}) and (\\ref{9}) is also chosen using this argument, i. e. the finite part in the UV \nlimit of the diagrams can not contain terms which are not proportional to $m^2$ because they break scale invariance in the \nlimit $m\\rightarrow0$. These considerations complies with the definition of Naturalness imposed by 't Hooft \\cite{Hooft}.\n\nA second argument from the phenomenological standpoint in favor of this choice is to assure Naturalness of the SM and the consequent absence of the fine-tuning problem. This \nchoice appears to be more appealing than the one made in refs. \\cite{Veltman} and \\cite{Castro}, since now the quadratic energy scale vanishes without the \nneed to enforce a constraint among masses of different particles which may vary differently with the energy scale. \n\nThere remain only the terms proportional to $m^2\\ln\\frac{\\Lambda}{m}$ which do not break scale symmetry in the limit $m\\rightarrow0$. These terms come from \nthe integrals $I_{quad}(m ^2)$ and $I_{log}(m ^2)$ rewritten using eqs. (\\ref {8}) and (\\ref {9}). So, the correction to the Higgs mass is given by  \n\\begin{equation}\n  \\delta m ^ 2 = \\frac{3 m^ 2_H}{16 \\pi^2 \\upsilon^2}[2m_{t}^2+2m_{W}^2+m^2_H-m_{Z}^2] \\ln \\frac {\\Lambda^ 2}{m^2_H} +... \n  \\label{12}\n\\end{equation}\nwhere we have used $m^2_H=m^2+O(\\frac{3m^2}{16\\pi^2\\upsilon^2})$ to replace $m$ by $m_H$. The neglected terms are of order of two-loop corrections.\n\nIf we consider only contributions of the order $\\ln\\frac{\\Lambda}{m_H}$, we can use experimental data for the masses ($m_{t}=173 GeV$, $m_{W}= 80.2 \nGeV$ and $m_{Z}= 91.2 GeV$), the vev value ($\\upsilon=246\\ GeV$) and the currently accepted value for the Higgs mass ( $m_H=125\\ GeV$), to \nestimate the limit of validity of perturbation theory. To do this, we ask where perturbation theory starts to fail, i. e. where $\\delta m= O(m_H)$. We \nget $\\Lambda{\\approx} 10^{10} GeV$. \n\nWe can obtain a more refined estimate if we consider running masses in equation (\\ref {12}). \nTo include this information in that equation, we use the usual Standard Model relations between masses and coupling constants. Then we solve the beta functions of ref. \\cite{Chaichian} in order to find the running coupling constants. We have\n\\begin{equation}\n\\frac{\\delta m^2}{m^2_H}=\\frac{3}{8\\pi^2}[\\frac{m^2_H}{\\upsilon^2}+\\frac{1}{4}g^2(\\Lambda)-\\frac{1}{4}g'^2(\\Lambda) +g^2_t(\\Lambda)]\\ln\\frac{\\Lambda}{m_H}\n\\label{13}\n\\end{equation}\nwhere $g(\\Lambda)$, $g'(\\Lambda)$ and $g_t(\\Lambda)$ are the solutions of one loop beta functions for the SU(2), U(1) and Yukawa \ncouplings, respectively. We use as initial values $g^2(m_Z)= 0.42$, $g'^2(m_Z)= 0.13$ and $g_t(m_t)= 1.01$. \n\n\\begin{figure}[!ht]\n\\centering\n    \\subfigure[]\n    {   \\includegraphics[scale=0.8]{figure3a.pdf}\n        \\label{fig3a} }\n    \\subfigure[]\n{   \\includegraphics[scale=0.8]{figure3b.pdf}\n        \\label{fig3b}}\n\\caption{(a)Excluded values for the Higgs mass: the pink bound is the {\\it vacuum stability bound} \\cite{Regiao2} and the dark green one is the {\\it triviality \nbound} \\cite{Regiao1}. The blue, brown and light green bounds are the ones which exclude all values of $m_H$ and $\\Lambda$ which lead to corrections $\\delta m^2$ \nlarger than or equal to $35\\%$, $40\\%$ and $45\\%$ of $m^2_H$, respectively.(b)Excluded values for the Higgs boson mass considering $\\Lambda^2$ dependence: the brown \nbound is the {\\it vacuum stability bound} \\cite{Regiao2} and the pink one is the {\\it triviality bound} \\cite{Regiao1}. The blue bound is the one which excludes all \nvalues of $m_H$ and $\\Lambda$ which lead to corrections $\\delta m^2$ larger than or equal to $m^2_H$.}\n\\label{fig3}\n\\centering\n\\end{figure}\n\nThe estimate now (for $\\upsilon=246\\ GeV$ and $m_H=125\\ GeV$) is $\\Lambda{\\approx} 10^{16} GeV$. At this scale $\\delta m$ starts to be equal or \nhigher than $m_H$. That limit would be of order $1\\ TeV$ should quadratic divergences be included \\cite\n{{Chaichian},{Espinosa}}.\n\nInstead of assuming that perturbation theory remains valid up to the scale of $10^{16} GeV$ , one can alternatively present estimates for $\\delta m^2\/m^2_H$ in the range $0$ to $1$ where perturbation theory is still acceptable. A more complete and adequate analysis is to use eq. (\\ref {13}) plus \nthe finite part in the UV limit to exclude a set of values of $m_H$ and $\\Lambda$ that yields values of corrections outside the $\\delta m^2\/m^2_H$ value we have chosen. The result is a region of excluded values of the Higgs mass in a certain energy scale.\nFigure \\ref {fig3a} shows a blue, a brown and a light green bound \nwhich excludes all values \nof $m_H$ and $\\Lambda$ whose correction $\\delta m^2$ is larger than or equal to $35\\%$, $40\\%$  and $45\\%$ of the Higgs mass $m_H$, respectively. There \nis a pink bound which \ncorresponds to the {\\it vacuum stability bound} \\cite{Regiao2} that excludes all values of $m_H$ and $\\Lambda$ that lead to a negative Higgs \nself-coupling. Our bounds also coincide in part with the {\\it triviality bound} \\cite{{Regiao1}}(dark green bound) which excludes all values of $m_H$ \nand $\\Lambda$ which trespass the Landau pole. For $\\delta m^2\/m^2_H\\sim 0.1$ we obtain bounds which essentially overlap with the vacuum stability bound and\nprevent the existence of allowed values for the Higgs mass. On the other hand, a bound $\\delta m\/m_H \\rightarrow 1$ approaches the triviality bound and\nlarge ranges for $m_H$ are obtained.\n\nFor a given value of the mass correction, we can determine a Higgs mass range or a Higgs mass value in a certain energy scale. The crossing of the {\\it vacuum stability bound} with a $\\delta m^2\/m_H^2$ percentage curve yields the minimal mass correction that one can achieve at a certain scale. Its crossing with the blue mass region means that for a $35\\%$ correction and in a energy around $10^3\\ TeV$, the \nHiggs mass value is around $119\\ GeV$. For the brown region, we have a Higgs mass \nvalue around $124\\ GeV$ in an energy around $10^4\\ TeV$, if the mass correction squared is around $40\\%$. That means that if the cutoff of the SM\nis around that order we still have a Higgs boson mass near the expected one \\cite{Higgs}. The same interpretation can be given for the light green bound. In that \ncase, we obtain a Higgs mass around $129 GeV$ for energies around $10^6\\ TeV$. We see that the greater the correction value, the greater the allowed regions for the Higgs mass. For example, the crossing of the bound obtained for a $20\\%$ correction with the\nvacuum stability bound occurs at $10\\ TeV$ and we have a smaller white region than the ones of figure \\ref{fig3a}.\n\n\nThe use of a consistent scale symmetry breaking to fix an arbitrary parameter that controls the quadratic divergence excludes the fine-tuning problem and consequently, we have a reliable perturbation \ntheory. The choices $\\delta m^2<m^2_H$ are also in agreement with the Renormalization Group because those bounds do not trespass the {\\it triviality bound}. \nIn the scenario where fine-tuning is used, the Higgs mass should approach $160\\ GeV$ at an energy of $100\\ TeV$ \\cite{Regiao3}. However,\nthe Higgs mass correction of ref. \\cite{Regiao3} violates perturbation theory and the consistency of the scale symmetry breaking.\n\nFurthermore, these bounds give another argument against the $\\Lambda^2$ dependence. If that dependence exists, the measured value of $125 GeV$ is excluded for all energy scales and for all $\\frac{\\delta m^2}{m^2_H}<1$. We draw the bound corresponding to $\\frac{\\delta m^2}{m^2_H}=1$ (the maximum perturbation theory allows) considering the $\\Lambda^2$ dependence in figure \\ref{fig3b}. That means if we want the Higgs mass around the expected value we must allow some fine-tuning, i.e. $\\delta m^2\/m^2_H>1$ , as we can see in ref. \\cite{Regiao3}. On the other hand, the result of figure \\ref {fig3a} allows Higgs masses around the expected value for $\\delta m^2\/m^2_H<1$, where perturbation theory is acceptable.\n\n\\section {Concluding remarks}\n\nWe showed that regularization dependent arbitrariness which appear in quadratic divergences can be parametrized as shown in section \\ref{s2} and fixed on symmetry grounds to restore Naturalness and perturbation theory for the Higgs boson mass in the SM. \nWe have analyzed the quantum corrections to the Higgs boson mass together with the vacuum stability bound.\nAlthough the correction values are not known, it is possible to establish bounds if we choose certain values for  $\\delta m^2\/m^2_H$. These bounds provide phenomenological arguments against the existence of $\\Lambda^2$ dependence in the Higgs boson mass correction. \n\n\n\\vspace{0.5cm}\n\n{\\bf Acknowledgments}\n\nResearch supported by Funda\\c{c}\\~{a} o para a Ci\\^ecia e Tecnologia, \nproject CERN\/FP\/116334\/2010, developed under the iniciative QREN, financed by\nUE\/FEDER through COMPETE - Programa Operacional Factores de Competitividade. \nThis research is part of the EU Research Infrastructure Integrating Activity \nStudy of Strongly Interacting Matter (HadronPhysics3) under the 7th Framework \nProgramme of EU: Grant Agreement No. 283286. \n\nThe authors thank CNPq and FAPEMIG for financial support and Dr. Luis Cabral\nfor fruitful discussions.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{Sec:Intro}\n\n\n\nRace-track microtron (RTM) is a specific type of electron accelerator\nwith beam recirculation combining properties of the linear accelerator\nand a circular machine~\\cite{KapitzaMelekhin1978,Rand1984}.\nFor applications in which a modest beam power at a relatively\nhigh beam energy is required the RTM turns out to be\nthe most optimal source of electron beams. \nThis is the case of applications like the cargo inspection or\nIntraoperative Radiation Therapy for which RTMs allow\nto get pulsed and continuous beams in a quite cost and energy effective\nway with the most optimal dimensions of the machine.\nAn example of such accelerator is a compact 12 MeV RTM\nwhich is under construction at the Technical University of Catalonia\nin collaboration with the Moscow State University and CIEMAT\n(Spain)~\\cite{Aloev_etal2010,Kubyshin_etal2009,Vladimirov_etal2014}.\n\nIn the design of a particle accelerator its main parameters are optimized\nfor some reference particle, usually referred to as synchronous particle.\nReal particles of the beam perform transverse and longitudinal oscillations\nwith respect to this synchronous particle.\nOne of the important issues of the RTM design is to assure the stability\nof these oscillations,\nin particular to avoid an uncontrolled growth of their amplitude that\nleads to the loss of the beam.\nIn the RTM beam dynamics the main role is played by longitudinal oscillations\nof individual electrons with respect to the synchronous\nparticle~\\cite{Veksler1945,Lidbjork2001}.\nSuch oscillations are usually referred to as phase motion,\nand the region of initial states in the phase space which give rise to\nstable oscillations is called acceptance.\n\nStability of the phase oscillations of particles in the beam is a\nmatter of primary concern both at the stage of the accelerator design\nand during its operation. \nIn particular,\nthe size of the acceptance determines the efficiency of caption into\nacceleration of particles injected from a source of electrons,\nusually an electron gun. \nThe shape and size of the acceptance depend on the phase of\nthe synchronous particle, also called synchronous phase and\ndenoted as $\\phi_{\\rm s}$ along this paper. \nThe definition of these notions will be given in Section~\\ref{Sec:RTMModel}.\nA specific feature of the RTM is that the range of values of $\\phi_{\\rm s}$ for\nwhich the acceptance exists is quite narrow and therefore this parameter\nmust be carefully chosen and controlled during the RTM operation.\nThe main criteria here is to keep the synchronous phase equal or close\nto the value for which the size of the acceptance is maximal and to\navoid resonant values of $\\phi_{\\rm s}$. \nThe latter is important\nbecause resonant phase oscillations lead to a buildup of beam instabilities,\nrapid growth of the amplitude of the phase oscillations and eventually\nto a loss of the beam. \nTherefore, a good understanding of the longitudinal\ndynamics of the beam in RTMs in general and the phase oscillations\nof electrons in a particular machine is important.\n\nThe phase oscillations of the beam in an RTM are described by a system\nof nonlinear difference equations~\\cite{Henderson_etal1953,Rand1984} that\nwill be derived in Section~\\ref{Sec:RTMModel}.\nIt gives rise to an analytic area preserving map which is the object of study\nin the present paper. \nThese difference equations cannot be approximated by differential\nequations without loss of accuracy, since RTMs have large energy gain\nper turn and high frequency of phase oscillations. \nThe description of the acceptance in the linear approximation\nis well known~\\cite{Rand1984}.\nThe change of the acceptance under the variation of the synchronous phase,\nthe appearance of stable regions,\nand the emergence of the stochastic regime is studied\nin~\\cite{Melekhin1972} by looking at the normal form of\nthe difference equations up to order four. \nFor instance, it was shown that there is a shift of the center of\noscillations and that the oscillation frequency depends on the amplitude. \nWe will reproduce these known results at the beginning of our analysis.\n\nIn the present paper we will study the phase motion of the beam in\nRTMs using modern Dynamical Systems methods. \nOur goal is to characterize the acceptance,\ncalled stability domain in Dynamical Systems,\nas a function of the synchronous phase. \nIn particular,\nwe will establish the intervals of values of $\\phi_{\\rm s}$ for which\nthe acceptance and its central connected component exist,\nanalyze their geometry, shape and size, and calculate their area for\nthe whole range of $\\phi_{\\rm s}$.\nOur main theoretical tools are the Moser twist theorem and several\nSim\\'o's stability results~\\cite{Simo1982}. \nWe will also approximate some rotational invariant curves of\nthe RTM beam longitudinal dynamics by level curves of suitable Hamiltonians. \nFinally, we will carry out a global study of the acceptance and\nits connected component using results and algorithms, like the orbit method,\ndeveloped in~\\cite{LuqueVillanueva2013,Vieiro2009}.\n\nThe article is organized as follows. \nWe give a short introduction into the RTM beam longitudinal dynamics,\ndefine the notion of synchronous trajectory and synchronous phase\nand derive the map describing the longitudinal beam dynamics that\nwe will refer to as RTM map in Section~\\ref{Sec:RTMModel}.\nWe introduce the notion of the acceptance and its connected component,\nwe find fixed points of the RTM map and analyze their type in the linear\napproximation in Section~\\ref{Sec:MainResult}. \nWe also formulate a theorem about the local stability of\nthe synchronous trajectory and give a summary of further results on\nthe behavior of the acceptance as a function of the synchronous phase. \nThe proof of the theorem is given in Section~\\ref{Sec:LocalStability}.\nSection~\\ref{Sec:HamiltonianApproximations} is devoted to the approximation\nof the RTM map by the integrable dynamics of certain Hamiltonians\nand correspondingly the approximation of the invariant curves around\nthe synchronous trajectory by means of Hamiltonian level curves. \nThe global stability of the synchronous trajectory is studied\nin Section~\\ref{Sec:GlobalStability},\nalso a detailed characterization of the acceptance is given there.\nThese results are obtained numerically by using the orbit method.\nThe invariant curves of hyperbolic points are analyzed\nin Section~\\ref{Sec:InvariantCurves}. \nA summary and a discussion of the obtained results\nare given in Section~\\ref{Sec:Conclusions}.\n\n\n\n\\section{The RTM model}\n\\label{Sec:RTMModel}\n\n\n\nThe operation of a race-track microtron (RTM) is illustrated\nin Fig.~\\ref{fig:MicrotronScheme}.\nThe initial beam is injected from an electron source\n(electron gun or external pre-accelerator)\ninto an accelerating structure (AS) consisting of a few resonant cavities.\nThe longitudinal electric component of a high frequency electromagnetic\nwave (usually a standing wave) accelerates the electrons in the AS.\nThen the beam is bent by the magnetic field of a $180^{\\circ}$ bending magnet,\ncalled \\emph{end magnet}, travels through a free space,\nusually referred to as \\emph{drift space}, follows along a circular trajectory\ninside the second end magnet and returns to the AS. \nIn this way the beam makes a number of recirculations through the RTM with\nthe energy being increased at each pass through the AS.\nOnce the beam gets the final design energy,\nit is deflected by an extraction magnet and is\ndirected towards the accelerator beam exit.\n\nWe approximate the AS by a zero length accelerating gap and consider\na stationary regime with a constant amplitude of the accelerating field. \nThen the energy gain of an electron passing through the AS is equal to\n$\\Delta_{\\max} \\cos\\phi$, where $\\Delta_{\\max}$ is the maximum energy gain\nin the AS and $\\phi$ is the phase of the accelerating field. \nIn accelerator physics this parameter is usually referred to\nas particle phase and is used to characterize the longitudinal position\nof the particle along the orbit with respect to the accelerating gap\n(in a real machine with respect to, say, the exit, of the last cavity\nof the AS in the direction of the beam motion).\nAnother idealization in our study is that the end magnets will be considered\nas hard-edge dipole magnets so that the fringe-field effects\nare not taken into account.\n\nLet us consider the longitudinal (phase) motion of electrons in an\nRTM with magnetic field induction $B$ in the end magnets\nand separation $l$ between the magnets (straight section length).\nWe assume that the injected electrons are already ultra-relativistic,\nso that in the formulas below the velocity of the particles in the\nbeam is equal to the velocity of light $c$.\n\nOur dynamical variables are the full particle energy $E_n$ and\nits phase $\\phi_n$ at the $n$-th turn at some point of the orbit.\nFor the sake of convenience, we choose this point to be the exit of the AS. \nLet $\\phi_0$ and $E_0$ be the particle phase and energy at the injection;\nthat is, just before its first passage through the AS. \nWe would like to note that, in some pulsed RTM,\nthe electrons reverse their trajectory in special reverse field magnets\nafter the injection and first acceleration at the\nAS~\\cite{Kubyshin_etal2009,Vladimirov_etal2014}.\nIn that case,\n$E_0$ is the energy after the first acceleration and reflection of the beam\nin the end magnet, before the second passage through the AS,\nbut we will still use the term injection energy.\n\n\\begin{figure}\n\\includegraphics*[width=85mm]{RTM.pdf}\n\\caption{Schematic view of our RTM model:\n1) Accelerating structure,\n2) Drift space,\n3) End magnets,\n4) Electron gun, and\n5) Extraction magnet.}\n\\label{fig:MicrotronScheme}\n\\end{figure}\n\nFor the RTM model described above,\nthe duration of the $n$-th revolution of the beam is \n\\begin{equation}\\label{eq:TimeEnergy}\nT_n =\n\\frac{2 l + 2 \\pi r_n}{c} =\n\\frac{2l}{c} + \\frac{2 \\pi E_n}{e c^2 B},\n\\end{equation}\nwhere $e$ is the elementary charge and $r_n$ is the beam trajectory radius\nin the end magnets. \nWe have used that $E_n = e c B r_n$ in bending\nmagnets for ultrarelativistic beams~\\cite{Rand1984}. \nRelation~(\\ref{eq:TimeEnergy}) allows us to work indistinctly with\ntime variables $T_n$ or energy variables $E_n$. \nThe beam dynamics in the phase-energy variables, for a zero-length AS,\nis governed by the difference equations\n\\begin{equation}\\label{eq:Dynamics_in_phi_E}\n\\phi_{n+1} = \\phi_n + 2 \\pi T_n\/T_{{\\rm RF}}, \\qquad\nE_{n+1} = E_n + \\Delta_{\\max} \\cos\\phi_{n+1},\n\\end{equation}\n$T_{{\\rm RF}}$ being the period of the accelerating electromagnetic field,\nusually called \\emph{radiofrequency (RF) field} in the context\nof accelerator beam dynamics.\nSee~\\cite{Rand1984}.\n\nThe design of a particle accelerator requires a proper choice of physical\nand technical parameters in order to guarantee the existence of a\nreference trajectory, called \\emph{synchronous trajectory},\nso that an ideal particle following this trajectory is accelerated\nin the most optimal way.\nNamely, it can be tuned in resonance with the accelerating field.\nLet us explain the choice of parameters in case of an RTM. \nWe fix two positive integers $m$ and $k$.\nThe resonance conditions in the case of an RTM are\n\\begin{equation}\\label{eq:ResonanceConditionsInTime}\n\\frac{T_{1,{\\rm s}}}{T_{\\rm RF}} = m, \\qquad\n\\frac{T_{n+1,{\\rm s}}-T_{n,{\\rm s}}}{T_{\\rm RF}} = k.\n\\end{equation}\nHenceforth, the subindex ``${\\rm s}$'' indicates quantities related\nto the synchronous trajectory. \nThe integer $m$ is the number of RF field periods during the first turn\nand defines the synchronicity condition at this turn,\nwhereas $k$ is the increase of the multiplicity factor due to the increase\nof the  period of revolution of the reference particle in each turn.\nThe ratio \n\\[\nj_n := T_{n,{\\rm s}}\/T_{\\rm RF} = m + (n-1)k\n\\]\nis called \\emph{harmonic number}. \nRTMs are accelerators with variable harmonic numbers,\nsince $j_n$ depends on the $n$-th turn.\n\nIf we rewrite the resonance conditions~(\\ref{eq:ResonanceConditionsInTime})\nusing the energy variables $E_{n,{\\rm s}}$ and relation~(\\ref{eq:TimeEnergy}),\nwe get that $E_{n,{\\rm s}} = E_{\\rm s} + n \\Delta_{\\rm s}$, where \n\\[\nE_{\\rm s}= \\left( \\frac{m}{k} - 1 - \\frac{2l}{k\\lambda} \\right) \\Delta_{\\rm s}, \\qquad\n\\Delta_{\\rm s} = \\frac{e c B \\lambda k}{2 \\pi},\n\\]\nand $\\lambda = c T_{\\rm RF}$ is the wavelength of the RF field.\nFinally, it is straightforward to check that if the injection\nphase for the synchronous trajectory\n$\\phi_{0,{\\rm s}} = \\phi_{1,{\\rm s}} = \\phi_{\\rm s}$ satisfies relation \n\\[\n\\Delta_{\\rm s} = \\Delta_{\\max} \\cos\\phi_{\\rm s},\n\\]\nthen we get a particular solution $(\\phi_{n,{\\rm s}},E_{n,{\\rm s}})$ of\nequations~(\\ref{eq:TimeEnergy})--(\\ref{eq:Dynamics_in_phi_E}), whose\nenergy undergoes a constant gain $\\Delta_{\\rm s}$ at each turn: \n\\begin{equation}\\label{eq:SynchronousDynamics}\n\\phi_{n,{\\rm s}} = \\phi_{\\rm s} + 2\\pi i_n,\\qquad\nE_{n,{\\rm s}} = E_{\\rm s} + n \\Delta_{\\rm s},\n\\end{equation}\nwith $i_n = (n-1)m + (n-1)(n-2)k\/2$.\nWe note that $i_n \\in \\mathbb{Z}$,\nso this synchronous particle passes through the AS in the same phase\nof the RF field at each turn:\n$\\phi_{n,{\\rm s}} = \\phi_{\\rm s}$ (mod $2\\pi$) for all $n$.\nWe say that $\\phi_{\\rm s}$ is the \\emph{synchronous phase}.\n\nOnce we realize that\nthe synchronous trajectory~(\\ref{eq:SynchronousDynamics}) exists,\nwe wonder whether the oscillations of other trajectories around it are stable.\nThis depends on the synchronous phase $\\phi_{\\rm s}$.\n\nWe study this dependence by introducing the variables \n\\begin{equation}\\label{eq:psi-w-defn}\n\\psi_n = \\phi_n-\\phi_{\\rm s}, \\qquad\nw_n = 2 \\pi k (E_n-E_{n,s})\/\\Delta_{\\rm s},\n\\end{equation}\nthat describe the phase and energy deviation of an arbitrary trajectory\nfrom the synchronous one.\nThen the beam dynamics is modeled by the map\n$(\\psi_{n+1},w_{n+1})=f(\\psi_n,w_n)$, where\n\\[\n\\left\\{\n\\begin{array}{ccl}\n\\psi_{n+1} & = & \\psi_n+w_n, \\\\\nw_{n+1} & = & w_n + 2 \\pi k \\big( \\cos(\\psi_{n+1} + \\phi_{\\rm s} )\/\\cos\\phi_{\\rm s}-1 \\big).\n\\end{array}\n\\right.\n\\]\nFor simplicity, we will assume that $k=1$ in the rest of the paper.\nThe study of other values can be carried out in a similar way.\n\nWe end this section by stressing that the model of longitudinal oscillations\naround the synchronous trajectory for non-ultra-relativistic beams\nis more complicated~\\cite{Kubyshin_etal2008}.\n\n\n\n\\section{Terminology and main results}\n\\label{Sec:MainResult}\n\n\n\nUnder the assumption $k = 1$, our model for the nonlinear oscillations around\nthe synchronous trajectory~(\\ref{eq:SynchronousDynamics}) is\nthe analytic area preserving diffeomorphism $(\\psi_1,w_1) = f(\\psi,w)$, defined by\n\\begin{equation}\\label{eq:f}\n\\left\\{\n\\begin{array}{ccl}\n\\psi_1 & = & \\psi + w,\\\\\nw_1    & = & w + 2\\pi (\\cos \\psi_1 -1) - \\mu \\sin \\psi_1.\n\\end{array}\n\\right.\n\\end{equation}\nHere, $\\mu = 2\\pi \\tan \\phi_{\\rm s}$ is a parameter, and\n$\\psi$ (respectively, $w$) is the deviation of\nthe phase (respectively, energy) of an arbitrary particle from the\nphase (respectively, energy) of the synchronous trajectory,\nwhich corresponds to the fixed point\n\\[\np_{\\rm s} = (\\psi_{\\rm s},w_{\\rm s}) = (0,0).\n\\]\nThe angular coordinate $\\psi$ is defined modulo $2\\pi$,\nso our phase space is $\\mathbb{T} \\times \\mathbb{R}$, with $\\mathbb{T} = \\mathbb{R}\/2\\pi\\mathbb{Z}$.\nFor brevity, we will write $p = (\\psi,w)$ and\n$p_n = (\\psi_n,w_n) = f^n(\\psi,w) = f^n(p)$.\nWe look for initial conditions that give rise to particles with\na bounded energy deviation from the synchronous trajectory.\nMore precisely, we will estimate the size and the shape of\n\\[\n\\mathcal{A} =\n\\left\\{ p \\in \\mathbb{T} \\times \\mathbb{R} : \\mbox{$(w_n)_{n \\in \\mathbb{Z}}$ is bounded} \\right\\}.\n\\]\nThis domain is called \\emph{(longitudinal) acceptance} in\nAccelerator Physics, and \\emph{stability domain} in Dynamical Systems.\nWe will also study its connected component $\\mathcal{D} \\subset \\mathcal{A}$\nthat contains $p_{\\rm s}$.\n\n\\begin{remark}\\label{rem:PhaseDeviation}\nWe have numerically checked that\n\\[\n\\mathcal{A} \\subset [-0.45,0.35] \\times [-0.8,0.8],\\qquad\n\\forall \\mu > 0.\n\\]\nHence, if $(\\psi_n,w_n) \\in \\mathbb{T} \\times \\mathbb{R}$ is the phase-energy\ndeviation at the $n$-th turn of a trajectory contained in $\\mathcal{A}$\nand $(\\tilde{\\psi}_n,w_n) \\in \\mathbb{R} \\times \\mathbb{R}$ denotes the lift of\nthis trajectory determined by $-\\pi \\le \\tilde{\\psi}_0 < \\pi$,\nthen $-\\pi \\le \\tilde{\\psi}_n < \\pi$ for all $n$.\nThis means that trajectories with bounded energy deviation,\nhave also bounded phase deviation respect to the synchronous one.\n\\end{remark}\n\nThe map $f$, its acceptance $\\mathcal{A}$, and the connected component\n$\\mathcal{D}$ depend on $\\mu$, but we will frequently omit this dependence.\nOtherwise, we will write $f_\\mu$, $\\mathcal{A}_\\mu$,\nand $\\mathcal{D}_\\mu$, respectively.\n\nWe say that the synchronous trajectory is:\n\\begin{itemize}\n\\item\n\\emph{Globally stable} when $\\mathcal{D}$ is a neighborhood of $p_{\\rm s}$;\n\\item\n\\emph{Locally stable} when $\\forall \\epsilon > 0$ there exists\n$\\delta > 0$ such that\n\\[\n\\| p-p_{\\rm s} \\| < \\delta \\Rightarrow\n\\|p_n - p_{\\rm s} \\| < \\epsilon ,\\ \\forall n \\in \\mathbb{Z};\n\\]\n\\item\n\\emph{Linearly stable} when the sequence $(M_{\\rm s}^n)_{n \\in \\mathbb{Z}}$\nis bounded, where $M_{\\rm s}$ is the linear part of the map $f$\nat $p_{\\rm s}$; that is, when the linearized map $p  \\mapsto M_{\\rm s} p$\nis locally stable.\n\\end{itemize}\nOtherwise, it is \\emph{globally}, \\emph{locally}, or \\emph{linearly unstable}.\n\nLocal stability implies global stability,\nbut local instability does not imply global instability.\nThe case $\\mu \\gtrsim 4$ is a sample of this claim.\nSee Fig.~\\ref{fig:AfterParabolic4}.\nOur map~(\\ref{eq:f}) shows the four possible combinations of\nlocal stability\/instability and linear stability\/instability.\n\nThe synchronous trajectory is \\emph{hyperbolic},\n\\emph{parabolic}, or \\emph{elliptic} when the eigenvalues of\nthe matrix $M_{\\rm s}$ are real of modulus different from one, real of modulus\nequal to one, or non-real of modulus equal to one, respectively.\nWe note that $\\det[M_{\\rm s}] = 1$, since $f$ is an area preserving map.\nTherefore, the behavior in the linear approximation of\nthe synchronous trajectory only depends\non the trace $T_{\\rm s} = \\mathop{\\rm tr}\\nolimits[M_{\\rm s}]$.\nConcretely, it is hyperbolic, parabolic or elliptic if and only if\n$|T_{\\rm s}| > 2$, $|T_{\\rm s}| = 2$, or $|T_{\\rm s}| < 2$, respectively.\n\nThe linear type and the local stability are related as follows.\nThe hyperbolic type implies local instability,\nwhereas the elliptic type is generically locally stable,\nbut local instability may take place in degenerate cases~\\cite{SiegelMoser1995}.\nThe parabolic type is the hardest one, but it can be\nstudied using results from~\\cite{LeviCivita1901,Simo1980,Simo1982}.\n\nThe linear part of the map~(\\ref{eq:f}) at the fixed point $p_{\\rm s}$ is\n\\[\nM_{\\rm s} =\n\\left.\n\\frac{\\partial(\\psi_1,w_1)}{\\partial(\\psi,w)}\n\\right|_{(\\psi,w) = (\\psi_{\\rm s},w_{\\rm s})} =\n\\left(\n\\begin{array}{rc}\n1 & 1 \\\\ -\\mu & 1 - \\mu\n\\end{array}\n\\right),\n\\]\nso $T_{\\rm s} = \\mathop{\\rm tr}\\nolimits[M_{\\rm s}] = 2 - \\mu = 2 - 2\\pi \\tan \\phi_{\\rm s}$.\nThus, the linear type of $p_{\\rm s}$ depends on the parameter $\\mu$\nas follows.\nIt is hyperbolic, parabolic or elliptic if and only if\n$\\mu \\not \\in [0,4]$, $\\mu \\in \\{0,4\\}$, and $\\mu \\in (0,4)$,\nrespectively.\n\nIf $\\mu \\in (0,4)$,\nthen the eigenvalues of $M_{\\rm s}$ have the form\n\\[\n\\lambda_{\\rm s} = {\\rm e}^{\\mathrm{i} \\theta},\\qquad\n\\lambda_{\\rm s}^{-1} = \\bar{\\lambda_{\\rm s}} = {\\rm e}^{-\\mathrm{i} \\theta},\n\\]\nfor some $\\theta \\in (0,\\pi)$ such that\n\\begin{equation}\\label{eq:RotationNumber}\n\\cos \\theta =\n(\\lambda_{\\rm s} + \\bar{\\lambda_{\\rm s}})\/2 =\n\\mathop{\\rm tr}\\nolimits[M_{\\rm s}] \/2 = 1 -\\mu\/2 =\n1 - \\pi \\tan \\phi_{\\rm s}.\n\\end{equation}\nThis means that the linear dynamics around $p_{\\rm s}$ is\nconjugated to a rotation by angle $\\theta$.\n\nIf $p_n = f^n(p)$ is an orbit of the map,\nthen we say that it is \\emph{unbounded} when $(p_n)_{n \\in \\mathbb{Z}}$\nis unbounded in $\\mathbb{T} \\times \\mathbb{R}$,\nand we say that it is \\emph{homoclinic} to the fixed\npoint $p_{\\rm s}$ when $\\lim_{n \\to \\pm \\infty} p_n = p_{\\rm s}$,\nbut $p_n \\neq p_{\\rm s}$.\nAny unbounded orbit is placed outside the acceptance.\n\nThe map~(\\ref{eq:f}) has two fixed points:\n$p_{\\rm s} = (0,0)$ and\n\\[\np_{\\rm h} = (\\psi_{\\rm h},w_{\\rm h}) := (-2\\phi_{\\rm s},0).\n\\]\nThe linear part of the map~(\\ref{eq:f}) at the fixed point $p_{\\rm h}$ is\n\\[\nM_{\\rm h} =\n\\left.\n\\frac{\\partial(\\psi_1,w_1)}{\\partial(\\psi,w)}\n\\right|_{(\\psi,w) = (\\psi_{\\rm h},w_{\\rm h})} =\n\\left(\n\\begin{array}{cc}\n1 & 1 \\\\ \\mu & 1 + \\mu\n\\end{array}\n\\right),\n\\]\nso $T_{\\rm h} = \\mathop{\\rm tr}\\nolimits[M_{\\rm h}] = 2 + \\mu$.\nHence, $p_{\\rm h}$ is hyperbolic for any $\\mu > 0$.\nIndeed, if $\\mu > 0$, then the eigenvalues of $M_{\\rm h}$ have the form\n\\[\n\\lambda_{\\rm h} = {\\rm e}^h, \\qquad \\lambda^{-1}_{\\rm h} = {\\rm e}^{-h},\n\\]\nfor some $h > 0$ such that\n\\[\n\\cosh h = (\\lambda_{\\rm h} + \\lambda^{-1}_{\\rm h})\/2 =\n\\mathop{\\rm tr}\\nolimits[M_{\\rm h}]\/2 = 1+ \\mu\/2 =\n1 + \\pi \\tan \\phi_{\\rm s}.\n\\]\nThis means that the linear dynamics around $p_{\\rm h}$\nexpands the unstable direction by a factor $\\lambda_{\\rm h}$ and\ncontracts the stable direction by a factor $\\lambda^{-1}_{\\rm h}$.\n\nThe quantity $\\theta\/2\\pi$ is called \\emph{tune} in\nAccelerator Physics~\\cite{Rand1984}.\nNevertheless, following a standard terminology in Dynamical Systems,\nwe will say that $\\theta\/2\\pi$ is the \\emph{rotation number} of\nthe elliptic fixed point $p_{\\rm s}$.\nThe elliptic point $p_{\\rm s}$ is called \\emph{$(m,n)$-resonant} when\n$\\theta = 2\\pi m\/n$ for some relatively prime integers $m$ and $n$\nsuch that $1 \\le m \\le n\/2$, which implies that $\\lambda_{\\rm s}^n = 1$.\nBesides, $n$ is the \\emph{order} of the resonance.\nThe quantity $h$ is the \\emph{characteristic exponent} of\nthe hyperbolic fixed point $p_{\\rm h}$.\n\nThe quantities $\\theta$ and $h$ carry the main local information around\nthe fixed points when $\\mu \\in (0,4)$.\n\n\\emph{Moser twist theorem} is the standard tool to prove\nthat fixed points of analytic area preserving maps are locally\nstable~\\cite{SiegelMoser1995}.\nIf the dynamics around the fixed point satisfies some\n\\emph{twist condition}, Moser twist theorem implies that\nany neighborhood of the fixed point contains infinitely many\nclosed invariant curves.\nWe say that such curves are \\emph{rotational\ninvariant curves (RICs)} since they surround the fixed point and\ntheir internal dynamics is conjugated to a rigid rotation.\nThe local stability follows from the fact that the domain\nenclosed by any RIC is invariant.\n\nWe will prove the following characterization\nin Section~\\ref{Sec:LocalStability}.\n\n\\begin{thm}\\label{thm:LocalStability}\nThe synchronous trajectory is locally stable\nif and only if $\\mu \\in (0,4] \\setminus \\{3\\}$.\n\\end{thm}\n\nThe range of local stability in terms of $\\theta$ and $h$\nis given in Table~\\ref{tab:Relations}.\n\n\\begin{table}\n\\centering\n\\begin{tabular}{ll}\n\\hline\nRelation & Local stability \\\\\n\\hline\n$\\mu = 2\\pi \\tan \\phi_{\\rm s}$ & $\\mu \\in (0,4]\\setminus\\{3\\}$ \\\\\n$\\cos \\theta = 1 - \\pi \\tan \\phi_{\\rm s}$ & $\\theta \\in (0,\\pi]\\setminus \\{ 2\\pi\/3\\}$ \\\\\n$\\cosh h = 1 + \\pi \\tan \\phi_{\\rm s}$ & $h \\in (0,h_{\\rm p}] \\setminus \\{ h_{\\rm u} \\}$ \\\\\n\\hline\n\\end{tabular}\n\\caption{\\label{tab:Relations}Relations among\nthe synchronous phase $\\phi_{\\rm s} \\in \\mathbb{T}$,\nthe parameter $\\mu \\in \\mathbb{R}$,\nthe rotation number $\\theta\/2\\pi$,\nand the characteristic exponent $h>0$.\nHere, the values $h_{\\rm p}$ and $h_{\\rm u}$ are\ngiven by $\\cosh h_{\\rm p} = 3$ and $\\cosh h_{\\rm u} = 5\/2$.}\n\\end{table}\n\nOnce we have a clear understanding of the local dynamics,\nwe focus on the global picture, which rises several questions.\nWhich is the ``last'' RIC (LRIC)?\nHow can we compute it?\nDoes it coincide with the border of the connected component $\\mathcal{D}$?\n\nIf the border $\\mathcal{C} = \\partial \\mathcal{D}$ is an analytic curve\nand its internal dynamics is conjugated to a rigid rotation,\nthen $\\mathcal{C}$ is the LRIC that we are looking for and we can numerically\ncompute it by using the algorithm described in~\\cite{SimoTreschev1998}.\nThese two hypotheses about $\\mathcal{C}$ are not so restrictive\nas they may look.\nIndeed, all RICs obtained from the Moser's twist theorem satisfy them.\nThis means that even when $\\mathcal{C} = \\partial \\mathcal{D}$ does not\nsatisfy them, probably there are many Moser-like RICs close to $\\mathcal{C}$,\nwhich are the objects that our computations will find.\n\nA summary of the phenomena that take place when $\\mu$\nmoves is displayed in Fig.~\\ref{fig:SummaryOfDynamics}.\nIf there exist RICs around the origin,\nwe plot some of them, the last one (the LRIC),\nand one unbounded orbit very close to the LRIC.\nIf the origin is globally unstable, we plot suitable unbounded orbits\nto highlight it.\nThe invariant curves (ICs) displayed in Fig.~\\ref{fig:BeforeParabolic0}\nare no RICs for the synchronous trajectory,\nsince they do not surround the origin.\n\nFig.~\\ref{fig:SummaryOfDynamics} shows the basic skeleton from which\nan expert reader can deduce the main dynamical changes.\nLet us describe them.\n\n\\begin{figure*}\n\\centering\n\\subfigure[ICs and two unbounded orbits for $\\mu = -0.1$]{\n\\label{fig:BeforeParabolic0}\n\\iffigures\n\\includegraphics[height=1.62in]{dynamics_mu_m0p1.pdf}\n\\else\n\\vspace{1.62in}\n\\fi\n}\n\\hfill\n\\subfigure[Five unbounded orbits for $\\mu = 0$]{\n\\label{fig:Parabolic0}\n\\iffigures\n\\includegraphics[height=1.62in]{dynamics_mu_0.pdf}\n\\else\n\\vspace{1.62in}\n\\fi\n}\n\\hfill\n\\subfigure[RICs and one unbounded orbit for $\\mu = 0.1$]{\n\\label{fig:AfterParabolic0}\n\\iffigures\n\\includegraphics[height=1.62in]{dynamics_mu_0p1.pdf}\n\\else\n\\vspace{1.62in}\n\\fi\n}\n\\subfigure[RICs and one unbounded orbit for $\\mu = 1$]{\n\\label{fig:mu_1}\n\\iffigures\n\\includegraphics[height=1.62in]{dynamics_mu_1.pdf}\n\\else\n\\vspace{1.62in}\n\\fi\n}\n\\hfill\n\\subfigure[RICs and one unbounded orbit for $\\mu = 2$]{\n\\label{fig:FourthOrderResonance}\n\\iffigures\n\\includegraphics[height=1.62in]{dynamics_mu_2.pdf}\n\\else\n\\vspace{1.62in}\n\\fi\n}\n\\hfill\n\\subfigure[RICs and one unbounded orbit for $\\mu = \\mu_{\\rm r} \\simeq 2.53$]{\n\\label{fig:RootOfTheTwistCoefficient}\n\\iffigures\n\\includegraphics[height=1.62in]{dynamics_mu_root.pdf}\n\\else\n\\vspace{1.62in}\n\\fi\n}\n\\subfigure[RICs and one unbounded orbit for $\\mu = 2.9$]{\n\\label{fig:BeforeThirdOrderResonance}\n\\iffigures\n\\includegraphics[height=1.62in]{dynamics_mu_2p9.pdf}\n\\else\n\\vspace{1.62in}\n\\fi\n}\n\\hfill\n\\subfigure[One unbounded orbit for $\\mu = 3$]{\n\\label{fig:ThirdOrderResonance}\n\\iffigures\n\\includegraphics[height=1.62in]{dynamics_mu_3.pdf}\n\\else\n\\vspace{1.62in}\n\\fi\n}\n\\hfill\n\\subfigure[RICs and one unbounded orbit for $\\mu = 3.1$]{\n\\label{fig:AfterThirdOrderResonance}\n\\iffigures\n\\includegraphics[height=1.62in]{dynamics_mu_3p1.pdf}\n\\else\n\\vspace{1.62in}\n\\fi\n}\n\\subfigure[RICs and one unbounded orbit for $\\mu = 3.9$]{\n\\label{fig:BeforeParabolic4}\n\\iffigures\n\\includegraphics[height=1.62in]{dynamics_mu_3p9.pdf}\n\\else\n\\vspace{1.62in}\n\\fi\n}\n\\hfill\n\\subfigure[RICs and one unbounded orbit for $\\mu = 4$]{\n\\label{fig:Parabolic4}\n\\iffigures\n\\includegraphics[height=1.62in]{dynamics_mu_4.pdf}\n\\else\n\\vspace{1.62in}\n\\fi\n}\n\\hfill\n\\subfigure[One RIC, one unbounded orbit and one\nhomoclinic (to the origin) orbit for $\\mu = 4.05$]{\n\\label{fig:AfterParabolic4}\n\\iffigures\n\\includegraphics[height=1.62in]{dynamics_mu_4p05.pdf}\n\\else\n\\vspace{1.62in}\n\\fi\n}\n\\caption{Race-track microtron phase dynamics in $(\\psi,w)$-coordinates\nfor several values of the parameter $\\mu = 2\\pi \\tan \\phi_{\\rm s}$.\nIn the electronic version one can magnify the plots to check\ndetails of the invariant curves: proximity to the unbounded orbits,\nregularity at their ``corners'', etcetera.\nThis also applies to many other figures.}\n\\label{fig:SummaryOfDynamics}\n\\end{figure*}\n\nThe elliptic fixed point $p_{\\rm s}$ and the hyperbolic fixed point $p_{\\rm h}$\nmerge in a parabolic point as $\\mu \\to 0^+$;\nsee Figs.~\\ref{fig:BeforeParabolic0}--\\ref{fig:AfterParabolic0}.\nThis scenario corresponds to a\nsaddle-center bifurcation~\\cite{Broer_etal1996,Gelfreich2000}.\nThe stable and unstable invariant curves (called \\emph{separatrices})\nof the hyperbolic fixed point seem to form a small loop\naround the elliptic fixed point when $0 < \\mu \\ll 1$.\nThe LRIC around the elliptic point is exponentially close\n(in the parameter $\\mu$) to the separatrices.\nSee Fig.~\\ref{fig:AfterParabolic0}.\nWe will use this fact to see that\n\\begin{equation}\\label{eq:AsymptoticSaddleCenterBifurcation}\n|\\mathcal{A}_\\mu|,|\\mathcal{D}_\\mu| = 6 \\mu^{5\/2}\/5\\pi^2 + \\mathop{\\rm O}\\nolimits(\\mu^3),\n\\qquad \\mbox{as $\\mu \\to 0^+$}.\n\\end{equation}\nTherefore, both regions are almost the same\nwhen $\\mu \\to 0 ^+$.\n(If $\\mathcal{R}$ is a subset of $\\mathbb{T} \\times \\mathbb{R}$,\n then $|\\mathcal{R}|$ denotes its area.)\n\nThe LRIC grows in size, changes its shape, and moves away from\nthe separatrices as $\\mu$ increases; see Fig.~\\ref{fig:mu_1}.\nWe reach the fourth order resonance $\\theta = \\pi\/2$ at $\\mu = 2$.\nThe elliptic point is locally stable at this resonance,\nand the RICs near it look like a Latin cross,\nwhose arms have a width of the order of the square of its length;\nsee Fig.~\\ref{fig:FourthOrderResonance}.\nThe H\\'enon map displays exactly the same behavior at the\nfourth order resonance~\\cite{Simo1980}.\n\nThe twist coefficient (also called first Birkhoff coefficient)\ngoes from negative to positive at $\\mu = \\mu_{\\rm r} \\simeq 2.537706$\nand from positive to negative at $\\mu = 3$,\nwhere the third order resonance $\\theta = 2\\pi\/3$ takes place.\nThis is one of the two typical behaviors for the twist\ncoefficient described in~\\cite{Moeckel1990}.\nIt means that the orbits around the origin rotate slower\n(resp., faster) as they go away from the origin\nwhen $\\mu \\in (0,\\mu_{\\rm r})\\cup (3,4)$ (resp., $\\mu \\in (\\mu_{\\rm r},3)$).\nFigs.~\\ref{fig:BeforeThirdOrderResonance}--\\ref{fig:AfterThirdOrderResonance}\nshow that behavior,\nsince we see a three-periodic orbit near the elliptic point\nfor both $\\mu \\lesssim 3$ and $\\mu \\gtrsim 3$.\n\nThe elliptic point is locally and globally unstable at\nthe third order resonance; see Fig.~\\ref{fig:ThirdOrderResonance}.\nIf $\\mu = 3 + \\epsilon$ with $0 < |\\epsilon| \\ll 1$,\nthe LRIC is approximately a triangle of vertices\n\\begin{equation}\\label{eq:TriangleVertices}\n\\frac{\\epsilon}{\\pi}(1,0),\\qquad \\frac{\\epsilon}{\\pi}(1,-3),\\qquad\n\\frac{\\epsilon}{\\pi}(-2,3).\n\\end{equation}\nIn particular,\n\\begin{equation}\\label{eq:AsymptoticThirdOrderResonace}\n|\\mathcal{D}_{3+\\epsilon}| = 9\\epsilon^2\/2\\pi^2 + \\mathop{\\rm O}\\nolimits(\\epsilon^3),\n\\qquad \\mbox{as $\\epsilon \\to 0$}.\n\\end{equation}\n\nThe LRIC grows in size and its triangular shape changes to\na ``banana-like'' shape as the parameter moves from the three order resonance\nat $\\mu = 3$ to the second order resonance at $\\mu = 4$.\nThe fixed point $p_{\\rm s}$ is locally stable at this last resonance,\nand the RICs near it looks like ``bananas'' whose width is of\nthe order of the square of its length. See Fig.~\\ref{fig:Parabolic4}.\n\nFinally, a period-doubling bifurcation takes place after\nthe second order resonance.\nTo be precise, the point $p_{\\rm s}$ becomes a saddle for $\\mu > 4$,\nbeing parabolic with reflection at $\\mu = 4$.\nOne then expects to find an elliptic two-periodic orbit for $\\mu > 4$.\nThe \\emph{separatrices} of the hyperbolic fixed point $p_{\\rm s}$\nform two small loops around the two elliptic two-periodic points.\nNevertheless, the saddle $p_{\\rm s}$ remains globally stable when $\\mu \\gtrsim 4$,\nsince some of the RICs around $p_{\\rm s}$ still persist.\nSee Fig.~\\ref{fig:AfterParabolic4}.\n\nIn spite of all the above comments, the expulsion of some\nresonance is a more relevant phenomenon than any resonance at the elliptic\nfixed point, but the third order resonance.\nThe reason is that only the first phenomenon changes\ndrastically $\\mathcal{A}$ and $\\mathcal{D}$,\nsince the second one takes place inside $\\mathcal{D}$.\nWe will check that $|\\mathcal{A}|$ and $|\\mathcal{D}|$ experiment a\njump for each primary resonance that is thrown away.\nBesides,\nthese primary jumps are smaller in $|\\mathcal{A}|$ than in $|\\mathcal{D}|$.\nHowever, $|\\mathcal{A}|$ displays many more jumps, the secondary ones,\nwhich are related to resonances inside elliptic islands\nthat are already outside $\\mathcal{D}$.\nWe will visualize such properties in Section~\\ref{Sec:GlobalStability}.\n\n\n\n\\section{Local stability of the synchronous trajectory}\n\\label{Sec:LocalStability}\n\n\n\nThis section has a purely local character.\nIt contains the proof of Theorem~\\ref{thm:LocalStability}.\nThe hyperbolic type always implies local instability,\nso we only study the range $0 \\le \\mu \\le 4$.\n\n\n\\subsection{Local instability in the saddle-center bifurcation}\n\\label{Ssec:SaddleCenterBifurcation}\n\n\nSet $\\mu = 0$.\nThen the map~(\\ref{eq:f}) has the form\n\\[\n\\left\\{\n\\begin{array}{ccl}\nw_1    & = & w - \\pi (\\psi+w)^2 + \\mathop{\\rm O}\\nolimits_4(\\psi,w), \\\\\n\\psi_1 & = & \\psi + w.\n\\end{array}\n\\right.\n\\]\nOn the other hand,\nthe Levi-Civita criterion~\\cite{LeviCivita1901} implies that\nthe origin is unstable under any analytic map of the form\n\\[\n\\left\\{\n\\begin{array}{ccl}\nw_1 & = & w + \\mathop{\\rm O}\\nolimits_2(\\psi,w), \\\\\n\\psi_1 & = & \\psi + w + \\mathop{\\rm O}\\nolimits_2(\\psi,w),\n\\end{array}\n\\right.\n\\]\nwith $\\frac{\\partial^2 w_1}{\\partial \\psi^2}(0,0) \\neq 0$.\nIn our concrete map,\n$\\frac{\\partial^2 w_1}{\\partial \\psi^2} (0,0) = -2\\pi \\neq 0$.\nThus, the synchronous trajectory is locally unstable if $\\mu = 0$.\n\n\n\\subsection{Local stability in the elliptic case: Generic values}\n\\label{Ssec:LocalStabilityElliptic}\n\n\nLet $\\mu \\in (0,4)$.\nThen the point $p_{\\rm s}$ is elliptic with rotation number $\\theta\/2\\pi$.\nThe relation between $\\theta$ and $\\mu$ is given in Table~\\ref{tab:Relations}.\n\nFirst, we bring the linear part of map~(\\ref{eq:f}) around\nthe elliptic fixed point $p_{\\rm s}$ into its real Jordan normal form\n\\[\nR_\\theta = \n\\left( \\begin{array}{rr}\n\\cos \\theta & -\\sin \\theta \\\\ \\sin \\theta & \\cos \\theta\n\\end{array} \\right) \n\\]\nby means of an area preserving linear change $(\\psi,w) \\mapsto (x,y)$.\nWe write our map in these new variables as\n\\[\n\\left( \\begin{array}{c} x_1 \\\\ y_1 \\end{array} \\right) =\nR_\\theta \n\\left( \\begin{array}{c} x \\\\ y \\end{array} \\right) + \\mathop{\\rm O}\\nolimits_2(x,y).\n\\]\n\nThen we introduce the complex variables $z = x + \\mathrm{i} y$\nand $\\bar{z} = x - \\mathrm{i} y$.\nThe map reads in these variables as\n\\begin{equation}\\label{eq:ComplexVariable}\nz_1 = \\lambda_{\\rm s} z + \\mathop{\\rm O}\\nolimits(|z|^2) = \n\\lambda_{\\rm s} \\left(\nz + \\sum_{\\substack{2 \\le j+k \\le 3 \\\\ j,k \\ge 0}} c_{jk} z^j \\bar{z}^k\n\\right) + \\mathop{\\rm O}\\nolimits(|z|^4),\n\\end{equation}\nfor some complex coefficients $c_{jk}$,\nwhich depend on the original parameter $\\mu$.\nWe recall that $\\lambda_{\\rm s} = {\\rm e}^{\\mathrm{i} \\theta}$.\n\nIf $\\lambda_{\\rm s}^n = 1$ for some integer $n$ such that $1 \\le n \\le 4$,\nthen we say that the elliptic point $p_{\\rm s}$ is \\emph{strongly resonant}.\nIt is known~\\cite{SiegelMoser1995} that if the elliptic point $p_{\\rm s}$\nis not strongly resonant, then there exists an area preserving polynomial\nchange of variables which brings the map~(\\ref{eq:ComplexVariable})\ninto its third order \\emph{Birkhoff normal form}\n\\begin{equation}\\label{eq:BirkhoffNormalForm}\nz_1 = \\lambda_{\\rm s} \\left(z + \\mathrm{i} \\tau z^2 \\bar{z} \\right) + \\mathop{\\rm O}\\nolimits(|z|^4)\n= {\\rm e}^{\\mathrm{i} (\\theta + \\tau |z|^2)} z + \\mathop{\\rm O}\\nolimits(|z|^4),\n\\end{equation}\nfor some real coefficient $\\tau$,\ncalled \\emph{first Birkhoff coefficient} or \\emph{twist coefficient}.\nIt turns out that\n\\[\n\\tau = \\Im c_{21} +\n\\frac{2(4 \\cos \\theta +1)\\sin\\theta}\n     {(\\cos \\theta -1)(2\\cos \\theta + 1)}|c_{20}|^2,\n\\]\nwhere $\\Im c_{21}$ denotes the imaginary part of $c_{21}$,\nsee~\\cite{Moeckel1990,KamphorstPinto2005}.\nThe coefficients $c_{21}$ and $c_{20}$ also depend on $\\theta$,\nvia the parameter $\\mu$.\nAfter some tedious computations~\\cite{Larreal2012},\none gets the expression\n\\begin{equation}\\label{eq:TwistCoefficient}\n\\tau(\\theta) =\n\\frac{2 \\cos^3 \\theta - 3 \\cos^2 \\theta + 4\\pi^2 \\cos \\theta + 1 + \\pi^2}\n{(\\cos \\theta - 1)(2\\cos \\theta + 1)\\sin^2 \\theta}.\n\\end{equation}\nWe observe that $\\lim_{\\theta \\to 0^+} \\theta^4 \\tau(\\theta)$,\n$\\lim_{\\theta \\to 2\\pi\/3} (\\theta-2\\pi\/3)\\tau(\\theta)$, and\n$\\lim_{\\theta \\to \\pi^-} (\\theta-\\pi)^2 \\tau(\\theta)$ exist and are negative.\nThat is, $\\tau(\\theta)$ has a negative fourth-order pole at $\\theta = 0$,\na negative simple pole at $\\theta = 2\\pi\/3$,\nand a negative second-order pole at $\\theta = \\pi$.\nThese properties agree with the generic behavior of twist coefficients\nof families of area-preserving maps described in~\\cite{Moeckel1990}.\n\nConsequently, we deduce that $\\tau(\\theta)$ has at least one root\nin the interval $(0,2\\pi\/3)$.\nLet us prove that it has no more real roots.\nThe numerator in~(\\ref{eq:TwistCoefficient}) is the polynomial\n$a \\zeta^3 + b \\zeta^2 + c \\zeta + d$ in the variable $\\zeta = \\cos \\theta$,\nwith $a = 2$, $b = -3$, $c = 4\\pi^2$, and $d = 1 + \\pi^2$.\nThis cubic polynomial has one real root and two complex conjugated roots,\nbecause its discriminant is negative:\n\\[\n\\Delta =\nb^2 c^2 - 4 a c^3 - 4 b^3 d - 27 a^2 d^2 + 18 a b c d \\simeq\n-536135.\n\\]\nThis means that $\\tau(\\theta)$ has just one root $\\theta_{\\rm r}$\nin the interval $(0,\\pi)$.\nIt was numerically computed in~\\cite{Larreal2012} that\n\\[\n\\theta_{\\rm r} \\simeq 1.842998343412199023198246043 \\in (0,2\\pi\/3).\n\\]\nWe have plotted the twist coefficient $\\tau$ versus $\\theta$\nin Fig.~\\ref{fig:twist_coefficient}. \n\n\\begin{figure}\n\\iffigures\n\\centering\n\\psfrag{0}{\\footnotesize{$0$}}\n\\psfrag{pi}{\\footnotesize{$\\pi$}}\n\\psfrag{pi\/3}{\\footnotesize{$\\pi\/3$}}\n\\psfrag{2*pi\/3}{\\footnotesize{$2\\pi\/3$}}\n\\includegraphics[height=2.4in]{twist_coefficient.pdf}\n\\else\n\\vspace{2.4in}\n\\fi\n\\caption{The twist coefficient $\\tau$ versus $\\theta$.}\n\\label{fig:twist_coefficient}\n\\end{figure}\n\nFinally, we recall that the Moser twist theorem~\\cite{SiegelMoser1995}\nimplies that the origin is a stable elliptic fixed point of any analytic map\nof the form~(\\ref{eq:BirkhoffNormalForm}) when $\\tau \\neq 0$.\nHence,\nthe synchronous trajectory is locally stable\nwhen $\\theta \\in (0,\\pi) \\setminus \\{\\pi\/2,\\theta_{\\rm r},2\\pi\/3\\}$;\nor, equivalently, when $\\mu \\in (0,4) \\setminus \\{2,\\mu_{\\rm r},3\\}$, where\n\\[\n\\mu_{\\rm r} = 2-2\\cos \\theta_{\\rm r} \\simeq  2.537706055658189018165133406.\n\\]\n\n\n\\subsection{Local stability in the fourth order resonance}\n\\label{Ssec:LocalStabilityFourthOrder}\n\n\nSet $\\mu = 2$.\nThen the map~(\\ref{eq:f}) takes the form\n\\[\n\\left\\{\n\\begin{array}{ccl}\n\\psi_1 & = & \\psi + w,\\\\\nw_1    & = & -2\\psi -w - a(\\psi_1),\n\\end{array}\n\\right.\n\\]\nwhere $a(\\psi_1) = 2\\pi(1 - \\cos \\psi_1) - 2(\\psi_1 - \\sin \\psi_1) =\n\\mathop{\\rm O}\\nolimits(\\psi_1^2)$.\n\nWe consider the area preserving linear change of variables\n\\begin{equation}\\label{eq:CoordinatesFourthOrder}\nx = \\psi + w,\\qquad y = \\psi.\n\\end{equation}\nThis change brings the linear part of the map at the origin\ninto its real Jordan normal form $R_{\\pi\/2}$.\nIndeed, the map becomes\n\\begin{equation}\\label{eq:Map_At_4Resonance}\n\\left( \\begin{array}{c} x_1 \\\\ y_1 \\end{array} \\right) =\n\\left( \\begin{array}{c} -y - a(x) \\\\ x \\end{array} \\right) =\nR_{\\pi\/2}\n\\left( \\begin{array}{c} x \\\\ y + a(x) \\end{array} \\right).\n\\end{equation}\nLet $\\sum_{j \\ge 2} a_j x^j$ be the Taylor expansion of $a(x)$.\nWe know from Corollary 4.2 in~\\cite{Simo1982} that if $a_2 \\neq 0$\nand $a_3 \\neq 0$, then the origin is locally unstable\nunder the analytic map~(\\ref{eq:Map_At_4Resonance}) if and only if\n\\[\n0 < a_3 \\le a_2^2.\n\\]\nIn our concrete map, $a_2 = \\pi$ and $a_3 = -1\/3$.\nTherefore, the synchronous trajectory is locally stable when $\\mu = 2$,\nand we will analytically study the shape of its RICs\nin Section~\\ref{Ssec:HamiltonianFourthOrder}.\n\n\n\\subsection{Local stability for the root of the twist coefficient}\n\\label{Ssec:LocalStabilityRoot}\n\n\nIf $\\mu = \\mu_{\\rm r}$, then the third order Birkhoff normal\nform~(\\ref{eq:BirkhoffNormalForm}) does not provide much information\nbecause the first Birkhoff coefficient $\\tau_1 = \\tau$ vanishes.\nThus, we should compute the fifth order Birkhoff normal form\n\\[\nz_1 =  {\\rm e}^{\\mathrm{i} (\\theta_{\\rm r} + \\tau_1 |z|^2 + \\tau_2 |z|^4)} z + \\mathop{\\rm O}\\nolimits(|z|^6),\n\\]\nwhere $\\tau_1 = 0$ and\n$\\tau_2 \\in \\mathbb{R}$ is the \\emph{second Birkhoff coefficient}.\nThe analytical computation of $\\tau_2$ is cumbersome,\nso we have taken a simpler numerical approach.\nFirst, we have computed the rotation number $\\rho(p)$\nof the points of the form $p = (\\psi,0)$ using the algorithm described\nin subsection~\\ref{Ssec:RotationNumber}.\nNext, we have checked that\n\\begin{equation}\\label{eq:FlatBehavior}\n\\rho(\\psi,0) = \\theta_{\\rm r}\/2\\pi + \\rho_2 \\psi^4 + \\mathop{\\rm O}\\nolimits(\\psi^5),\n\\end{equation}\nfor some non-zero coefficient $\\rho_2 \\approx -200$,\nwhich implies that $\\tau_2 \\neq 0$.\nThen the Moser twist theorem implies that the origin is a\nstable elliptic fixed point~\\cite{SiegelMoser1995}.\n\n\\begin{figure}\n\\iffigures\n\\centering\n\\includegraphics[height=2.4in]{RotationNumber_mu_root.pdf}\n\\else\n\\vspace{2.4in}\n\\fi\n\\caption{The rotation number $\\rho(\\psi,0)$ versus $\\psi$ for $\\mu = \\mu_{\\rm r}$.}\n\\label{fig:RotationNumber_mu_root}\n\\end{figure}\n\nThe flat behavior~(\\ref{eq:FlatBehavior}) is clearly observed\nin Fig.~\\ref{fig:RotationNumber_mu_root}.\nWe will explain the meaning of the small gaps that appear in\nFig.~\\ref{fig:RotationNumber_mu_root} at the end of\nsubsection~\\ref{Ssec:RotationNumber}.\n\n\n\\subsection{Local instability in the third order resonance}\n\\label{Ssec:LocalUnstabilityThirdOrder}\n\n\nSet $\\mu = 3$.\nThen the map~(\\ref{eq:f}) takes the form\n\\[\n\\left\\{\n\\begin{array}{ccl}\n\\psi_1 & = & \\psi + w,\\\\\nw_1    & = & -3\\psi -2w - \\pi (\\psi+w)^2 + \\mathop{\\rm O}\\nolimits_3(\\psi,w).\n\\end{array}\n\\right.\n\\]\nThe map $(\\psi_3,w_3) = f^3(\\psi,w)$ is close to the identity,\nsince\n\\[\n\\left\\{\n\\begin{array}{ccl}\n\\psi_3 & = & \\psi - \\pi( 3 \\psi^2 + 2 \\psi w ) + \\mathop{\\rm O}\\nolimits_3(\\psi,w),\\\\\nw_3    & = & w + \\pi (6\\psi^2+6\\psi w + w^2) + \\mathop{\\rm O}\\nolimits_3(\\psi,w).\n\\end{array}\n\\right.\n\\]\nWe determine the local stability of the origin under the map $f^3$\n(and so, under the map $f$) by applying Sim\\'o's criterion~\\cite{Simo1982}.\nThat criterion states that the origin is locally stable under an\nanalytic area preserving map of the form\n\\[\n\\left\\{\n\\begin{array}{ccl}\nx_1 & = & x + \\mathop{\\rm O}\\nolimits_2(x,y),\\\\\ny_1 & = & y + \\mathop{\\rm O}\\nolimits_2(x,y),\n\\end{array}\n\\right.\n\\]\nif and only if $G(x_1,y)$ has a strict extremum at the origin,\nwhere $x_1 y + G(x_1,y)$ is the generating function of the map;\nthat is, $G(x_1,y)$ is a function determined by the implicit equations\n\\begin{equation}\\label{eq:GeneratingFunctionEquations}\nx_1 = x + \\frac{\\partial G}{\\partial y}(x_1,y)\n\\qquad \\mbox{and} \\qquad\ny_1 = y - \\frac{\\partial G}{\\partial x_1}(x_1,y).\n\\end{equation}\nThus, we look for a function $G(w_3,\\psi) = O_3(w_3,\\psi)$ such that\n\\[\nw_3 = w + \\frac{\\partial G}{\\partial \\psi}(w_3,\\psi) \\qquad \\mbox{and}\\qquad\n\\psi_3 = \\psi - \\frac{\\partial G}{\\partial w_3}(w_3,\\psi).\n\\]\nAfter a straightforward computation, we obtain that\n\\begin{eqnarray*}\nG(w_3,\\psi) & = & \\pi (2\\psi^3 + 3\\psi^2 w_3 + \\psi w_3^2)\n+ \\mathop{\\rm O}\\nolimits_4(\\psi,w_3) \\\\\n& = & \\pi \\psi (\\psi + w_3) (2\\psi + w_3) + \\mathop{\\rm O}\\nolimits_4(\\psi,w_3).\n\\end{eqnarray*}\nThis function has no strict extremum at the origin.\nTherefore, the synchronous trajectory is unstable when $\\mu = 3$.\nWe will give more details about the dynamics around the origin\nnear this third order resonance in Section~\\ref{Ssec:HamiltonianThirdOrder}.\n\n\n\\subsection{Local stability in the second-order resonance}\n\\label{Ssec:SecondOrderResonance}\n\n\nSet $\\mu = 4$. Then the map~(\\ref{eq:f}) takes the form\n\\[\n\\left\\{\n\\begin{array}{ccl}\n\\psi_1 & = & \\psi + w,\\\\\nw_1    & = & -4\\psi - 3 w + b(\\psi_1),\n\\end{array}\n\\right.\n\\]\nwhere $b(\\psi_1) = 2\\pi(\\cos \\psi_1 - 1) + 4(\\psi_1 - \\sin \\psi_1)  =\n\\mathop{\\rm O}\\nolimits(\\psi_1^2)$.\n\nWe consider the area preserving linear change of variables\n\\[\nx = 2\\psi + w,\\qquad y = \\psi,\n\\]\nwhich brings the linear part of the map at the origin\ninto its Jordan normal form.\nIndeed, the map becomes\n\\begin{equation}\\label{eq:Map_At_2Resonance}\n\\left\\{\n\\begin{array}{ccl}\nx_1 & = & -x + b(x-y),\\\\\ny_1 & = & x - y.\n\\end{array}\n\\right.\n\\end{equation}\nLet $\\sum_{j \\ge 2} b_j u^j$ be the Taylor expansion of $b(u)$.\nWe know from Lemma 1.4 in~\\cite{Simo1982} that if\n$b_2 \\neq 0$, $b_3 \\neq 0$, and $2 b_3 + b_2^2 > 0$,\nthen the origin is locally stable under the\nanalytic map~(\\ref{eq:Map_At_2Resonance}).\n\nIn our concrete map, $b_2 = -\\pi$ and $b_3 = 2\/3$.\nTherefore, the synchronous trajectory is locally stable when $\\mu = 4$.\n\nThis completes the proof of Theorem~\\ref{thm:LocalStability}.\n\n\\section{Hamiltonian approximations}\n\\label{Sec:HamiltonianApproximations}\n\n\nThis section contains a semi-local study of the RICs around\nthe synchronous trajectory.\nWe will approximate them by the level curves of suitable\nHamiltonians in three different scenarios.\nWe will also prove the asymptotic\nformulas~(\\ref{eq:AsymptoticSaddleCenterBifurcation})\nand~(\\ref{eq:AsymptoticThirdOrderResonace}).\n\n\n\n\\subsection{Near the saddle-center bifurcation}\n\\label{Ssec:HamiltonianSaddleCenterBifurcation}\n\n\nLet us study the size and shape of $\\mathcal{D}_\\mu \\subset \\mathcal{A}_\\mu$\nwhen $\\mu \\to 0^+$.\n\nThe elliptic fixed point $p_{\\rm s} = (\\psi_{\\rm s},0) = (0,0)$\nand the hyperbolic fixed point $p_{\\rm h} = (\\psi_{\\rm h},0)$\nof the map~(\\ref{eq:f}) collapse as $\\mu \\to 0^+$.\nThe following rough quantitative estimates are typical for\nsaddle-center bifurcations.\nThe size of $\\mathcal{A}_\\mu$ in the $\\psi$-coordinate\nis $\\mathop{\\rm O}\\nolimits(|\\psi_{\\rm s}-\\psi_{\\rm h}|) = \\mathop{\\rm O}\\nolimits(\\mu)$.\nIts size in the $w$-coordinate is\n$\\mathop{\\rm O}\\nolimits(|\\psi_{\\rm s}-\\psi_{\\rm h}|) \\times \\mathop{\\rm O}\\nolimits(\\mu^{1\/2}) = \\mathop{\\rm O}\\nolimits(\\mu^{3\/2})$,\nsince the angle between the eigenvectors of the matrix $M_{\\rm h}$\nis $\\mathop{\\rm O}\\nolimits(\\mu^{1\/2})$.\nConsequently,\n$\\mathcal{A}_\\mu = \\mathop{\\rm O}\\nolimits(\\mu) \\times \\mathop{\\rm O}\\nolimits(\\mu^{3\/2}) = \\mathop{\\rm O}\\nolimits(\\mu^{5\/2})$.\nFinally, $|\\mathcal{A}_\\mu| \\asymp |\\mathcal{D}_\\mu|$ when $\\mu \\to 0^+$.\nWe will confirm and refine these rough estimates.\n\nFollowing the above comments, \nwe scale the $\\psi$-coordinate (respectively, $w$-coordinate) by\na factor of order $\\mu$ (respectively, order $\\mu^{3\/2}$).\nTo be precise, we consider the change of scales\n\\begin{equation}\\label{eq:Scaling_SaddleCenterBifurcation}\nx = \\psi\/\\mu,\\qquad y = w\/\\mu^{3\/2}.\n\\end{equation}\n\nIf $0 < \\mu \\ll 1$, then the map~(\\ref{eq:f}) is transformed under this change\ninto a map $(x_1,y_1) = \\tilde{f}(x,y)$ of the form\n\\begin{equation}\\label{eq:ftilde_SaddleCenter}\n\\left\\{\n\\begin{array}{ccl}\nx_1 & = & x + \\mu^{1\/2} y, \\\\\ny_1 & = & y - \\mu^{1\/2}(x + \\pi x^2) + \\mathop{\\rm O}\\nolimits(\\mu).\n\\end{array}\n\\right.\n\\end{equation}\nThe map $\\tilde{f}$ is close to the identity map\n${\\rm I}(x,y) = (x,y)$, since\n\\[\n\\tilde{f} = {\\rm I} + \\mu^{1\/2} \\tilde{f}_1 + \\mathop{\\rm O}\\nolimits(\\mu),\\qquad\n\\tilde{f}_1(x,y) = (y, -x - \\pi x^2).\n\\]\nThe area-preserving character of $\\tilde{f}$ implies that the term\n$\\mu^{1\/2} \\tilde{f}_1$ is a Hamiltonian vector field.\nConcretely,\n\\[\n\\mu^{1\/2} \\tilde{f}_1 =\n\\mu^{1\/2} \\left(\\frac{\\partial \\tilde{H}_1}{\\partial y},\n               -\\frac{\\partial \\tilde{H}_1}{\\partial x}\n\\right),\n\\]\nwhere the Hamiltonian $\\tilde{H}_1:\\mathbb{R}^2 \\to \\mathbb{R}$ is given by\n\\begin{equation}\\label{eq:H1_SaddleCenterBifurcation}\n\\tilde{H}_1(x,y) =\n\\frac{x^2 + y^2}{2} + \\frac{\\pi}{3} x^3 - \\frac{1}{6\\pi^2}.\n\\end{equation}\nWe have subtracted the constant $1\/6\\pi^2$ for convenience.\nWe denote by $\\phi_H^t$ the $t$-time flow of a Hamiltonian $H$.\nThen\n\\begin{equation}\\label{eq:f_H1}\n\\tilde{f} =\n{\\rm I} + \\mu^{1\/2} \\tilde{f}_1 + \\mathop{\\rm O}\\nolimits(\\mu) =\n\\phi^{\\mu^{1\/2}}_{\\tilde{H}_1} + \\mathop{\\rm O}\\nolimits(\\mu) =\n\\phi^1_{\\mu^{1\/2} \\tilde{H}_1} + \\mathop{\\rm O}\\nolimits(\\mu),\n\\end{equation}\nsince the Euler method has a second order\nlocal truncation error.\n\nTherefore, the integrable dynamics of the Hamiltonian $\\tilde{H}_1$\napproximates the dynamics of the map $\\tilde{f}$ in the limit\n$\\mu \\to 0^+$.\nWe say that $\\tilde{H}_1$ is the \\emph{limit Hamiltonian} associated\nto the saddle-center bifurcation.\nIt has two equilibrium points.\nNamely, the elliptic point $\\tilde{q}_{\\rm s} = (0,0)$\nand the saddle point $\\tilde{q}_{\\rm h} = (-1\/\\pi,0)$.\nBesides, the unstable and stable invariant curves of the saddle point\ncoincide along one branch, giving rise to the separatrix\n\\[\n\\tilde{\\mathcal{S}} =\n\\left\\{\n\\tilde{q}=(x,y) \\in \\mathbb{R}^2 : \\tilde{H}_1(\\tilde{q}) = 0, \\ x > -1\/\\pi\n\\right\\}.\n\\]\nLet $\\tilde{\\mathcal{R}}$ be the domain enclosed by\n$\\tilde{\\mathcal{S}} \\cup \\{\\tilde{q}_{\\rm h}\\}$;\nthat is,\n\\[\n\\tilde{\\mathcal{R}} =\n\\left\\{\n\\tilde{q}=(x,y) \\in \\mathbb{R}^2 : \\tilde{H}_1(\\tilde{q}) \\le 0,\\ x \\ge -1\/\\pi\n\\right\\}.\n\\]\nThe phase portrait of the limit Hamiltonian~(\\ref{eq:H1_SaddleCenterBifurcation})\nis sketched in Fig.~\\ref{fig:H1_SaddleCenterBifurcation}.\nOnly the points inside the domain $\\tilde{\\mathcal{R}}$\ngive rise to bounded trajectories.\nThis implies that $\\tilde{\\mathcal{R}}$ is a good approximation of the scaled versions\nof $\\mathcal{A}_\\mu$ and $\\mathcal{D}_\\mu$ when $0 < \\mu \\ll 1$.\n\n\\begin{figure}\n\\iffigures\n\\centering\n\\psfrag{qh}{$\\tilde{q}_{\\rm h}$}\n\\psfrag{qs}{$\\tilde{q}_{\\rm s}$}\n\\psfrag{ar}{$\\blacktriangleright$}\n\\psfrag{al}{$\\blacktriangleleft$}\n\\includegraphics[height=2.4in]{H1_SaddleCenterBifurcation.pdf}\n\\else\n\\vspace{2.4in}\n\\fi\n\\caption{The separatrix (thick line), some level curves (thin lines),\nand the two equilibrium points (small circles) of the limit\nHamiltonian~(\\ref{eq:H1_SaddleCenterBifurcation})\nassociated to the saddle-center bifurcation.\nThe black arrows show the Hamiltonian dynamics on the separatrix.}\n\\label{fig:H1_SaddleCenterBifurcation}\n\\end{figure}\n\nThe separatrix $\\tilde{\\mathcal{S}}$ is described by a homoclinic trajectory\n$\\tilde{q}(t) = (x(t),y(t))$ to the saddle point:\n$\\lim_{t \\to \\pm \\infty} \\tilde{q}(t) = \\tilde{q}_{\\rm h}$.\nIf we impose the initial condition $y(0) = 0$ and\nsolve the Hamiltonian equations on the separatrix $\\tilde{\\mathcal{S}}$,\nwe get that\n\\begin{equation}\\label{eq:HomoclinicOrbit}\nx(t) = \\frac{3}{2\\pi \\cosh^2(t\/2)}-\\frac{1}{\\pi},\\qquad\ny(t) = \\frac{3 \\sinh (t\/2)}{2\\pi \\cosh^3(t\/2)}.\n\\end{equation}\nTherefore,\n\\[\n|\\tilde{\\mathcal{R}}| = \\oint_{\\tilde{S}} y {\\rm d} x =\n\\int_{-\\infty}^{+\\infty} y(t) x'(t) {\\rm d} t =\n\\frac{6}{5\\pi^2}.\n\\]\nThe first identity follows from Green's theorem,\nthe last one follows from the residue's theorem.\nThe asymptotic estimate~(\\ref{eq:AsymptoticSaddleCenterBifurcation})\nfollows from the change of scales~(\\ref{eq:Scaling_SaddleCenterBifurcation}).\n\nFig.~\\ref{fig:AreaStabilityRegion_close_to_0} shows a strong agreement\nbetween the numerically computed values of $|\\mathcal{A}_\\mu|$\nand $|\\mathcal{D}_\\mu|$ and their asymptotic\nestimate $\\mu \\mapsto 6 \\mu^{5\/2}\/5\\pi^2$,\neven for relatively big values of $\\mu$.\n\nNext, we compute better Hamiltonian approximations near\nthe saddle-center bifurcation.\nThe Lie's series method is the standard tool to find them,\nhowever we will follow a generating function method that\nfits perfectly with the map~(\\ref{eq:ftilde_SaddleCenter}).\n\nWe recall the following formal result from~\\cite[Remark 1]{Simo1982}.\n\nLet $(x_1,y_1) = \\tilde{f}(x,y)$ be an analytic area preserving map that is,\nin some sense, close to the identity.\nLet $\\tilde{G}(x,y)$ be a function such that $x_1 y + \\tilde{G}(x_1,y)$\nis a generating function of $\\tilde{f}$; that is,\nit satisfies the implicit equations~(\\ref{eq:GeneratingFunctionEquations}).\nWe note that $\\| \\tilde{G} \\| \\ll 1$.\nThe 1-time flow of the Hamiltonian $\\tilde{H} = \\sum_{j\\ge 1} \\hat{H}_j$\ndefined by\n\\begin{eqnarray*}\n\\hat{H}_1 & = & \\tilde{G}, \\\\\n\\hat{H}_2 & = & \\textstyle{\\frac{1}{2}} \\tilde{G}_x \\tilde{G}_y, \\\\\n\\hat{H}_3 & = & \\textstyle{\\frac{1}{12}}(\\tilde{G}_{xx} \\tilde{G}_y^2 +\n                4\\tilde{G}_{xy} \\tilde{G}_x \\tilde{G}_y + \\tilde{G}_{yy} \\tilde{G}_x^2),\\\\\n\\hat{H}_4 & = & \\textstyle{\\frac{1}{12}}(\\tilde{G}_{xxy} \\tilde{G}_y +\n                \\tilde{G}_{xyy} \\tilde{G}_x + \\tilde{G}_{xx} \\tilde{G}_{yy} +\n                3\\tilde{G}_{xy}^2)\\tilde{G}_x \\tilde{G}_y \\\\\n          &   & \\textstyle{\\frac{1}{12}} \\tilde{G}_{xy}(\\tilde{G}_{xx} \\tilde{G}_y^2\n                + \\tilde{G}_{yy} \\tilde{G}_x^2),\n\\end{eqnarray*}\nand so on, coincides with the map $\\tilde{f}$ at a \\emph{formal} level.\nThere is a small misprint in~\\cite{Simo1982};\nthe numerical factors in the fourth term are $\\frac{1}{12}$, not $\\frac{1}{2}$.\nBesides, we must skip the minus signs in front of the even terms that\nappear in~\\cite{Simo1982}, because our generating function has the\nform $\\tilde{G}(x_1,y)$, instead of $\\tilde{G}(x,y_1)$.\nHere, subindexes in $\\tilde{G}$ mean partial derivatives.\nWe note that $\\hat{H}_j = \\mathop{\\rm O}\\nolimits(\\| \\tilde{G} \\|^j)$.\nThe series $\\tilde{H} = \\sum_{j\\ge 1} \\hat{H}_j$ is generically divergent.\nOtherwise, $\\tilde{f} = \\phi^1_{\\tilde{H}}$ would be integrable,\nwhich is an exceptional situation.\n\nNow, we come back to our concrete problem.\n\nWe write the map~(\\ref{eq:ftilde_SaddleCenter}) as\n\\[\n\\left\\{\n\\begin{array}{ccl}\nx_1 & = & x + \\mu^{1\/2} y, \\\\\ny_1 & = & y - \\mu^{1\/2} c(x_1),\n\\end{array}\n\\right.\n\\]\nwhere\n\\[\nc(x_1) =\n\\frac{2\\pi (1 - \\cos(\\mu x_1)) + \\mu \\sin(\\mu x_1)}{\\mu^2} =\nx_1 + \\pi x^2_1 + \\mathop{\\rm O}\\nolimits(\\mu^2).\n\\]\nThen it is easy to check that the function\n\\[\n\\tilde{G}(x_1,y) =\n\\mu^{1\/2} \\left( y^2\/2 + \\int_{-1\/\\pi}^{x_1} c(s) {\\rm d} s \\right) =\n\\mu^{1\/2} \\tilde{H}_1(x_1,y) + \\mathop{\\rm O}\\nolimits(\\mu^{5\/2}),\n\\]\nsatisfies the implicit equations~(\\ref{eq:GeneratingFunctionEquations}),\nwhere $\\tilde{H}_1(x,y)$ is the limit\nHamiltonian~(\\ref{eq:H1_SaddleCenterBifurcation}).\nLet $\\sum_{j\\ge 1} \\hat{H}_j$ be the formal series associated to this\ngenerating function.\nWe note that $\\tilde{G} = \\mathop{\\rm O}\\nolimits(\\mu^{1\/2})$,\nand so $\\hat{H}_j = \\mathop{\\rm O}\\nolimits(\\mu^{j\/2})$ for all $j \\ge 1$.\nTherefore,\nif we retain just the first $n$ terms of the formal series,\nwe get an approximating Hamiltonian\n$\\tilde{H}^{[n]}(x,y;\\mu) = \\sum_{j = 1}^n \\hat{H}_j(x,y;\\mu)$ such that\n\\[\n\\tilde{f} = \\phi^1_{\\tilde{H}^{[n]}} + \\mathop{\\rm O}\\nolimits(\\mu^{(n+1)\/2}).\n\\]\nIt is interesting to compare this formula with~(\\ref{eq:f_H1}).\nOn the other hand, using the stronger estimate\n$\\tilde{G} = \\mu^{1\/2} \\tilde{H}_1 + \\mathop{\\rm O}\\nolimits(\\mu^{5\/2})$,\nwe get that $\\hat{H}_j(x,y;\\mu) = \\mu^{j\/2} \\tilde{H}_j(x,y)$\nfor some function $\\tilde{H}_j(x,y)$ that does not depend on $\\mu$\nfor all $j \\le 4$.\nIndeed,\n\\begin{eqnarray*}\n\\tilde{H}_2(x,y) & = &\n(\\tilde{H}_1)_x (\\tilde{H}_1)_y\/2 = (x + \\pi x^2)y\/2,\\\\\n\\tilde{H}_3(x,y) & = &\n\\left(\n(\\tilde{H}_1)_{xx} (\\tilde{H}_1)_y^2 + (\\tilde{H}_1)_{yy} (\\tilde{H}_1)_x^2\n\\right)\/12 \\\\\n& = & (x + \\pi x^2)^2\/12 + (1 + 2\\pi x)y^2\/12, \\\\\n\\tilde{H}_4(x,y) & = &\n(\\tilde{H}_1)_{xx} (\\tilde{H}_1)_{yy} (\\tilde{H}_1)_x (\\tilde{H}_1)_y \/12 \\\\\n& = & (1 + 2\\pi x)(x+\\pi x^2)y\/12,\n\\end{eqnarray*}\nsince all mixed partial derivatives of $\\tilde{H}_1$ vanish.\n\n\\begin{figure}\n\\iffigures\n\\centering\n\\includegraphics[height=2.4in]{AreaSRCC_close_to_0.pdf}\n\\else\n\\vspace{2.4in}\n\\fi\n\\caption{$|\\mathcal{A}_\\mu|$ (continuous line),\n$|\\mathcal{D}_\\mu|$ (dashed line), and the asymptotic\nestimate $6\\mu^{5\/2}\/5\\pi^2$ (dotted line) versus $\\mu$.}\n\\label{fig:AreaStabilityRegion_close_to_0}\n\\end{figure}\n\n\\begin{figure}\n\\iffigures\n\\centering\n\\includegraphics[height=2.4in]{H4_SaddleCenterBifurcation.pdf}\n\\else\n\\vspace{2.4in}\n\\fi\n\\caption{Some level curves of the fourth order approximating Hamiltonian\nassociated to the saddle-center bifurcation (thin lines),\nthe LRIC (thick line), and some RICs of the microtron map\n(small circles) for $\\mu = 1$.}\n\\label{fig:LevelCurvesHamiltonian_mu_1}\n\\end{figure}\n\nWe have plotted some level curves of the approximating Hamiltonian\n\\[\nH^{[4]}(\\psi,w;\\mu) =\n\\tilde{H}^{[4]}(\\psi\/\\mu,w\/\\mu^{3\/2};\\mu) =\n\\sum_{j=1}^4 \\mu^{j\/2} \\tilde{H}_j(\\psi\/\\mu,w\/\\mu^{3\/2})\n\\]\nfor $\\mu = 1$ in Fig.~\\ref{fig:LevelCurvesHamiltonian_mu_1}.\nWe have used the original coordinates $(\\psi,w)$ for the sake of comparison,\ninstead of the scaled ones~(\\ref{eq:Scaling_SaddleCenterBifurcation}).\nWe have also plotted some RICs of the microtron map~(\\ref{eq:f}).\nWe stress that, in spite of the relatively big value of $\\mu$,\nthe RICs fit surprisingly well with the level curves,\nbeing this fitting better close to the elliptic fixed point.\nThe fitting improves when $\\mu$ is smaller.\nThe LRIC is represented in a thick line.\nThe ``last'' level curve correspond to the value\n$H^{[4]}(\\psi,w) \\simeq -0.00465$.\n\nFinally, let us discuss briefly how to refine the asymptotic estimate\n$|\\mathcal{D}_\\mu| \\asymp 6 \\mu^{5\/2}\/5\\pi^2$,\nalthough we do not carry out the associated computations.\nThey are too cumbersome.\n\nIf $0 < \\mu \\ll 1$, the Hamiltonian $\\tilde{H}^{[n]}(x,y;\\mu)$ has,\nbesides the elliptic point $(0,0)$,\na hyperbolic saddle point close to $\\tilde{q}_{\\rm h}=(-1\/\\pi,0)$ whose\nunstable and stable invariant curves coincide along one branch.\nLet $\\tilde{\\mathcal{R}}^{[n]}_\\mu$ be the domain enclosed by\nthe corresponding separatrix.\nThen $|\\tilde{\\mathcal{R}}^{[n]}_\\mu| =\n\\sum_{j = 0}^{n-1} \\alpha_j \\mu^{j\/2} + \\mathop{\\rm O}\\nolimits(\\mu^{n\/2})$ for some\ncoefficients $\\alpha_0,\\ldots,\\alpha_{n-1}$.\nWe have already seen that $\\alpha_0 = 6\/5\\pi^2$.\nIn this way, we may get the refined asymptotic estimates\n\\begin{eqnarray*}\n|\\mathcal{D}_\\mu| & = & \\mu^{5\/2} |\\tilde{\\mathcal{D}}_\\mu | \\asymp\n\\mu^{5\/2} \\left(|\\tilde{\\mathcal{R}}^{[n]}_\\mu| + \\mathop{\\rm O}\\nolimits(\\mu^{n\/2}) \\right) \\\\\n& \\asymp &\n6\\mu^{5\/2}\/5\\pi^2 + \\cdots + \\alpha_{n-1} \\mu^{n\/2 + 2} + \\mathop{\\rm O}\\nolimits(\\mu^{(n+5)\/2}). \n\\end{eqnarray*}\nAnyway, we stress that $|\\mathcal{D}_\\mu|$ is not smooth in the parameter $\\mu$.\nRather the opposite happens;\n$|\\mathcal{D}_\\mu|$ has a fractal self-similar structure with infinitely\nmany jumps, although they become very small when $\\mu \\to 0^+$.\nSee~\\cite{SimoTreschev1998,SimoVieiro2009}, Fig.~\\ref{fig:AreaStabilityRegion_close_to_0},\nand Fig.~\\ref{fig:AreaStabilityRegion}.\n\n\n\\subsection{At the fourth order resonance}\n\\label{Ssec:HamiltonianFourthOrder}\n\n\nLet $\\tilde{f}$ be the transformed map in the\ncoordinates~(\\ref{eq:CoordinatesFourthOrder}),\nwhich were the suitable ones for $\\mu = 2$.\n\nThe map $\\tilde{f}^4$ is very close to the identity\nin a neighborhood of the origin when $\\mu = 2$.\nTo be precise, $\\tilde{f}^4 = {\\rm I} + O_3(x,y)$.\nHence, given any order $n \\ge 4$,\nthere exists a unique approximating Hamiltonian of the form\n\\[\n\\tilde{H}^{[n]}(x,y) = \\tilde{H}_4(x,y) + \\cdots + \\tilde{H}_n(x,y),\n\\]\nbeing $\\tilde{H}_j(x,y)$ a homogeneous polynomial of degree $j$, such that\n\\[\n\\tilde{f}^4 = \\phi^1_{\\tilde{H}^{[n]}} + O_n(x,y).\n\\]\nA rather tedious computation, which we have the good taste to omit,\nleads to the following formulas:\n\\begin{eqnarray*}\n\\tilde{H}_4(x,y) & = & -(x^4+y^4)\/6 - \\pi^2 x^2 y^2, \\\\\n\\tilde{H}_5(x,y) & = & -\\pi^3 (x^4 y + x y^4) - \\pi(x^3 y^2 + x^2 y^3)\/3,\\\\\n\\tilde{H}_6(x,y) & = & \\alpha (x^6 + y^6) + \\beta (x^4 y^2 + x^2 y^4) + \\gamma x^3 y^3,\n\\end{eqnarray*}\nwith $\\alpha = 1\/180 - \\pi^4\/3$, $\\beta = -5\\pi^2\/12$,\n$\\gamma = 1\/9 - 2\\pi^4$.\n\nAll approximating Hamiltonians are symmetric:\n\\[\n\\tilde{H}^{[n]} (x,y) = \\tilde{H}^{[n]} (y,x), \\qquad \\forall n \\ge 4.\n\\]\nThis is a consequence of the fact that the reversor $r_0$ that we will\ngive in~(\\ref{eq:Reversors}) becomes $\\tilde{r}_0(x,y) = (y,x)$\nin the coordinates~(\\ref{eq:CoordinatesFourthOrder}).\n\n\\begin{figure}\n\\iffigures\n\\centering\n\\includegraphics[height=2.4in]{H6_FourthOrderResonance.pdf}\n\\else\n\\vspace{2.4in}\n\\fi\n\\caption{Some level curves of the polynomial Hamiltonian\nof degree six associated to the fourth order resonance (thin lines),\nthe LRIC (thick line), and some RICs of the microtron map (small circles)\nfor $\\mu = 2$.}\n\\label{fig:LevelCurvesHamiltonian_mu_2}\n\\end{figure}\n\nWe have plotted some level curves of the Hamiltonian\n\\[\nH^{[6]}(\\psi,w) = \\tilde{H}^{[6]}(\\psi+w,\\psi)\n\\]\nin Fig.~\\ref{fig:LevelCurvesHamiltonian_mu_2}.\nWe have used the original coordinates $(\\psi,w)$.\nThe RICs of the microtron map fit quite well with the level curves.\nOf course, the fitting improves close to the origin. \nThe LRIC is represented in a thick line.\nThe ``last'' level curve correspond to the value $H^{[6]}(\\psi,w) \\simeq -0.0022$.\n\n\n\\subsection{Near the third order resonance}\n\\label{Ssec:HamiltonianThirdOrder}\n\n\nLet us study the size and shape of the connected component\n$\\mathcal{D}_\\mu$ of the acceptance when $\\mu = 3 + \\epsilon$\nwith $0 < |\\epsilon| \\ll 1$.\nWe will see that $\\mathcal{D}_{3+\\epsilon}$ is approximately\na triangle of vertices~(\\ref{eq:TriangleVertices}),\nwhich implies that the asymptotic\nestimate~(\\ref{eq:AsymptoticThirdOrderResonace}) holds.\nWe scale the variables according to these claims,\nwhich suggest that all the interesting dynamics takes place in a\n$O(\\epsilon)$-neighborhood of the origin.\nTo be precise, we consider the change of scale\n\\begin{equation}\\label{eq:Scaling_ThirdOrder}\nx = \\pi \\psi\/\\epsilon,\\qquad y = \\pi w\/\\epsilon.\n\\end{equation}\nIf $\\mu = 3+\\epsilon$ with $|\\epsilon| \\ll 1$,\nthen the map~(\\ref{eq:f}) is transformed under this change\ninto a map $(x_1,y_1) = \\tilde{f}(x,y)$ of the form\n\\[\n\\left\\{\n\\begin{array}{ccl}\nx_1 & = & x + y, \\\\\ny_1 & = & -3x - 2y - x_1(x_1+1)\\epsilon + \\mathop{\\rm O}\\nolimits(\\epsilon^2).\n\\end{array}\n\\right.\n\\]\nThe map $\\tilde{f}^3$ is close to the identity:\n$\\tilde{f}^3 = {\\rm I} + \\epsilon \\tilde{f}_1 + \\mathop{\\rm O}\\nolimits(\\epsilon^2)$,\nwith\n\\[\n\\tilde{f}_1(x,y) = ( 3x + 2y - 3x^2 - 2xy, -6x - 3y + 6x^2 + 6xy + y^2 ).\n\\]\nFollowing Section~\\ref{Ssec:HamiltonianSaddleCenterBifurcation},\nwe realize that the first order term $\\epsilon \\tilde{f}_1$ is\na Hamiltonian vector field with Hamiltonian $\\epsilon \\tilde{H}_1$,\nwhere\n\\begin{eqnarray}\n\\nonumber\n\\tilde{H}_1(x,y) & = & 3x^2 + 3xy + y^2 - 2 x^3 - 3 x^2 y - xy^2 - 1 \\\\\n\\label{eq:H1_ThirdOrderResonance}\n& = & (1-x)(x+y-1)(2x+y+1).\n\\end{eqnarray}\nTherefore, $\\tilde{f}^3 = \\phi^1_{\\epsilon \\tilde{H}_1} + \\mathop{\\rm O}\\nolimits(\\epsilon^2)$,\nand the integrable dynamics of the Hamiltonian $\\tilde{H}_1$ approximates\nthe dynamics of the map $\\tilde{f}^3$ when $\\mu \\simeq 3$.\nThus, $\\tilde{H}_1$ is the \\emph{limit Hamiltonian} associated\nto the third order resonance.\nIt has four equilibrium points.\nOne elliptic point: $\\tilde{q}_{\\rm s} = (0,0)$, and three saddle points:\n\\[\n\\tilde{q}_1 = (1,0), \\qquad\n\\tilde{q}_2 = (1,-3), \\qquad\n\\tilde{q}_3 = (-2,3).\n\\]\nLet $\\tilde{\\mathcal{R}}$ be the triangle whose vertices are\nthese three points.\nClearly, $|\\tilde{\\mathcal{R}}| = 9\/2$.\nEach side of $\\tilde{\\mathcal{R}}$ is both the stable invariant curve\nof a saddle, and the unstable invariant curve of another saddle.\nThat is, $\\partial \\tilde{\\mathcal{R}}$ is formed by the three saddles\nand the three straight separatrices connecting them.\nSee Fig.~\\ref{fig:H1_ThirdOrderResonance}.\n\n\\begin{figure}\n\\iffigures\n\\centering\n\\psfrag{qs}{\\footnotesize $\\tilde{q}_{\\rm s}$}\n\\psfrag{q1}{\\footnotesize $\\tilde{q}_1$}\n\\psfrag{q2}{\\footnotesize $\\tilde{q}_2$}\n\\psfrag{q3}{\\footnotesize $\\tilde{q}_3$}\n\\includegraphics[height=2.4in]{H1_ThirdOrderResonance.pdf}\n\\else\n\\vspace{2.4in}\n\\fi\n\\caption{The three separatrices (thick lines), some level curves (thin lines),\nand the four equilibrium points (small circles) of the the limit\nHamiltonian~(\\ref{eq:H1_ThirdOrderResonance})\nassociated to the third order resonance.}\n\\label{fig:H1_ThirdOrderResonance}\n\\end{figure}\n\nOnly the points inside the closed domain $\\tilde{\\mathcal{R}}$\ngive rise to bounded trajectories,\nso $\\tilde{\\mathcal{R}}$ is a good approximation of the scaled version\nof $\\mathcal{D}_\\mu$ for $\\mu \\simeq 3$.\nThus, we get the quadratic asymptotic estimate~(\\ref{eq:AsymptoticThirdOrderResonace}).\nWe have displayed a comparison between the numerically computed area\n$|\\mathcal{D}_{3+\\epsilon}|$ and its asymptotic estimate $9\\epsilon^2\/2\\pi^2$\nin Fig.~\\ref{fig:AreaStabilityRegion_close_to_3}.\nWe note that $|\\mathcal{A}_3| > 0$,\nbecause of the traces of the $(1,3)$-periodic chain of elliptic islands\nthat was thrown away from the main connected component $\\mathcal{D}_\\mu$\nat some value $\\mu_\\star \\approx 2.85$.\nWe will visualize these traces in Fig.~\\ref{fig:StabilitySection}.\n\n\\begin{figure}\n\\iffigures\n\\centering\n\\includegraphics[height=2.4in]{AreaSRCC_close_to_3.pdf}\n\\else\n\\vspace{2.4in}\n\\fi\n\\caption{$|\\mathcal{A}_\\mu|$ (continuous line),\n$|\\mathcal{D}_\\mu|$ (dashed line), and the asymptotic\nestimate $9(\\mu-3)^2\/2\\pi^2$ (dotted line) versus\nthe parameter $\\mu$.}\n\\label{fig:AreaStabilityRegion_close_to_3}\n\\end{figure}\n\n\n\n\n\\section{Global stability of the synchronous trajectory}\n\\label{Sec:GlobalStability}\n\n\n\nThis section has a global and experimental character.\nWe will numerically study the stability domain $\\mathcal{A}$\nand its connected component $\\mathcal{D}$ in the range $0 < \\mu < 4.6$.\nWe will also describe some ideas behind the algorithms.\n\n\n\\subsection{The reversors}\n\\label{Ssec:Reversors}\n\n\nA map is \\emph{reversible} when each orbit is related to its time\nreverse orbit by a symmetry transformation, called a \\emph{reversor}.\nIf a map is reversible, many of their periodic and homoclinic points\nare located on certain \\emph{symmetry lines}, and many of their\ninvariant objects are invariant under the reversors~\\cite{Devaney1976}.\nTwo paradigmatic examples of such invariant objects are RICs around\nelliptic points and stable and unstable invariant curves of hyperbolic points.\nWe will use these facts to simplify some computations.\n\nThe map~(\\ref{eq:f}) can be written as the composition $f = r_1 \\circ r_0$,\nwhere $r_0,r_1 : \\mathbb{T}\\times\\mathbb{T}  \\to \\mathbb{T}\\times\\mathbb{R}$ are the involutions\n\\begin{equation}\\label{eq:Reversors}\nr_0(\\psi,w) = (\\psi + w,-w),\\qquad\nr_1(\\psi,w) = (\\psi,\\eta(\\psi)-w),\n\\end{equation}\nand $\\eta(\\psi) = 2\\pi (\\cos \\psi - 1) - \\mu \\sin \\psi$.\nThis means that the map $f$ is reversible with reversors $r_0$ and $r_1$.\nSee~\\cite{LambRoberts1998}.\nThe symmetry lines of these reversors are their sets of fixed points;\nthat is,\n\\begin{eqnarray}\\label{eq:FixedSet}\n\\mathop{\\rm Fix}\\nolimits(r_0) & = & \\left\\{ (\\psi,w) \\in \\mathbb{T} \\times \\mathbb{R} : w = 0 \\right\\}, \\\\\n\\nonumber\n\\mathop{\\rm Fix}\\nolimits(r_1) & = &\n\\left\\{ (\\psi,w) \\in \\mathbb{T} \\times \\mathbb{R} : w = \\eta(\\psi)\/2 \\right\\}.\n\\end{eqnarray}\nAll the RICs displayed in Fig.~\\ref{fig:SummaryOfDynamics} are invariant under\nthe reversors $r_0$ and $r_1$, and so is the stability domain $\\mathcal{A}$.\nSee Fig.~\\ref{fig:StabilityRegionColor_FourthOrderResonance}.\n\nIn particular, if we consider the decomposition\n\\[\n\\mathcal{A} = \\mathcal{A}^- \\cup \\mathcal{A}^+,\\qquad\n\\mathcal{A}^\\pm =\n\\mathcal{A} \\cap \\{ (\\psi,w)\\in \\mathbb{T} \\times \\mathbb{R} : \\pm w \\ge 0 \\},\n\\]\nthen $\\mathcal{A}^\\pm = r_0(\\mathcal{A}^\\mp)$.\nThis halves the computational effort to find $\\mathcal{A}$.\nThe connected component $\\mathcal{D}$ satisfies the same property.\n\nThe symmetry lines of any reversible map $f = r_1 \\circ r_0$ have\nmany more useful properties.\nLet us recall the characterization of \\emph{symmetric periodic orbits}\n(SPOs) given in~\\cite{LambRoberts1998}.\nSPOs are the periodic orbits that are invariant under both reversors $r_0$ and $r_1$.\nAn orbit of $f$ is an SPO if and only if it has exactly two points on\n$\\mathop{\\rm Fix}\\nolimits(r_0) \\cup \\mathop{\\rm Fix}\\nolimits(r_1)$, in which case it has a point\non each symmetry line if and only if it has odd period.\nWe have displayed in Fig.~\\ref{fig:StabilityRegionColor_FourthOrderResonance}\na couple of $(1,4)$-SPOs.\nFour is even, so one of these orbits has two points on $\\mathop{\\rm Fix}\\nolimits(r_0)$,\nand the other orbit has two points on $\\mathop{\\rm Fix}\\nolimits(r_1)$.\n\n\n\\subsection{The rotation number}\n\\label{Ssec:RotationNumber}\n\n\nThe points inside the stability domain $\\mathcal{A}$\nrotate around the elliptic fixed point $p_{\\rm s} = (0,0)$ when $0 < \\mu < 4$.\nThe points infinitesimally close to $p_{\\rm s}$ give $\\theta\/2\\pi$ turns\nper iteration, where $\\theta$ is the angle defined in~(\\ref{eq:RotationNumber}).\nWe recall that $\\theta\/2\\pi$ is the \\emph{rotation number} of the elliptic\npoint $p_{\\rm s}$, and we write $\\rho(p_{\\rm s}) = \\theta\/2\\pi$.\nNext, we try to define the rotation number $\\rho(p)$ for any point\n$p \\in \\mathcal{A} \\setminus\\{ p_{\\rm s} \\}$\nfollowing a standard approach~\\cite{KatokHasselblatt1995}.\n\nGiven any $p = (\\psi,w) \\in \\mathcal{A}$, let $\\varphi$ be its ``argument''.\nThat is, $\\varphi$ is the angle between the segment $[p_{\\rm s},p]$ and the\nsemi-straight line $\\{(\\psi,0) : \\psi> 0 \\}$.\nAnalogously, let $\\varphi_n \\in \\mathbb{R}$ be the ``argument'' of the\n$n$-th iterate $p_n = f^n(p)$.\nWe consider these arguments on the universal cover $\\mathbb{R}$,\nnot on $\\mathbb{T} = \\mathbb{R}\/2\\pi\\mathbb{Z}$.\nThen we wonder whether the limit\n\\begin{equation}\\label{eq:LimitRotationNumber}\n\\rho = \\rho(p) := \\frac{1}{2\\pi} \\lim_{n \\to +\\infty} \\frac{\\varphi_n - \\varphi}{n}\n\\end{equation}\nexists.\nIf so, we say that~(\\ref{eq:LimitRotationNumber}) is the \\emph{rotation number}\nof the point $p$ under the map $f$ around the elliptic point $p_{\\rm s}$.\nWe may be tempted to use the crude numerical approximation\n\\[\n\\rho \\approx \\frac{\\varphi_N - \\varphi_0}{2\\pi N}\n\\]\nfor big enough values of $N$,\nbut it has an $O(1\/N)$-error in the most common situations.\nFortunately, it can be refined using the following algorithm.\nSee~\\cite{SearaVillanueva2006,LuqueVillanueva2013} for details.\n\nGiven two integers $0 < P < Q$, we set $N = 2^Q$\nand compute\n\\[\nS^1_n = \\sum_{j=1}^n (\\varphi_j - \\varphi_0),\\quad\nS^p_n = \\sum_{j=1}^n S^{p-1}_j,\\quad\n\\tilde{S}^p_q = {2^q +p \\choose p+1}^{-1} S^p_{2^q},\n\\]\nfor $1 \\le n \\le N$, $1 \\le p \\le P$, and $1 \\le q \\le Q$.\nThen, under some mild hypotheses, the refined approximation\n\\[\n\\rho \\approx \\Theta(P,Q) := \\sum_{p=0}^P\n(-1)^{p-1} \\frac{2^{p(p+1)\/2}}{\\delta_p \\delta_{P-p}} \\tilde{S}^P_{Q-P+p},\\quad\n\\delta_p = \\prod_{j=1}^p (2^j -1),\n\\]\nhas an $\\mathop{\\rm O}\\nolimits(1\/N^{P+1})$-error.\nFurthermore, we have the following empirical\nbound of the error in the analytic setting:\n\\begin{equation}\\label{eq:ErrorRotationNumber}\n| \\rho - \\Theta(P,Q)| \\lessapprox\n2^{-(P+1)} |\\Theta(P,Q) - \\Theta(P,Q-1)|.\n\\end{equation}\n\nWe are interested in rotation numbers because they allow us to\ndistinguish the three main bounded dynamical behaviors in analytic\narea-preserving\ndiffeomorphisms~\\cite{KatokHasselblatt1995,SimoTreschev1998,SearaVillanueva2006}:\n\\begin{itemize}\n\\item\nIf $\\mathcal{C}$ is a Moser-like RIC,\nthen the limit~(\\ref{eq:LimitRotationNumber}) exists for all $p \\in \\mathcal{C}$,\nand it does not depend on $p$, so we write $\\rho(\\mathcal{C})$.\nBesides, $\\rho(\\mathcal{C})$ is, generically, a \\emph{Diophantine number}\n(that is, it is badly approximated by rational numbers),\nand the previous refined algorithm works quite well.\n\\item\nA $(m,n)$-\\emph{periodic chain of elliptic islands} is an invariant\nregion with several connected components such that each\nof them surrounds a $(m,n)$-periodic elliptic point.\nThe $(1,4)$-periodic chain around around $p_{\\rm s}$ is displayed\nin Fig.~\\ref{fig:StabilityRegionColor_FourthOrderResonance}\nfor two values of $\\mu$.\nIt is formed by the four big green domains.\nEach of them is mapped onto the next one in clockwise sense,\nso $\\rho(p) \\equiv 1\/4$ for any point $p$ inside them.\nSimilarly,\nif $p$ is inside an $(m,n)$-periodic chain of elliptic islands,\nthe limit~(\\ref{eq:LimitRotationNumber}) exists, $\\rho(p) = m\/n \\in \\mathbb{Q}$,\nand the refined algorithm also works quite well.\n\\item\nA \\emph{chaotic sea} (or \\emph{Birkhoff instability zone})\nis the region between two adjacent RICs minus the\nstable elliptic islands.\nIf $p$ is inside a chaotic sea,\nthen the limit~(\\ref{eq:LimitRotationNumber}) generically does not exist,\nand the empirical bound~(\\ref{eq:ErrorRotationNumber}) does not decrease\nwhen $P$ and $Q$ increase.\n\\end{itemize}\n\nFrom now on, we paint the points inside chaotic seas in blue,\nthe ones inside elliptic islands in green,\nand the ones on RICs in red.\nFig.~\\ref{fig:StabilityRegionColor_FourthOrderResonance} and\nFig.~\\ref{fig:StabilityRegionColor_LowOrderResonances} are samples\nof that convention.\nTo be precise, given any point $p \\in \\mathcal{A}$,\nwe apply the refined algorithm with $P = 7$ and $Q = 15$ to compute\nits rotation number.\nIf the empirical bound~(\\ref{eq:ErrorRotationNumber}) is bigger than\nthe tolerance $\\delta = 10^{-10}$, then we paint $p$ in blue.\nOtherwise, we paint $p$ in red\/green when $\\rho(p)$\nis irrational\/rational,\nwhich is decided by looking at its continued fraction;\nsee~\\cite{Khinchin1964} for details.\n\nFinally, we can understand the meaning of the small gaps that appear\nin the graph of the function $\\psi \\mapsto \\rho(\\psi,0)$ displayed in\nFig.~\\ref{fig:RotationNumber_mu_root}.\nThey simply correspond to the sections of some chaotic seas with\nthe symmetry line $\\mathop{\\rm Fix}\\nolimits(r_0) = \\{ w = 0 \\}$.\nThe rotation number is not well defined on those sections.\n\n\\begin{figure*}\n\\iffigures\n\\centering\n\\includegraphics[height=4.4in]{StabilityRegionColor_mu_2p037.pdf}\n\\includegraphics[height=4.4in]{StabilityRegionColor_mu_2p038.pdf}\n\\else\n\\vspace{4.5in}\n\\fi\n\\caption{The stability domain for $\\mu = 2.037$ (left)\nand $\\mu = 2.038$ (right). Blue corresponds to chaotic seas,\ngreen to periodic elliptic islands, and red to RICs.\nThe elliptic and hyperbolic $(1,4)$-SPO orbits are marked\nwith solid black circles and solid black squares, respectively. \nThe symmetry lines $\\mathop{\\rm Fix}\\nolimits(r_0)$ and $\\mathop{\\rm Fix}\\nolimits(r_1)$\nare displayed as dashed black lines.\nEach SPO of even period has exactly two points\non a single symmetry line.\nA short part of the stable and unstable invariant curves of the\nhyperbolic $(1,4)$-SPO is drawn with continuous black lines.\nThese stable and unstable invariant curves split\n(that is, they do not coincide),\nbut only the outer splitting can be seen at this scale.\nSee Section~\\ref{Sec:InvariantCurves} for details.\nIn the electronic version one can magnify the plots.}\n\\label{fig:StabilityRegionColor_FourthOrderResonance}\n\\end{figure*}\n\n\\begin{figure*}\n\\iffigures\n\\centering\n\\includegraphics[height=3.25in]{StabilityRegionColor_mu_1p539.pdf}\n\\hspace{-0.15in}\n\\includegraphics[height=3.25in]{StabilityRegionColor_mu_2p853.pdf}\n\\hspace{-0.15in}\n\\includegraphics[height=3.25in]{StabilityRegionColor_mu_3p735.pdf}\n\\else\n\\vspace{3.25in}\n\\fi\n\\caption{The stability domain for $\\mu = 1.539$ (left),\n$\\mu = 2.853$ (center), and $\\mu = 3.735$ (right).\nColors, solid black circles, solid black squares, dashed black lines,\nand continuous black lines have the same meaning as\nin Figure~\\ref{fig:StabilityRegionColor_FourthOrderResonance},\nbut here the SPOs are $(1,5)$-periodic (left),\n$(1,3)$-periodic (center), and $(2,5)$-periodic (right).\nEach SPO of odd period has exactly one point on each symmetry line.}\n\\label{fig:StabilityRegionColor_LowOrderResonances}\n\\end{figure*}\n\n\n\\subsection{The stability domain}\n\\label{Ssec:StabilityDomain}\n\n\nWe compute the stability domain of the microtron map~(\\ref{eq:f})\nusing the \\emph{orbit method} described in~\\cite{Vieiro2009}.\nVisual examples of several stability domains of the H\\'enon map provided by\nthis method can be found in~\\cite{SimoVieiro2009,SimoVieiro2011,SimoVieiro2012}.\nOur figures show a strong resemblance with those ones.\n\n\\begin{figure*}\n\\iffigures\n\\centering\n\\includegraphics[height=2.5in]{AreaSRCC.pdf}\n\\includegraphics[height=2.5in]{AreaSRCC_zoom.pdf}\n\\else\n\\vspace{2.5in}\n\\fi\n\\caption{Left: $|\\mathcal{A}_\\mu|$ (continuous line) and\n$|\\mathcal{D}_\\mu|$ (dashed line) versus $\\mu$.\nRight: A zoom of the previous figure to visualize its self-similar structure.}\n\\label{fig:AreaStabilityRegion}\n\\end{figure*}\n\n\\begin{figure*}\n\\iffigures\n\\includegraphics[height=5.6in]{StabilitySectionColor0.pdf}\n\\includegraphics[height=5.6in]{StabilitySectionColor1.pdf}\n\\else\n\\vspace{5.7in}\n\\fi\n\\caption{The fractal sets $\\mathcal{S}_0$ (left) and $\\mathcal{S}_1$ (right)\nin the $(\\psi,\\mu)$-plane. Compare with Fig.~2 in~\\cite{Henon1969}.}\n\\label{fig:StabilitySection}\n\\end{figure*}\n\nLet us describe our implementation of the orbit method.\n\nFirst, we take a fine rectangular grid in a suitable rectangle\n\\[\n[\\psi_{\\min},\\psi_{\\max}] \\times [0,w_{\\max}]\n\\]\nthat contains the upper half $\\mathcal{A}^+$ of the stability domain.\nSecond, we paint in white all grid points such that some of their first\n$1000$ iterates escape from the control region\n\\[\n\\left\\{ (\\psi,w) \\in \\mathbb{T} \\times \\mathbb{R}: |w| \\le 1 \\right\\}.\n\\]\nThis is a fast step, since $1000$ is a relatively small number\nfor any modern computer.\nThird, we consider the $n$-th iterates with $-10^7 \\le n \\le 10^7$\nof all not-yet-white grid points adjacent to some already-white one.\nIf some of these iterates escapes from the control region,\nwe paint in white all grid points ``visited'' by\nits corresponding unbounded orbit or by its\n$r_0$-symmetric orbit.\nWe do not use the reversor $r_1$,\nsince $r_0$ is computationally cheaper and $r_1 = f \\circ r_0$.\nWe repeat this process until no more points escape from the control region.\nThis is the hardest step.\nFourth, we determine the color of each non-white grid point by\ncomputing its rotation number as explained before.\nThis step of the algorithm is not ``orbitally coherent''\nin the sense of~\\cite{Vieiro2009}, but we feel that it is the right choice,\nsince there is no clear way to choose the color in disputed cases.\nFinally, we get the lower half of the stability domain from the\nidentity $\\mathcal{A}^- = r_0 (\\mathcal{A}^+)$.\n\nWe stress that all grid points in the interior (respectively, ``on'' the border)\nof the stability domain have been iterated only 1000\n(respectively, $2 \\cdot 10^7$) times.\nThis is good, since the interior contains much more\ngrid points than the border.\n\nFig.~\\ref{fig:StabilityRegionColor_FourthOrderResonance} shows a couple\nof stability domains computed with this algorithm.\nEach pixel is the center of a square with side $\\ell = 1\/4000$,\nso the area of the stability domain is approximately equal to $\\ell^2$\ntimes the number of colored pixels.\nWe also count the number of pixels in the connected component\ncontaining the origin following a standard algorithm in Computer Vision.\nSee, for instance,~\\cite[pags.~72--75]{ShapiroStockman2002}.\nIn that way, we obtain that\n$|\\mathcal{A}_\\mu| \\approx 1.1657\\cdot 10^{-1}$ and\n$|\\mathcal{D}_\\mu| \\approx 1.1029 \\cdot 10^{-1}$ for $\\mu = 2.037$,\nbut $|\\mathcal{A}_\\mu| \\approx 7.6105 \\cdot 10^{-2}$ and\n$|\\mathcal{D}_\\mu| \\approx 1.5067 \\cdot 10^{-2}$ for $\\mu = 2.038$.\n\nThese jumps in $|\\mathcal{A}_\\mu|$ and $|\\mathcal{D}_\\mu|$ have\na simple explanation.\nWe recall that the fourth order resonance takes place at $\\mu = \\mu_\\bullet = 2$.\nAfter that value is crossed,\nfour $(1,4)$-periodic elliptic islands surrounded by a chaotic sea\nemanate from the elliptic point $p_{\\rm s}$.\nThis structure (elliptic islands plus chaotic sea) moves\naway from $p_{\\rm s}$ as $\\mu$ grows,\nbut remains inside $\\mathcal{D}_\\mu \\subset \\mathcal{A}_\\mu$\nwhile some RIC surrounds it.\nHowever, the LRIC surrounding it disappears\nat some value $\\mu = \\mu_\\star \\in (2.037,2.038)$.\nAfter that value is crossed,\nboth the elliptic islands and the chaotic sea are thrown away\nfrom the connected component $\\mathcal{D}_\\mu$,\nalthough any elliptic island is, by definition, part\nof $\\mathcal{A}_\\mu$.\nThus,\nthe jump in $|\\mathcal{A}_\\mu|$ only takes into account the loss of\nthe chaotic sea,\nwhereas the jump in $|\\mathcal{D}_\\mu|$ also takes into account the loss\nof many elliptic islands.\n\nSimilar jumps take place for any periodic elliptic island,\nalthough the greater is the order of the expelled island,\nthe smaller is the jump in both areas.\nFor instance,\nwe see in Fig.~\\ref{fig:StabilityRegionColor_LowOrderResonances} the\nstability domains for $\\mu = 1.539$, $\\mu = 2.853$, and $\\mu = 3.735$,\nwhich are parameter values smaller than, but very close to, the values\nat which the periodic elliptic islands of orders three and five\nare thrown away from the connected component $\\mathcal{D}$.\nHence, the stability domains corresponding to $\\mu = 1.540$, $\\mu = 2.854$,\nand $\\mu = 3.736$ are significantly smaller,\nbecause the more extern chaotic sea is lost. \n\nFig.~\\ref{fig:AreaStabilityRegion} shows $|\\mathcal{A}_\\mu|$\nand $|\\mathcal{D}_\\mu|$ as a function of $\\mu$.\nThe maximal value $|\\mathcal{A}_\\mu| \\approx 1.718 \\cdot 10^{-1}$ is\nattained for $\\mu \\approx 1.912$,\n$|\\mathcal{D}_\\mu| = 0$ for all $\\mu \\gtrsim 4.08$,\nand $|\\mathcal{A}_\\mu| = 0$ for all $\\mu \\gtrsim 4.53$.\nObviously, $|\\mathcal{D}_\\mu| \\le |\\mathcal{A}_\\mu|$,\nsince $\\mathcal{D}_\\mu \\subset \\mathcal{A}_\\mu$.\nWe note that $|\\mathcal{A}_\\mu|$ displays many more jumps than $|\\mathcal{D}_\\mu|$,\nbecause it is affected by \\emph{secondary resonances}\n(resonances inside the islands).\nThe graph of $|\\mathcal{A}_\\mu|$ has a self-similar fractal structure\ncaused by those secondary resonances, as the displayed magnification shows.\n\nWe also list in Table~\\ref{tab:Summary} the exact value\n\\begin{equation}\\label{eq:mu0}\n\\mu_\\bullet := 2 - 2 \\cos(2\\pi m\/n) \\in [0,4]\n\\end{equation}\nat which the elliptic fixed point $p_{\\rm s}$ becomes $(m,n)$-resonant,\nso that the $(m,n)$-periodic chain of elliptic islands is created from $p_{\\rm s}$,\njointly with the numerically approximated value $\\mu = \\mu_\\star$ at which\nthe $(m,n)$-periodic chain of elliptic islands is thrown away from\n$\\mathcal{D}$, for all $(m,n)$-resonances of order $n < 10$.\nFor instance, the $(2,7)$-periodic chain is thrown away at some value\n$\\mu_\\star \\in (2.526,2.527)$, which explains the seven ``holes'' delimited\nby the unbounded orbit displayed in Fig.~\\ref{fig:RootOfTheTwistCoefficient}\nfor $\\mu_{\\rm r} \\approx 2.538$.\n\nWe observe that $\\mu_\\bullet < \\mu_\\star$, but in the $(1,3)$-resonance,\nwhich comes as no surprise, since the twist coefficient is positive\nif and only if $2.538 \\approx \\mu_{\\rm r} < \\mu < 3$,\nsee Subsection~\\ref{Ssec:LocalStabilityElliptic}.\n\n\\begin{table}\n\\centering\n\\begin{tabular}{lll}\n\\hline\n$(m,n)$ & Emanate at & Escape at $\\mu_\\star$ with \\\\\n\\hline\n$(1,9)$ & $\\mu_\\bullet \\approx 0.468$                            & $0.859 < \\mu_\\star < 0.860$ \\\\\n$(1,8)$ & $\\mu_\\bullet = 2 - \\sqrt{2} \\simeq 0.586$              & $0.948 < \\mu_\\star < 0.949$ \\\\\n$(1,7)$ & $\\mu_\\bullet \\approx 0.753$                            & $1.071 < \\mu_\\star < 1.072$ \\\\\n$(1,6)$ & $\\mu_\\bullet = 1$                                      & $1.251 < \\mu_\\star < 1.252$ \\\\\n$(1,5)$ & $\\mu_\\bullet = \\frac{1}{2}(5 - \\sqrt{5}) \\simeq 1.382$ & $1.539 < \\mu_\\star < 1.540$ \\\\\n$(2,9)$ & $\\mu_\\bullet \\approx 1.653$                            & $1.835 < \\mu_\\star < 1.836$ \\\\\n$(1,4)$ & $\\mu_\\bullet = 2$                                      & $2.037 < \\mu_\\star < 2.038$ \\\\\n$(2,7)$ & $\\mu_\\bullet \\approx 2.445$                            & $2.526 < \\mu_\\star < 2.527$ \\\\\n$(1,3)$ & $\\mu_\\bullet = 3$                                      & $2.853 < \\mu_\\star < 2.854$ \\\\\n$(3,8)$ & $\\mu_\\bullet = 2 + \\sqrt{2} \\simeq 3.414$              & $3.589 < \\mu_\\star < 3.590$ \\\\\n$(2,5)$ & $\\mu_\\bullet = \\frac{1}{2}(5 + \\sqrt{5}) \\simeq 3.618$ & $3.735 < \\mu_\\star < 3.736$ \\\\\n$(3,7)$ & $\\mu_\\bullet \\approx 3.802$                            & $3.942 < \\mu_\\star < 3.943$ \\\\\n$(4,9)$ & $\\mu_\\bullet \\approx 3.879$                            & $4.023 < \\mu_\\star < 4.024$ \\\\\n$(1,2)$ & $\\mu_\\bullet = 4$                                      & $4.080 < \\mu_\\star < 4.081$ \\\\\n\\hline\n\\end{tabular}\n\\caption{\\label{tab:Summary}Exact values $\\mu_\\bullet$ at which\nthe main resonances emanate from $p_{\\rm s}$,\nand approximated values $\\mu_\\star$ at which they escape from $\\mathcal{D}$.}\n\\end{table}\n\nWe end with a couple of warnings regarding\nthe accuracy of our pictures.\n\nOn the one hand, we know that the red domains displayed\nin Fig.~\\ref{fig:StabilityRegionColor_FourthOrderResonance},\nand in the left picture of Fig.~\\ref{fig:StabilityRegionColor_LowOrderResonances}\nare not completely filled with RICs.\nIndeed, KAM theory implies that the set of all RICs has a\ncomplicated Cantorian structure, whose gaps are filled with resonances.\nLet us explain why we do not see those resonances.\nWe restrict our explanation to the case of\nFig.~\\ref{fig:StabilityRegionColor_FourthOrderResonance},\nsince the phenomenon is the same in both cases.\nWe do not see any resonance inside the red zone displayed in\nFig.~\\ref{fig:StabilityRegionColor_FourthOrderResonance} because\n\\[\n\\rho_\\star := \\rho(p_{\\rm s}) =\n\\theta\/2\\pi = \\mathop{\\rm acos}\\nolimits(1-\\mu_\\star\/2)\/2\\pi \\approx 0.25302,\n\\]\nwhen $\\mu = \\mu_\\star \\in (2.037,2.038)$, see Table~\\ref{tab:Relations}.\nTherefore, all missed resonances have orders $n \\ge 83$,\nsince $21\/83$ is the rational number in the interval $(1\/4,\\rho_\\star)$\nwith the smallest denominator.\nSuch resonances are too small to be detected with our pixel resolution.\nWe recall that a generic $(m,n)$-resonance in an\n$\\mathop{\\rm O}\\nolimits(\\eta)$-neighborhood of an elliptic fixed point of an analytic\narea preserving map has an $\\mathop{\\rm O}\\nolimits(\\eta^{n\/4})$-size~\\cite{SimoVieiro2009}.\n\nOn the other hand, the border of the connected component $\\mathcal{D}_\\mu$\nshould be a red curve (the LRIC), which is missing in the left picture\nof Fig.~\\ref{fig:StabilityRegionColor_FourthOrderResonance} and in the three\npictures of Fig.~\\ref{fig:StabilityRegionColor_LowOrderResonances}.\nThe reason is, once more, that the LRIC is too thin\nto be detected with our pixel resolution.\nIn the same way, probably there are some RICs in the middle of the blue zone,\nwhich would mean that that blue zone is composed by several chaotic seas,\ninstead of by a single big chaotic sea.\n\n\n\\subsection{The sections with the symmetry lines}\n\\label{Ssec:Sections}\n\n\nWe consider the sections of the stability domain with the symmetry\nlines~(\\ref{eq:FixedSet}) for two reasons.\nFirst, the stability domain is symmetric with respect to these lines.\nSecond,\nwe recall that each SPO has exactly two points on these lines,\nand such SPOs are the basic invariant objects that organize\nthe dynamics inside their resonance.\n\nWe look at Fig.~\\ref{fig:StabilityRegionColor_FourthOrderResonance}\nto understand how SPOs organize the resonant dynamics.\nWe see an elliptic $(1,4)$-SPO with two points on $\\mathop{\\rm Fix}\\nolimits(r_1)$ that\norganizes the stable dynamics inside the four big green elliptic islands,\nand a hyperbolic $(1,4)$-SPO with two points on $\\mathop{\\rm Fix}\\nolimits(r_0)$\nwhose stable and unstable invariant curves delimit almost perfectly\nthe red region ``filled'' with RICs.\n\nWe do not deal with each one-dimensional section as a separate object,\nbut we gather them into the two-dimensional sets:\n\\begin{eqnarray*}\n\\mathcal{S}_0 & = &\n\\left\\{ (\\psi,\\mu) \\in \\mathbb{T} \\times (0,+\\infty) :\n        (\\psi,0) \\in \\mathcal{A}_\\mu \\right\\}, \\\\\n\\mathcal{S}_1 & = &\n\\left\\{ (\\psi,\\mu) \\in \\mathbb{T} \\times (0,+\\infty) :\n        (\\psi,\\eta(\\psi)\/2) \\in \\mathcal{A}_\\mu \\right\\},\n\\end{eqnarray*}\nin order to visualize their evolution in the parameter $\\mu$.\n\nThese sets are represented in Fig.~\\ref{fig:StabilitySection}\nwith the usual color codes.\nWe see that the connected component $\\mathcal{D}_\\mu$\ncollapses to the elliptic fixed point at $\\mu = 3$,\nand undergoes its major loss at some $\\mu = \\mu_\\star \\in(2.037,2.038)$.\nThe collapse is associated to the local instability of the third\norder resonance.\nThe loss takes place when the fourth order resonance is thrown away.\nThese are the most relevant phenomena regarding the size of\nthe stability domain in generic families of area-preserving maps.\n\nWe appreciate a clear self-similar fractal structure in the\nsets $\\mathcal{S}_0$ and $\\mathcal{S}_1$.\nOne can magnify their pictures to appreciate several details.\nLet us describe the main ones.\n\nResonances emanate from the elliptic fixed point $p_{\\rm s}$ in the form\nof thin blue and green tongues, since they are sections of chaotic seas\nsurrounding elliptic islands.\nThese tongues begin at the points $(\\psi,\\mu) = (0,\\mu_\\bullet)$,\nwhere $\\mu_\\bullet$ is defined as in~(\\ref{eq:mu0}) for any rational\nnumber $m\/n \\in (0,1\/2)$, so there are infinitely many of such tongues.\nBesides, they look symmetric with respect to the vertical line\n$\\{\\psi = 0\\}$ in a neighborhood of that line. \nThe resonances, but the $(1,4)$-one, have a small width\nwhen they are close to $p_{\\rm s}$.\nHence, most of the tongues become visible only at some distance\nof $\\{ \\psi = 0 \\}$.\n\nThe elliptic islands inside a resonance grow in size when they move away\nfrom $p_{\\rm s}$.\nIf the resonance has even order,\nthen the green part becomes the biggest part of the tongue in the set $\\mathcal{S}_1$,\nbut not in the set $\\mathcal{S}_0$.\nThis means that each elliptic SPO with even period has two points on\n$\\mathop{\\rm Fix}\\nolimits(r_1)$, but none on $\\mathop{\\rm Fix}\\nolimits(r_0)$.\nFig.~\\ref{fig:StabilityRegionColor_FourthOrderResonance} is a sample\nof that empirical claim.\nOn the contrary, if the resonance has odd order,\nthen the green part becomes the biggest part of just one branch of the tongue,\nin both sets $\\mathcal{S}_0$ and $\\mathcal{S}_1$.\nThe other branch remains blue.\nThis is a completely expected behavior, since we know that all SPOs\nwith odd period have just one point on each symmetry line.\n\nWe warn that red and green regions should not be in direct contact,\nbut we have again difficulties to detect the blue chaotic seas\nbetween them with our current pixel resolution.\n\nResonances are thrown away from $\\mathcal{D}$,\nwhich is the reason for the saw-like border of\n$\\mathcal{S}_0$ and $\\mathcal{S}_1$.\nOnce a green tongue is separated from the main red body in\nthe $\\psi$-direction, it is no longer delimited by blue portions,\nbecause chaotic seas do not form part of $\\mathcal{A}$\nwhen their resonances are thrown away.\nSee Fig.~\\ref{fig:StabilityRegionColor_FourthOrderResonance}.\nBesides, the separated part of any green tongue shows a shape similar\nto the shape of the whole set, which is due to the secondary resonances.\nIndeed, we see the two above-mentioned phenomena\n(major loss in the stability domain and collapse of the\nstability domain to a point) along many of these tongues.\n\nSuch phenomena are clearly visible in the tongues of $\\mathcal{S}_1$\nthat cover the range $4.08 < \\mu < 4.53$.\nThe fixed point $p_{\\rm s}$ is already globally unstable in that range,\nbut there is still a locally stable elliptic two-periodic orbit on\nthe symmetry line $\\mathop{\\rm Fix}\\nolimits(r_1)$.\n\nH\\'enon studied some sets similar to $\\mathcal{S}_0$ and\n$\\mathcal{S}_1$ for the H\\'enon map more than\nforty years ago~\\cite{Henon1969}.\nHis computations already show many of the above-described phenomena,\nin spite of the limitations of the computers in that time.\nSuch limitations were recently overcome\nin~\\cite{Migueletal2013}.\n\n\n\n\\section{On the invariant curves of some hyperbolic points}\n\\label{Sec:InvariantCurves}\n\n\n\nThe stable and unstable invariant curves of hyperbolic\n(fixed or periodic) points organize the dynamics of area\npreserving maps in several ways.\nLet us just mention two of them.\n\nOn the one hand,\nsome of these invariant curves are approximate boundaries of\nstability domains~\\cite{Giovannazzi1996},\nin which case the area of the lobes between them is equal to the flux\nthrough certain closed curves composed by arcs of invariant\ncurves~\\cite{Meiss1992}.\nThese lobes have an exponentially small area in the analytic\ncase~\\cite{FontichSimo1990},\nso that the above-mentioned closed curves become \\emph{partial barriers}\nof the dynamics~\\cite{MacKayMP1984,Meiss1992}.\nIf the map is entire,\nthen the stable and unstable invariant curves never coincide~\\cite{Ushiki1980},\nso these partial barriers are never complete barriers.\n\n\\begin{figure}\n\\iffigures\n\\centering\n\\includegraphics[height=2.4in]{StabilityRegionColor_mu_0p859.pdf}\n\\else\n\\vspace{2.4in}\n\\fi\n\\caption{The stability domain for $\\mu = 0.859$.\nBlue corresponds to chaotic seas,\ngreen to periodic elliptic islands, and red to RICs.\nThe elliptic fixed point $p_{\\rm s} = (0,0)$ and\nthe hyperbolic fixed point $p_{\\rm h} = (-2\\phi_{\\rm s},0)$ are marked\nwith a solid black circle and a solid black square, respectively. \nThe symmetry lines $\\mathop{\\rm Fix}\\nolimits(r_0)$ and $\\mathop{\\rm Fix}\\nolimits(r_1)$\nare displayed as dashed black lines.\nA short part of the stable and unstable invariant curves\nof $p_{\\rm h}$ is shown as continuous black lines.\nThe primary intersections of these invariant curves with the\nsymmetry lines are marked with two solid black triangles.}\n\\label{fig:StabilityRegion_mu_0p859}\n\\end{figure}\n\nOn the other hand,\nif the unstable invariant curve of some\nperiodic orbit intersects the stable invariant curve of\nanother periodic orbit,\nthen there can be no RICs between both periodic orbits.\nThis \\emph{obstruction criterion} was established\nin~\\cite{OlveraSimo1987}.\n\nNext, we discuss these ideas in the setting of map~(\\ref{eq:f}).\n\n\\begin{figure}\n\\iffigures\n\\centering\n\\includegraphics[height=2.4in]{Zoom_mu_0p859.pdf}\n\\else\n\\vspace{2.4in}\n\\fi\n\\caption{A zoom of Figure~\\ref{fig:StabilityRegion_mu_0p859},\nbut without the stability domain.\nThe area of the lobe $\\mathcal{L}$ delimited by the separatrices\nbetween the two primary homoclinic points marked with\nsolid black triangles is\n$|\\mathcal{L}| \\approx 3.808194826948494 \\times 10^{-5}$.}\n\\label{fig:Zoom_mu_0p859}\n\\end{figure}\n\n\\subsection{Singular splitting near the saddle-center bifurcation}\n\\label{Ssec:SeparatricesSaddleCenterBifurcation}\n\nWe saw in Section~\\ref{Ssec:HamiltonianSaddleCenterBifurcation} that\nthe map~(\\ref{eq:f}) is approximated,\nafter the rescaling~(\\ref{eq:Scaling_SaddleCenterBifurcation}),\nby the $\\mu^{1\/2}$-time flow of the Hamiltonian~(\\ref{eq:H1_SaddleCenterBifurcation})\nwhen $0 < \\mu \\ll 1$.\nBesides, the Hamiltonian~(\\ref{eq:H1_SaddleCenterBifurcation})\nhas a separatrix that encloses a region which resembles the\nstability domain of the map when $0 < \\mu \\ll 1$.\nCompare the stability domain displayed in Fig.~\\ref{fig:AfterParabolic0}\nwith the phase portrait of the Hamiltonian~(\\ref{eq:H1_SaddleCenterBifurcation})\nsketched in Fig.~\\ref{fig:H1_SaddleCenterBifurcation}.\nThe separatrix is described by the homoclinic trajectory~(\\ref{eq:HomoclinicOrbit}),\nwhich is analytic in a complex strip of width $d_0 = \\pi$.\n\nNevertheless,\nthe stable and unstable invariant curves of the saddle point\n$p_{\\rm h} = (-2\\phi_{\\rm s},0)$ of our map~(\\ref{eq:f}) do not coincide,\nsince the map is entire.\nThis result goes back to Ushiki~\\cite{Ushiki1980}.\nWe have displayed the stability domain and the separatrices of\nthe saddle point $p_{\\rm h} = (-2\\phi_{\\rm s},0)$ for $\\mu = 0.859$ in\nFig.~\\ref{fig:StabilityRegion_mu_0p859}.\nWe check that the separatrices enclose the stability domain.\nThe reversibility of our map implies that the separatrices\nhave a primary homoclinic point on each symmetry line~(\\ref{eq:FixedSet}).\nLet $\\mathcal{L}$ be the region (such region is called \\emph{lobe})\ndelimited by the pieces of the separatrices between these\ntwo primary homoclinic points.\nFor instance, we display the lobe $\\mathcal{L}$ for $\\mu = 0.859$\n in figure~\\ref{fig:Zoom_mu_0p859}.\nIn that case, the lobe area is\n$|\\mathcal{L}| \\approx 3.808194826948494 \\times 10^{-5}$.\n\nFontich and Sim\\'o~\\cite{FontichSimo1990} proved that\nthe splitting of the separatrices for any close to the identity analytic\narea preserving map is exponentially small in the characteristic exponent\n$h$ of the saddle point.\nTo be precise, they established that the splitting size is smaller\nthan $\\mathop{\\rm O}\\nolimits({\\rm e}^{-2\\pi d\/h})$ for any $0 < d < d_0 = \\pi$.\nHere, $d_0$ is the width of the analyticity strip of the\nhomoclinic solution of the limit Hamiltonian.\nSince in our case $d_0 = \\pi$, we get the upper bound\n$|\\mathcal{L}| \\le \\mathop{\\rm O}\\nolimits({\\rm e}^{-c\/h})$ for any $0 < c < 2\\pi^2$.\nWe recall that $\\mu = 2(\\cosh h -1) = h^2 + \\mathop{\\rm O}\\nolimits(h^4)$, so\n$h \\asymp \\sqrt{\\mu}$ as $\\mu \\to 0^+$.\n\nTen years later,\nGelfreich~\\cite{Gelfreich2000} derived an asymptotic formula\nfor the splitting angle between the separatrices in analytic\nsaddle-center bifurcations, although he did not provide a\ncomplete proof.\nGelfreich's formula, once adapted to our map,\nsays that $|\\mathcal{L}| \\asymp a_0 {\\rm e}^{-2\\pi^2\/h}$\nas $h \\to 0^+$ for some constant $a_0 \\in \\mathbb{R}$.\n\nOur numerical experiments strongly suggest that there exist\nsome asymptotic coefficients $a_n \\in \\mathbb{R}$, $n \\ge 0$, such that\n\\begin{equation}\\label{eq:AsymptoticFormula}\n|\\mathcal{L}| \\asymp {\\rm e}^{-2\\pi^2\/h} \\sum_{n \\ge 0} a_n h^{2n},\n\\qquad (h \\to 0).\n\\end{equation}\nThis fits perfectly with both Fontich-Sim\\'o's upper bound,\nand Gelfreich's asymptotic formula.\nOur refined asymptotic formula~(\\ref{eq:AsymptoticFormula})\nmeans that if we retain only finitely many terms of the right-hand side,\nthen the error will be of the order of the first discarded term.\nSuch refined asymptotic formulas in singular splitting problems\nwere first presented in~\\cite{GelfreichLS1994} for the Standard map,\nand first proved in~\\cite{MartinSS2011} for the perturbed McMillan map.\n\nBesides, we have numerically seen that the first asymptotic\ncoefficient in formula~(\\ref{eq:AsymptoticFormula}) is non-zero:\n\\[\na_0 \\approx 1.42098502709189813726617259727 \\times 10^5,\n\\]\nwhereas the second asymptotic coefficient vanishes: $a_1 = 0$,\nso the approximation $|\\mathcal{L}| \\approx a_0 {\\rm e}^{-2\\pi^2\/h}$\nhas an $\\mathop{\\rm O}\\nolimits(h^4)$ relative error.\nWe have also checked that the asymptotic series\n$\\sum_{n \\ge 0} a_n h^{2n}$ is divergent,\nbut its Borel transform $\\sum_{n\\ge 0} a_n h^{2n}\/(2n)!$\nhas radius of convergence $2\\pi^2$.\nThis is a typical behaviour for many other maps,\nsee~\\cite{DelshamsRamirez1999,Ramirez2005,GelfreichSimo2008,Migueletal2013}.\n\nLet us consider the closed curve formed by the unstable\ninvariant curve from the saddle point $p_{\\rm h}$ to the primary\nhomoclinic point on some fixed symmetry line plus the\nstable invariant curve from that primary point to $p_{\\rm h}$.\nThis closed curve encloses a planar domain $\\mathcal{R}$\nslightly bigger than the stability domain $\\mathcal{A}$,\nsee Fig.~\\ref{fig:StabilityRegion_mu_0p859}.\nThe key observation is that this closed curve is an\n\\emph{effective barrier} when $0 < \\mu \\ll 1$.\nThe term \\emph{effective} means that the flux through this closed curve\nis so small that it looks like a true barrier for a very big number\nof iterates of the map.\nFor instance, if we set $\\mu=0.2$, then $h \\approx 0.44357$,\n\\[\n|\\mathcal{R}| > |\\mathcal{A}| \\approx 2.1455 \\times 10^{-3},\\quad\n|\\mathcal{L}| \\approx a_0 {\\rm e}^{-2\\pi^2\/h} \\approx 6.7000 \\times 10^{-15}.\n\\]\nBesides, we know that the lobe area $|\\mathcal{L}|$\nis an exact measure of the flux through $\\partial \\mathcal{R}$\nafter one iteration of the map, see~\\cite{MacKayMP1984,Meiss1992}.\nThis means that after $10^9$ iterates of the map $f$,\nless than three thousandths parts of the points\ninside $\\mathcal{R}$ have escaped.\nThus, one may approximate the stability domain $\\mathcal{A}$\nby the region $\\mathcal{R}$ in many practical situations.\n\nFinally, we note that the numerical computation of any exponentially\nsmall splitting quantity (angle, area, or distance)\ngets complicated by problems of precision, stability, and time.\nIn order to overcome them,\nSim\\'o proposed to use a multiple-precision arithmetic,\nto expand the invariant curves up to high order, \nand to take advantage of the reversor~\\cite{Simo1990}.\nThese ideas have been used\nin~\\cite{DelshamsRamirez1999,Ramirez2005,GelfreichSimo2008}.\nWe have also used them.\n\n\n\\subsection{Singular splitting near the third-order resonance}\n\\label{Ssec:SeparatricesThirdOrderResonance}\n\n\nWe saw in Section~\\ref{Ssec:HamiltonianThirdOrder} that the third power\nof the map~(\\ref{eq:f}) is approximated,\nafter the rescaling~(\\ref{eq:Scaling_ThirdOrder}), by the\n$\\epsilon$-time flow of the Hamiltonian~(\\ref{eq:H1_ThirdOrderResonance})\nwhen $\\mu = 3 + \\epsilon$ with $0 < |\\epsilon| \\ll 1$.\nBesides, the Hamiltonian~(\\ref{eq:H1_ThirdOrderResonance})\nhas three saddle points whose invariant curves coincide giving rise\nto the triangle sketched in Fig.~\\ref{fig:H1_ThirdOrderResonance}.\n\nIf $\\mu \\simeq 3$, then the stability domain of the map~(\\ref{eq:f}) has a\ncentral part with a triangular shape, that contains many RICs, and three\n``sheets'', that contain points with rotation number equal to $1\/3$,\nattached to the vertices of that ``triangle''.\nThe vertices of this ``triangle'' correspond to hyperbolic\nthree-periodic points whose stable and unstable invariant curves\ndo not coincide.\nThere are two different splitting phenomena in this setting.\nNamely,\nthe inner splitting (associated to the invariant curves that\nenclose the ``triangle'') and\nthe outer splitting (associated to the invariant curves that\nenclose the ``sheets'').\nEach splitting should be studied separately.\nThe inner one is generically much smaller than the outer\none~\\cite{SimoVieiro2009}.\n\nWe have displayed the stability domain for $\\mu = 2.853$\nin the central picture\nof Fig.~\\ref{fig:StabilityRegionColor_LowOrderResonances}.\nThe red part is the ``triangle'', the green parts are the three ``sheets'',\nand the continuous  black lines are the invariant curves of the hyperbolic\nthree-periodic points.\nWe stress that,\nalthough the value of $|\\epsilon| = |\\mu - 3|$ is not very small, \nthe inner splitting can not be detected even after a\nbig magnification of our picture.\nThis suggest that the inner splitting is exponentially small in\n$|\\epsilon| = |\\mu -3|$.\nG.~Moutsinas~\\cite{Moutsinas2016} has studied the inner splitting\nin analytic area-preserving maps close to the third-order resonance.\nHe deduced, under a generic assumption on the third-order Birkhoff\nnormal form around the elliptic fixed point at the exact third-order\nresonance, that the inner splitting is exponentially small\nin the characteristic exponent of the third iterate of the map\nat the hyperbolic three-periodic points.\nTo be more precise, he found that the Lazutkin homoclinic invariant\nassociated to some distinguished heteroclinic orbits has a\nrefined asymptotic formula of the form~(\\ref{eq:AsymptoticFormula}),\nbut now $h$ is the characteristic exponent of the map $f^3$\nat the three-periodic points instead of the characteristic exponent\nof the map $f$ at the origin.\n\nOn the contrary, the outer splitting in the central picture\nof Figure~\\ref{fig:StabilityRegionColor_LowOrderResonances}\ncan be perceived after a suitable magnification of a small\nneighborhood of a hyperbolic three-periodic point.\nThis visual inspection fits with the results given\nin~\\cite[Section~6.1]{SimoVieiro2009},\nwhere it is established that the outer splitting associated to a\ngeneric third-order resonance does not tend to zero as we\napproach the resonance.\nThat is, the outer splitting is $\\mathop{\\rm O}\\nolimits(1)$.\n\nWe can extract two practical consequences of these results.\n\nFirst, set $\\mu \\simeq 3$ and\nlet $\\mathcal{R}^{(1,3)}_{\\rm inner}$ and $\\mathcal{R}^{(1,3)}_{\\rm outer}$\nbe the regions enclosed by suitable parts of the stable and\nunstable invariant curves of the hyperbolic three-periodic points\nsuch that $\\mathcal{R}^{(1,3)}_{\\rm inner}$ contains the triangular shaped\npart of $\\mathcal{A}$ containing many RICs and\n$\\mathcal{R}^{(1,3)}_{\\rm outer}$ contains all the points with rotation number\nequal to $1\/3$.\nThen the flux through the effective barrier\n$\\partial \\mathcal{R}^{(1,3)}_{\\rm inner}$ is much smaller\nthan the flux through $\\partial \\mathcal{R}^{(1,3)}_{\\rm outer}$.\n\nSecond, let $\\mu_\\star \\in [2.853,2.854]$ be the value\nat which the third-order resonance is thrown away from $\\mathcal{D}$.\nThen $\\mathcal{R}^{(1,3)}_{\\rm inner}$ is a really good approximation\nof the connected component $\\mathcal{D}$ when $\\mu \\gtrsim \\mu_\\star$.\n\n\n\\subsection{Singular splitting near high-order resonances}\n\\label{Ssec:SeparatricesHiguerOrderResonance}\n\n\nThe singular splitting near resonances of order $n \\ge 4$\nshares several qualitative and quantitative features\nwith the singular splitting near the saddle-center bifurcation\nand near the third-order resonance.\nLet us explain this.\n\nLet $\\mu_\\bullet$ and $\\mu_\\star$ be the values at which\nthe $(m,n)$-resonance emanates from $p_{\\rm s}$ and is thrown\naway from $\\mathcal{D}$, respectively.\nLet $\\mathcal{R}^{(m,n)}_{\\rm inner}$\n(respectively, $\\mathcal{R}^{(m,n)}_{\\rm outer}$)\nbe the region enclosed by suitable parts of the inner\n(respectively, enclosed between suitable parts of the inner and outer)\nbranches of the stable and unstable invariant curves of\nthe hyperbolic $(m,n)$-periodic points.\nThe inner region usually looks like a red ``polygon'' with $n$\ncurved sides, because it is almost completely foliated by RICs.\nThe outer region contains the $(m,n)$-periodic chain of elliptic islands,\nand it also contains part of its surrounding chaotic sea before\nthe $(m,n)$-resonance is thrown away.\nSee Figures~\\ref{fig:StabilityRegionColor_FourthOrderResonance}\nand~\\ref{fig:StabilityRegionColor_LowOrderResonances} for several\npictures about the resonances\n\\[\n(m,n) = \\{(1,4), (1,5), (1,3), (2,5)\\}.\n\\]\n\nSince the flux through the borders of the inner and outer regions\ncan be geometrically interpreted as the area of certain\nlobes~\\cite{MacKayMP1984,Meiss1992},\nwe obtain the following information about the inner and outer flux.\nThe inner flux is smaller than the outer flux, and both of them\nare exponentially small in $|\\mu-\\mu_\\bullet|$~\\cite{SimoVieiro2009}.\nThe inner region is a good approximation of the connected component\n$\\mathcal{D}$ when $\\mu \\gtrsim \\mu_\\star$.\nSee, for instance, the right picture in\nFigure~\\ref{fig:StabilityRegionColor_FourthOrderResonance}.\nThe inner and outer regions are not completely contained\nin the stability domain when $\\mu > \\mu_\\star$, since there is a small,\nbut not zero, flux through their borders~\\cite{Ushiki1980}.\n\n\n\n\\subsection{On the obstruction criterion for the existence of RICs}\n\\label{Ssec:Obstructions}\n\n\\begin{figure}\n\\iffigures\n\\centering\n\\includegraphics[height=2.4in]{ObstructionCriterion_mu_2p9.pdf}\n\\else\n\\vspace{2.4in}\n\\fi\n\\caption{\nThe stable and unstable invariant curves of the hyperbolic fixed point $p_{\\rm h}$\nand the hyperbolic $(1,3)$-periodic orbit intersect transversally for $\\mu = 2.9$.\nThe hyperbolic\/elliptic fixed point is marked with a black rhombus\/triangle.\nThe hyperbolic\/elliptic 3-periodic points are marked with black squares\/circles.\nThe stable and unstable invariant curves of the hyperbolic fixed point\n(respectively, 3-periodic points) are displayed in red (respectively, in blue).}\n\\label{fig:ObstructionCriterion_mu_2p9}\n\\end{figure}\n\nWe recall the obstruction criterion for the existence of RICs\nstated in~\\cite{OlveraSimo1987}.\nIf an area-preserving twist diffeomorphism on the annulus $\\mathbb{T} \\times \\mathbb{R}$\nhas two hyperbolic periodic orbits of rotation numbers $m_1\/n_1 < m_2\/n_2$\nsuch that their stable and unstable invariant curves intersect transversally,\nthen the map has no RIC with a rotation number $\\rho \\in [m_1\/n_1,m_2\/n_2]$.\n\nLet $f$ be the map~(\\ref{eq:f}).\nThe point $p_{\\rm h} = ( -2\\phi_{\\rm s}, 0)$ is a hyperbolic fixed point\nor, equivalently, a hyperbolic $(0,1)$-periodic point.\nIf the invariant curves of $p_{\\rm h}$ intersect\ntransversally the invariant curves of a hyperbolic $(m,n)$-periodic orbit\nof the map, then the map has no RIC with rotation number $\\rho \\in [0,m\/n]$ and\nthe $(m,n)$-resonance has already escaped from the connected component $\\mathcal{D}$.\nTherefore, we should expect that the exact value $\\mu = \\mu_\\star$\nat which the $(m,n)$-resonance escapes from $\\mathcal{D}$ coincides with the\nbifurcation value at which the invariant curves of the hyperbolic\n$(m,n)$-periodic orbit have their first contact with the invariant curves\nof $p_{\\rm h}$.\n\nLet us present a concrete application of this idea.\nWe have already seen that the $(1,3)$-resonance escapes from $\\mathcal{D}$\nat some value $\\mu = \\mu_\\star \\in (2.853,2.854)$ by means of the\nbrute force method used in subsection~\\ref{Ssec:StabilityDomain}.\nNext, we study this escape with the obstruction criterion.\n\nWe have drawn the stable and unstable invariant curves of\n$p_{\\rm h}$ and the hyperbolic $(1,3)$-periodic orbit for\nseveral values of the parameter $\\mu$ in the interval $(2.8,3)$.\nFor instance, the $(1,3)$-resonance has already escaped\nfrom $\\mathcal{D}$ when $\\mu = 2.9$, since the invariant\ncurves drawn in Figure~\\ref{fig:ObstructionCriterion_mu_2p9}\nintersect transversally.\nWe have obtained similar pictures for $\\mu \\in \\{2.89, 2.88, 2.87\\}$,\nbut the closer we are to $\\mu = \\mu_\\star$,\nthe bigger part of the invariant curves we have to draw in order to\nfind intersections.\nIn particular,\npictures for $\\mu = 2.88$ and $\\mu = 2.87$ are far from pretty.\nWe have not found an intersection for $\\mu = 2.86$,\nbecause the computation and visualization of such a big part\nof the invariant curves is not an easy task.\n\n\n\n\\section{Conclusions}\\label{Sec:Conclusions}\n\n\n\nWe have studied the stability of longitudinal beam motion in RTMs.\nNamely, we have analyzed the stability domain $\\mathcal{A}$\n(and its central connected component $\\mathcal{D}$)\nof the area-preserving map that describes the phase oscillations\nusing standard Dynamical Systems tools. \nWe have found the range of values of the synchronous phase $\\phi_{\\rm s}$\nfor which $\\mathcal{A}$ and $\\mathcal{D}$ exist.\nWe have studied their structure and calculated their area as a function of\n$\\phi_{\\rm s}$.\n\nThe knowledge of $\\mathcal{A}$,\ncalled longitudinal acceptance in the theory of particle accelerators,\nis of much importance for the optimization of the beam motion in RTMs.\nIndeed, the adjustment of machine parameters for the efficient acceleration\nof the beam during its commissioning consists in matching the domain\nin the phase space occupied by the particles emitted by an injector\n(often an electron gun) to the acceptance for a given value of $\\phi_{\\rm s}$.\nThe optimal beam matching allows to minimize beam losses and undesired excess\nof strayed radiation produced by the accelerator and, so, maximize\nthe output beam current without increasing the current at the injection.\nFor the adjustment to be most efficient,\nthe acceptance area must be maximal and the shape of the phase domain\nof the injected beam must fit the acceptance shape.\nTherefore, our detailed analysis of the acceptance geometry could be useful.\n\nLet us comment on two ``empirical'' rules used in particle\naccelerators~\\cite{Rand1984}.\nThe first rule claims that the values of $\\phi_{\\rm s}$ for which an accelerator\ncan operate are contained in the interval of linear stability\nof the synchronous trajectory.\nThe second rule states that the optimal values of $\\phi_{\\rm s}$\nare close to the middle point of such interval.\nIn our study, we have checked that both rules are in fact quite precise\nwithin our RTM model, where the interval of linear stability is $(0,\\phi_{\\rm p})$,\nwith\n\\[\n\\phi_{\\rm p} := \\arctan(2\/\\pi) \\approx 32.5^{\\circ}.\n\\]\nFirst,\nwe have numerically seen that $|\\mathcal{D}| > 0$ for $0 < \\phi_{\\rm s} < 33^{\\circ}$,\nexcept for the value\n\\[\n\\phi_{\\rm u} := \\arctan(3\/2\\pi) \\approx 25.5^{\\circ}\n\\]\nthat corresponds to the third order resonance.\nSecond,\nwe have found that the acceptance area reaches its maximal\nvalue $|\\mathcal{A}| \\approx 0.17$ at $\\mu \\approx 1.912$,\nwhich roughly corresponds to\n\\[\n\\phi_{\\rm s} \\approx \\arctan(1.912\/2\\pi) \\approx 16.9^{\\circ} \\approx \\phi_{\\rm p}\/2.\n\\]\nIn fact, $|\\mathcal{A}|$ is sufficiently large for a rather wide\nrange of the values of $\\mu$.\nFor instance, $|\\mathcal{A}|\\ge 0.1$ if \n\\[\n\\mu \\in [1.027,1.071]\\cup[1.079,2.037] \\cup [2.245,2.827].\n\\]\nThird, we have studied $\\mathcal{A}$ and $\\mathcal{D}$\nin the vicinity of resonant values.\nIn particular, we have checked that $|\\mathcal{D}| = 0$ at\nthe third order resonance $\\phi_{\\rm s} = \\phi_{\\rm u}$,\nwhereas it reduces significantly, till $|\\mathcal{D}| \\approx 0.02$,\nat the fourth order resonance\n\\[\n\\phi_{\\rm s} = \\arctan(1\/\\pi) \\approx 17.7^{\\circ}.\n\\]\nOther resonances do not lead to so sharp decreases of $|\\mathcal{D}|$.\nThis data is quite important because one of the criteria of choosing\nthe design value of $\\phi_{\\rm s}$, or the working point of\nthe machine, is to avoid values close to resonant ones.\nOtherwise even a small natural drift of machine parameters may lead\nto $\\phi_{\\rm s}$ approaching one of the dangerous resonant values\nand consequently to excessive beam losses. In this respect the asymptotic\nformulas~(\\ref{eq:AsymptoticSaddleCenterBifurcation})\nand~(\\ref{eq:AsymptoticThirdOrderResonace})\nare of much interest. Another important aspect of the acceptance structure\nare the elliptic islands and chaotic seas like the ones displayed in\nFigs.~\\ref{fig:StabilityRegionColor_FourthOrderResonance}--\\ref{fig:StabilityRegionColor_LowOrderResonances}. For instance,\neach drastic change in $|\\mathcal{D}|$ is associated\nto the escape of a chain of elliptic islands from $\\mathcal{D}$.\n\nThe sizes of $\\mathcal{A}$ and $\\mathcal{D}$ along the\n$\\psi$ and $w$ axes are also important for the beam matching.\nIf the beam is previously bunched around $\\phi_{\\rm s}$ then the bunch\nlength in $\\psi$ must be shorter than the corresponding size of\nthe acceptance and the energy dispersion around $E_{n,s}$ measured\nin terms of $w$ ---see~(\\ref{eq:psi-w-defn})--- must be smaller\nthan its size in this variable.\nFor example, let us consider the case $\\mu = 2$.\nThen $\\psi_{\\rm max}-\\psi_{\\rm min} = 0.28$ for $w=0$,\nand $w_{\\rm max} -w_{\\rm min} = 0.4$ for $\\psi = 0$.\nSee Fig.~\\ref{fig:FourthOrderResonance}.\nThe bunches at the\ninjection should fit these sizes in order to avoid beam losses during\nthe acceleration (in practice, to minimize beam losses).\nThe latter means that \n\\[\n\\left| \\frac{E_0 - E_{0,s}}{\\Delta_{\\rm s}} \\right| < 0.064.\n\\]\n\nSmall accelerators do not have buncher and the beam is produced by\nan electron gun which emits particles continuously,\nso they occupy the whole\t interval $[0,2\\pi]$ in the phase variable $\\psi$\nat the AS entrance.\nOur results provide an estimate of the beam capture efficiency $\\epsilon$;\nthat is, the fraction of the initial beam that is successfully accelerated.\nFor $\\mu = 2$ this fraction is\n\\[\n\\epsilon = \\frac{\\psi_{\\rm max}-\\psi_{\\rm min}}{2\\pi} = 0.04.\n\\]\nThe numerical computations show that $\\epsilon \\le 0.13$ for all $\\mu$\nin our RTM model.\nSee Remark~\\ref{rem:PhaseDeviation} in Section~\\ref{Sec:MainResult}.\n\nHere, we understand stability in a mathematical sense.\nThat is, we are dealing with perpetual stability,\nalthough only $2 \\cdot 10^7$ turns were considered in our numerical\ncomputations of $\\mathcal{A}$.\nOn the contrary, the number of turns made by each particle is\ntypically of just a few tens in real RTMs.\nFor instance, the number of turns is roughly 90 in the RTM machine of the\nMAMI complex at the Institute for Nuclear Physics in Mainz, which\nis nowadays the largest RTM facility in operation~\\cite{Jankowiak_etal2008}.\nThus, the physical acceptance and the true capture efficiency\nare larger than the mathematical stability domain and the estimates\nof $\\epsilon$ given above, respectively.\nIn fact, the difference may not be that large due to the instabilities of\ntrajectories after just a few iterations.\n\nWe have studied the case of the multiplicity increase factor $k = 1$,\nsee Section~\\ref{Sec:RTMModel}.\nThe general case $k \\in \\mathbb{N}$ can be analyzed in a similar way,\nand the stability domains turn out to be smaller.\n\nOur detailed description of the acceptance is an essential widening\nand improvement of results reported in~\\cite{Melekhin1975}. To\nthe best of our knowledge our results give the first complete characterization\nof the stability domain of the full non-linear model of the beam longitudinal\nmotion in RTMs.\nWe would like to emphasize that the obtained results\nshow the importance of the non-linear effects in the RTM beam dynamics.\nThis feature was well known from the experience of operation of\nthis type of electron accelerators.\nMany properties of the RTM map are similar to those of the H\\'enon map.\nAlso, as it was pointed out in~\\cite{Melekhin1972}, similar maps appear\nin the theory of anharmonic oscillator and optical theory of open\nresonators.\n\nIn our study we assumed that the acceleration\ngap is of zero length and that the velocity of particle is equal to\nthe speed of light already at the injection. The latter is not the\ncase for compact RTMs with the injection from a standard electron\ngun. It would be useful to develop an approach in which these conditions\nare relaxed. A finite-size accelerating gap can be taken into account\nby introducing a transit-time factor~\\cite{Wiedemann2003}.\nThe non-relativistic dynamics,\nwhich is of practical importance for the RTM design,\ncan be considered by a corresponding modification of map~(\\ref{eq:f}).\nThe concept of generalized synchronous particle was introduced and\nphase oscillations and a corresponding map were studied in~\\cite{Kubyshin_etal2008}.\nIt was shown that the phase of the synchronous particles changes (slips)\nfrom turn to turn at the first orbits and this effect should be taken\ninto account in choosing the phase of electrons at the injection.\n\nWe have modeled the magnetic field in the RTM end magnets\nby a simplified hard-edge distribution,\nwithout taking into account neither the fringe field effect~\\cite{Wiedemann2003}\nnor more complicated field profiles~\\cite{Vladimirov_etal2014}.\nStudying the longitudinal dynamics in these cases is also of interest\nfor the RTM beam physics.\n\nFinally, let us note that in our study the phase oscillations of particles\nof the beam were considered as independent of the transverse oscillations,\nvertical and horizontal. This approximation is valid if the amplitudes\nof these oscillations are small.\nA coupling between all three oscillations should be included\nin a more precise and detailed analysis.\n\n\n\n\\section*{References}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe theoretical foundations of cosmology were laid by \nEinstein in 1915 with the discovery of General Relativity.\nIn this framework, it became possible to describe mathematically \nthe evolution of the universe and to address questions about its \nbeginning and its end. Subsequently, the world of cosmology \nopened up with Hubble's observation that distant galaxies are \nreceding, thus leading to the conclusion that the universe is \nexpanding. These historic discoveries marked the beginning of \nmodern observational cosmology and initiated detailed investigations \nof our universe. Today we can answer questions that earlier we could \nnot even imagine asking. \n\nThe observed expansion of the universe means that the younger\nuniverse was smaller and hotter. Using our current knowledge\nof physics, this leads to a picture of the universe when it\nwas only a few minutes old and at a temperature of $10^{10}$ K.\nRemarkably, this picture can be (and has been) tested, since the\nlight elements were ``cooked'' at this time and we can compare calculations\nof the cosmological fraction of elements like Hydrogen, Helium, Deuterium, \nand Lithium with their observed abundances. The success of ``Big Bang\nNucleosynthesis'' gives us confidence in our understanding\nof the universe from a few minutes after the big bang.\n\nIn accelerator experiments, we have studied matter up to energies\ncorresponding to temperatures of about $10^{15}$ K. The theoretical\ndescription of matter at such temperatures is given by the electroweak \nmodel due to Glashow, Salam and Weinberg. The triumph of the model was \nin the prediction of the existence of the $W^{\\pm}$ and $Z$ bosons which were \nlater discovered at CERN. Hence we feel fairly confident\nthat we understand the behaviour of matter up to $10^{15}$ K. \n\nThe standard model of cosmology that has been so successful in its \nbig bang nucleosynthesis predictions, when extrapolated\nback to a time of $10^{-10}$ s, predicts that the universe was at \na temperature of about $10^{15}$ K and so must have been the arena\nfor electroweak physics. Our confirmation of the electroweak model\nprovides us with some confidence in our understanding of the\nuniverse at an age of $10^{-10}$ s, though we do not yet have any \nmeans to directly probe the universe of that time.\nAt even earlier times, when the universe must\nhave been at a temperature of about $10^{29}$ K,\nparticle physicists believe the universe was the stage for the\nphysics of ``Grand Unified Theories'' (GUTs). Here, we do not yet\nhave a standard model of particle physics, but there are several\ncandidates. The exploration of the consequences of particle physics\n(and in particular, GUTs) for cosmology, and vice versa, has become \na subject in its own right.\n\nThe electroweak model and GUTs are based on a scheme called ``spontaneous\nsymmetry breaking'' which, in lay terms, is another name for\nphase transitions. If these descriptions of particle physics are\ncorrect, the unavoidable implication is that the early universe \nmust have seen phase transitions much like the freezing of water and \nthe magnetization of iron. Then, the consequences of phase transitions\nthat we observe in the laboratory can be expected to apply to \nthe universe as well. In particular, relics of the high temperature\nphase of condensed matter systems called ``topological defects'' are \nroutinely observed in the laboratory and similar relics of the \nearly high temperature universe could exist in the present universe.\nIn other words, these are possible fossils from the early universe.\n\nThe hunt for cosmic topological defects depends crucially on their\nproperties. The last two decades have seen extensive research on \ntopological defects and their potential role in cosmology \n\\footnote{In recent times, there has been discussion of whether the \nparticles that we know ({\\it eg.} electrons) are actually topological defects \n\\cite{rebbisoliani,tvdual}. This kind of idea has a\nlong history and the possibility that electrons are objects with \nstructure dates back to Abraham \\cite{abraham} and Lorentz \\cite{lorentz}. \nI will not discuss this very interesting aspect of topological defects \nin the present article.}. \nVery recently, the lack of experimental input has been relieved by enterprising \ncondensed matter physicists who have been performing experiments in the \nlaboratory to answer questions of great interest to cosmologists. But before \nexplaining the possible role of topological defects in the cosmos and the lab, \nI need to describe some basics of modern particle physics.\n\n\\section{Inside the Atom}\n\\label{inatom}\n\nToday, we observe four seemingly different forces in \nNature. First is the force that holds us on the Earth, \nnamely, gravity. Second is the force that keeps the atom\ntogether which is electromagnetism. Then there is the\n``weak'' nuclear force which causes radioactivity and the \n``strong'' nuclear force which holds the proton together.\n\nHistorically, electricity and magnetism were believed to be\ntwo different forces that were treated in a unified manner only \nonce Maxwell wrote his equations. In particular, this means that there \nis only one coupling constant that describes the strength of the \nelectric and magnetic forces. Today we understand electromagnetism\nas the simplest kind of ``gauge theory''. In fact, the known non-gravitational\nforces are ascribed to the exchange of spin 1 particles called gauge \nparticles which for the electromagnetic force is none other than the\nphoton. In mathematical language, the photon is a particle of a\ngauge field $A_\\mu (t, \\vec x )$. Now, it is well-known that there is\na symmetry of electromagnetism related to the gauge transformation:\n$$\nA_\\mu \\rightarrow A_\\mu ' = A_\\mu + \n{1\\over e} \\partial_\\mu \\Lambda (t, \\vec x )\n$$\nwhere, $\\Lambda$ is an arbitrary function and $e$ is a coupling\nconstant. This symmetry is described by rotations in a complex\nplane as can be seen if\nwe couple the photon to a complex scalar field $\\Phi$. Then  \nan interaction term that preserves the gauge symmetry is\n$$\n| D_\\mu \\Phi |^2 \\equiv |(\\partial _\\mu - ie A_\\mu )\\Phi |^2\n$$\nprovided we also perform the transformation\n\\be\n\\Phi \\rightarrow \\Phi ' = e^{i\\Lambda (t, \\vec x )} \\Phi \\ .\n\\label{gtphi}\n\\end{equation}\nThis transformation is a (space-time dependent) rotation in the complex \n$\\Phi$ plane, and hence\nelectromagnetism is invariant under rotations described by one angle\n$\\Lambda$. Such rotations form a group called $U(1)_Q$ (the group of\nunitary $1\\times 1$ matrices) where the subscript $Q$ is used to denote \nthat the charge associated with the symmetry is ordinary electric charge.\n\nThe $U(1)$ symmetry of the model can be ``broken'' or ``hidden'' in \nthe vacuum if $\\Phi$ takes on a non-vanishing, fixed value in the lowest \nenergy state.\nThis can happen if, for example, there is a potential term for $\\Phi$\nsuch as\n$$\nV(\\Phi ) = {\\lambda \\over 4} ( |\\Phi |^2 - \\eta^2 )^2 \\ .\n$$\nThen the lowest energy state is obtained with $| \\Phi | = \\eta$ which\nis non-zero, and we say\nthat $\\Phi$ has a ``Vacuum Expectation Value'' (VEV). As the VEV\nis not invariant under phase rotations, the $U(1)$ symmetry is said\nto be spontaneously broken. Furthermore, by calculating thermal effects\nit can be shown that at high temperature, $\\Phi =0$ is the lowest energy \nstate, while at low temperature $| \\Phi | = \\eta$ is preferred.\nSo, if we have a thermal bath of $\\Phi$ and $A_\\mu$ quanta, at high\ntemperatures the system will have a $U(1)$ symmetry which will be \nbroken upon cooling. This symmetry breaking is depicted as:\n$$\nU(1) \\rightarrow 1 \\ .\n$$\n\nThe reader unfamiliar with group theory might feel lost among the\nstrange symbols such as $U(1)$ and others to follow. It is best \nto simply think of these as shorthand notations for writing\ndown all the transformations of the fields in the model that leave\nthe physics of the system unchanged. So $U(1)$ is just a convenient\nway of saying that the transformations that leave Maxwell's equations\nunchanged correspond to rotations in the complex plane. Another example,\ncloser to everyday experience, is the set of continuous transformations \nthat leave a sphere unchanged. This is the group of all rotations in \nthree dimensions and is denoted by $SO(3)$. \n\nUsing a generalization of the gauge symmetry idea and spontaneous\nsymmetry breaking, electromagnetism and the weak force have now been \nunified in the Glashow-Salam-Weinberg electroweak model. \nThis unification is, however, different from that of \nelectricity and magnetism since the unified model still has two coupling \nconstants. The unification stems from the fact that the electromagnetic\nand weak forces are now described within a common framework.\nThe electroweak model is based on the gauge symmetry \n\\be\nSU(2)_L \\times U(1)_Y\n\\label{su2u1}\n\\end{equation}\nwhich means that the group elements are direct products of \nspecial (determinant equal to one), unitary, $2 \\times 2$ matrices, and, \nphase factors as in (\\ref{gtphi}).\nThe $L$ subscript means that the $SU(2)$ acts on certain\n(left-handed) fermions and the $Y$ subscript denotes that the\nassociated charge is ``hypercharge'' and serves to distinguish\nthe $U(1)$ symmetry from that of electromagnetism written as\n$U(1)_Q$. \nThere are four gauge fields in the electroweak model: three ($W^a_\\mu$,\n$a=1,2,3$) transforming under the $SU(2)_L$ and one ($Y_\\mu$) \nunder the $U(1)_Y$.\n\nAt this stage, it is not evident where electromagnetism is \ncontained in the electroweak theory since there is no sign of $U(1)_Q$ in\n(\\ref{su2u1}). Also, the theory with the symmetry group (\\ref{su2u1})\ncontains four different kinds of massless, spin 1 particles whereas\nwe only see one (the photon). What happened to three of the four bosons?\n\nLet us now introduce a Higgs (scalar) field, $\\Phi$,\nwhich transforms under the group elements in (\\ref{su2u1}) and is \nin the doublet representation of $SU(2)_L$ {\\it i.e.}\nit should be a two complex component vector. \n$\\Phi$ is now assumed to\nget a ``Vacuum Expectation Value'' (VEV), that is, $\\Phi = \\Phi_0 \\ne 0$.\nSo now transformations that change the value of $\\Phi$ are not allowed.\nThis means that the symmetry group in (\\ref{su2u1}) is no longer valid\nand one must find the subgroup that leaves the VEV unchanged. This\nsubgroup turns out to be a $U(1)$ group and is none other than $U(1)_Q$.\nTherefore, after spontaneous symmetry breaking,\nthere is only one gauge field ($A_\\mu$) that is massless just as we\nobserve, and there are\nthree gauge fields ($W^{\\pm}_\\mu$, $Z^0_\\mu$) that are massive. \nSo the massless photon can mediate long range forces,\nwhile the massive gauge bosons can only mediate short range (weak)\nforces. In this way, starting from a very symmetric situation one\nderives the vastly disparate electromagnetic and weak forces. \n\nIn this article, I will mainly be interested in the aspect of\nspontaneous symmetry breaking \nwhich in the electroweak model can be depicted as:\n$$\nSU(2)_L \\times U(1)_Y \\rightarrow U(1)_Q \\ .\n$$\nIn fact, this is not quite correct since the $SU(2)_L$\nand the $U(1)_Y$ factors contain two elements that are common.\n(This is the center of $SU(2)_L$ which contains the elements\n$\\pm {\\bf 1}$.) So the correct symmetry breaking is:\n\\be\n[ SU(2)_L \\times U(1)_Y ]\/Z_2 \\rightarrow U(1)_Q \\ .\n\\label{ewsymbreak}\n\\end{equation}\nThe precise structure of symmetry groups can be very important\nin the determination of the cosmological consequences of the\nmodel.\n\nSo far we have ignored the strong force.\nThe theory describing this force is called ``Quantum\nChromo Dynamics'' (QCD) and is based on an unbroken $SU(3)_c$ group \nwhere the index $c$ stands for the ``colour'' charge. So the standard\nmodel is based on a product of three groups, that is,\n$$\n[SU(3)_c \\times  SU(2)_L \\times U(1)_Y ]\/(Z_3 \\times Z_2) \n$$\nWith every group there is an associated gauge coupling\nconstant and so the model has three gauge coupling constants\nwhich are denoted by $g_3$, $g_2$ and $g_1$ for the strong, weak \nand hypercharge factors. \n\nIn field theory it is known that coupling constants ``run''. This\nmeans that the values of the coupling constants that one measures\ndepend on the energy at which the measurement is performed. The\nrate of the running is determined by the renormalization group\nequations which we will not discuss here. But the point is that\nthe three different coupling constants of the standard model seem\nto converge to the same value at an energy scale of about $10^{16}$ GeV \n(see Fig. \\ref{coupcons}). This suggests that there is only\none coupling constant at high energies and most likely only one\nsymmetry group. In other words, the suggestion is that there is\n``Grand Unification'' described in terms of a grand unified group.\n\n\\begin{figure}[tbp]\n\\caption{\\label{coupcons}\nSchematic depiction of the convergence of the three standard model\ncoupling constants at the grand unification energy scale. The\n$g_i$ ($i=1,2,3$) are the various coupling constants in the standard\nmodel, $g_G$ is the GUT coupling constant and $E$ is the energy\nat which the coupling constants are measured.\n}\n\\epsfxsize = \\hsize \\epsfbox{coupcons.ps}\n\\end{figure}\n\nLet us denote the grand unified group by $G$. Then, as in the\nelectroweak model, $G$ must break down to the standard model\ngroup which must be a subgroup of $G$:\n\\begin{equation}\nG \\rightarrow [SU(3)_c \\times SU(2)_L \\times U(1)_Y]\/(Z_3\\times Z_2) \\ .\n\\label{gutsymbreak}\n\\end{equation}\nTwo of the simpler examples of $G$ often seen in the literature \nare $SU(5)$ and $SO(10)$. \n\n\n\n\\section{A Change of Phase}\n\nAs we have seen, a central idea in modern theories of \nparticle physics is that there is spontaneous symmetry\nbreaking. However, the idea actually originated in \ncondensed matter physics in the context of phase transitions.\nTo understand the connection between spontaneous symmetry\nbreaking and phase transitions,\nconsider a very simple phase transition in which a gas\n(or liquid) freezes to form a solid. In the gaseous phase, the \nmolecules are flying around in random motion and the\n(infinite) container of gas is symmetric under \ntranslations:\n$$\n\\vec x \\rightarrow {\\vec x}' = \\vec x + \\vec \\delta\n$$\nwhere $\\vec \\delta$ can be any arbitrary vector. That is,\nthe symmetry group is that of all translations.\nNow, when the gas solidifies, the molecules are arranged\non some lattice and the residual symmetry transformations\nare restricted to \n$$\n\\vec \\delta = \\vec a\n$$\nwhere $\\vec a $ is a vector from any one lattice site to another. \nHence, translational\nsymmetry has been broken (reduced) by the change of phase.\n\nGoing back to particle physics, the very successful electroweak\ntheory is based on spontaneous symmetry breaking, and hence we\nare faced with the prospect of phase transitions in particle\nphysics. If, somehow, we were to heat up the particle physics \nvacuum, at some high temperature we would be likely to find a new \nphase. For the electroweak phase transition to occur, we expect to\nneed a temperature of about $10^{15}$ K. (The sun's interior is at \na mere ten million degrees.) In particle accelerators, we can\nachieve the corresponding energies, but only over a very small region\nand for a very short duration. So particle accelerators are not\ncurrently useful for studying the electroweak phase transition. \n(They are, however, being used to study the QCD phase transition\nat a temperature of $10^{10}$ K.) The GUT phase transition needs an\nexorbitant $10^{29}$ K and it would be hard to even dream of a\nmachine that could attain such energies.\nHowever, the early universe must have seen temperatures corresponding\nto the electroweak transition at the age of $10^{-10}$ s and the\nGUT phase transition at $10^{-35}$ s, making it the natural environment\nfor the study of high energy particle physics. At the same time,\nparticle processes in the early universe must have determined\nthe state of the current universe and so we would like to \nunderstand the cosmology of phase transitions. (For a review of\ncosmological phase transitions, see the article by M. Gleiser\n\\cite{mgleiser}.)\n\nAn obvious question at this stage is: how can we study something that\nhappened so long ago? To answer this, I must explain what topological\ndefects are.\n\n\\section{Topology and Frustration}\n\nLet us return to the solidification of a gas. During this\nphase transition, the molecules of the gas that are in\nrandom motion have to line up into a regular lattice. If\nthe gas is cooled quickly, each small volume of molecules\nstarts lining up but there is not enough time for the \ndistant parts of the gas to decide which line to choose.\nSo molecules in different parts of the gas line up in\na lattice but the orientation of the lattice is chosen\nindependently. If the orientations are chosen in a certain\nway it may become impossible for the entire gas to \nfreeze into a regular lattice. This can happen for topological\nreasons and the solidification might be frustrated. The end\nresult is a solid with defects in its lattice. Since these\ndefects are due to topological conditions, they are known as\n``topological defects''. \n(For reviews of topological defects in particle physics and\ncosmology, see \\cite{avps,mhtk,avpr}.)\n\nTo illustrate topological defects in the particle physics\ncontext, consider the $U(1)$ model described in Sec. \\ref{inatom}.\nSpontaneous symmetry breaking occurs in this model when \n$\\Phi (t, \\vec x )$ acquires a VEV (that is, becomes non-zero) at\nsome time. However, as described in a seminal paper by\nTom Kibble \\cite{tk},\nthe acquired value of $\\Phi$ at different spatial \npoints will, in general, be different. In particular, on a circle $C$\nin space, parametrized by an angle $\\theta$, we could have:\n\\be\n\\Phi \\bigr |_C = \\eta e^{i \\theta } \\ , \\ \\ \\  \\theta \\in [0,2\\pi ] .\n\\label{vevphi}\n\\end{equation}\nThere is a topological index associated with this VEV of $\\Phi$.\n(Basically, it is the number of times $\\Phi$ wraps around the circle\nin the complex plane as we go around $C$.) Next consider the disk \nbounded by the circle $C$ (see Fig. \\ref{wind}). With the value of $\\Phi$ \non $C$ given in (\\ref{vevphi}), because of the topology, it is possible \nto show that necessarily $\\Phi =0$ somewhere on the disk. But $\\Phi =0$ \nis the value of $\\Phi$ in the unbroken symmetry phase. Hence the \ncompletion of the phase transition is frustrated because of the topology \nin the model. Also, the spatial point where $\\Phi$ vanishes is not in the \nvacuum (because the vacuum corresponds to $\\Phi \\ne 0$) and hence, there \nis energy at this point. This energy configuration is called a topological \ndefect.\n\n\\begin{figure}[tbp]\n\\caption{ \\label{wind} \nThe winding of the field $\\Phi$ around the circle $C$ forces\n$\\Phi$ to vanish at a point on any surface spanned by $C$. \nBy considering different surfaces bounded by $C$, we see that\nthere is a one-dimensional locus of points\nat which $\\Phi =0$. Since $\\Phi \\ne 0$ in the vacuum, there is\nenergy in the neighbourhood of the curve on which $\\Phi =0$. This\nenergy is locked-in because to remove it, the field would have to\nbe rearranged over an infinite region of space.\nThe energy distribution around the curve with $\\Phi =0$ is a ``string''.\n}\n\n\\\n\n\\epsfxsize = \\hsize \\epsfbox{wind.ps}\n\\end{figure}\n\n\nIn the case of the $U(1)$ model, we could consider any surface\nbounded by the circle $C$, and since $\\Phi \\ne 0$ everywhere\non $C$, there will always be a point on the \nsurface with $\\Phi =0$. Therefore there will be a one-dimensional locus \nof points where the phase transition has been frustrated and has $\\Phi =0$. \nThis one-dimensional topological defect is called a ``string'' and was \nfirst theoretically described\nby Abrikosov in the condensed matter context \\cite{abrikosov}, and \nby Nielsen and Olesen in the particle physics context \\cite{hnpo}.\n\nVery similarly, phase transitions can get frustrated by topology in\nmore complicated models. This can result in two-dimensional topological\ndefects called ``domain walls'', strings with junctions in them, \npoint-like defects called ``monopoles'', and, many\nhybrids. A distinction is also\nmade between defects that have associated magnetic fields and those\nthat have none. The former are called ``local'' or ``gauge'' defects\nwhile the latter are called ``global'' defects. Domain walls are always\nglobal, but strings and monopoles can be global or local. Local monopoles\nwere discovered independently by `t Hooft \\cite{th} and by \nPolyakov \\cite{ap}. They are also known as ``magnetic monopoles'' and \nbehave just like isolated North or South poles of a bar magnet.\nIn addition to these topological defects, there is another defect called\na ``texture'' in which the field $\\Phi$ is forced to vanish at one point\nin space-time.\n\nHow can we determine the topological defects in any given model? The secret \nlies in the symmetry breaking pattern which in turn determines the topology\nof the vacuum manifold. The point is that, if a certain field configuration\nyields the lowest energy state of the system, transformations of this configuration\nby the elements of the symmetry group will also give the lowest energy state.\nFor example, if a spherically symmetric system has a certain lowest energy\nvalue, this value will not change if the system is rotated. More mathematically,\nif the group $G$ breaks to a subgroup $H$ (as, for example, in (\\ref{ewsymbreak}) \nor (\\ref{gutsymbreak})), and the system is in the lowest energy state which we \ndenote by $S$, transformations of $S$ by elements of $G$ will leave the energy\nunchanged. In addition, transformations of $S$ by elements of $H$ will leave\n$S$ itself (and not just the energy) unchanged. So the many {\\it distinct} ground\nstates of the system are given by all transformations of $G$ that are not\nrelated by elements in $H$. This space of distinct ground states is called\nthe ``vacuum manifold'' and is therefore given by the space of all elements\nof $G$ in which elements related by transformations in $H$ have been\nidentified. The space is denoted by $G\/H$ and mathematicians call it a \n``coset space''.\n\nThe outcome of the above discussion is that the symmetry breaking leads to\nthe determination of the vacuum manifold which is some surface in an abstract\nmathematical space. Now think of the vacuum manifold \nas a surface like the surface of a ball (two sphere), or, the surface of \na doughnut (torus). These surfaces have different topological properties. \nFor example, one can draw a closed path on a torus that cannot be continuously \nshrunk to a point while all closed paths on a two sphere can be. \nOne can also cover the two sphere with another two sphere\n(like an orange peel covers the orange)\nthat cannot then be shrunk to a point. It is these properties that are\ncrucial for the existence of topological defects.\n\nIf the vacuum manifold ({\\it i.e.} coset space) has incontractable one \nspheres (paths), the model will have string solutions. (With a little thinking, \nthe $U(1)$ example above can help to understand this claim.) If the vacuum manifold \nhas incontractable two and\/or three spheres, the model contains monopoles \nand\/or textures respectively. If the vacuum manifold is disconnected, \nwe will get domain wall solutions. The topology of various coset spaces has \nnow been determined and is given by what are called ``homotopy groups'' and \ndenoted by  $\\pi_n (G\/H)$. Mathematicians have prepared tables that give\nthe homotopy groups for different choices of $G$ and $H$.\n\nThe basic fact to remember is that the symmetry breaking pattern determines \nthe topology of the vacuum manifold and hence the topological defects. \nSo given $G$ and $H$ we can determine the topological defects present in the \nsystem.\n\n\nAn important feature of topological defects is that they cannot be\nremoved by locally rearranging the fields. In the string case,\nfor example, the circle $C$ could be chosen to be at infinity and\nthe removal of the string through the disk would require rearrangement\nof the field on an infinite portion of the disk. Any dynamical procedure\nto do this would need infinite energy and hence the string is permanently\nlocked in \\footnote{However, if there is a defect and an anti-defect in\nthe system, they can mutually annihilate.}.\n\nThe energy of a defect depends on the temperature at which it\nforms. Just to give an idea, monopoles formed at the GUT phase\ntransition would weigh $\\sim 10^{-8}$ gms, strings would have a linear\nenergy density of about $10^{22}$ gms\/cm, and, domain walls would\nhave a surface energy density of about $10^{52}$ gms\/sq-cm.\n\nNot all phase transitions lead to topological defects. A prime example\nof such a transition is the electroweak phase transition. (GUT phase\ntransitions always lead to magnetic monopoles.) Yet it should\nbe mentioned that there can still be field configurations in the absence\nof topology that closely resemble topological defects. Examples of\nsuch configurations include ``semilocal strings'' found by Ana Ach\\'ucarro\nand me \\cite{aatv} and ``electroweak strings'' first found by\nNambu \\cite{nambu}. Unlike topological defects, however, these \nconfigurations are not permanently locked in and can decay.\n\nThe possibility of topological defects in particle physics raises\nthe hope that some of these may have been produced in a cosmological\nphase transition and could be observed in the universe today by their\ninfluence on astronomical, astrophysical and cosmological processes.\n\n\n\n\n\\section{Cosmological Observations}\n\nCosmological surveys now cover a large fraction of our observable \nuniverse.  Astronomers have mapped the luminous structure in slices \nof the sky out to a distance of several hundred megaparsecs \n(see Fig. \\ref{lssobs}) \\cite{lcrs,cfa}. These maps of \nthe universe show that \ngalaxies are distributed on walls surrounding empty bubbles (voids). \nThis comes as somewhat of a surprise because one's first guess \nwould be that galaxies are spread randomly in space.\n\nRecently, another vital observational tool for the study of the\nearly universe has become available. This is the structure of the\ntemperature fluctuations in the\n``Cosmic Microwave Background Radiation'' (CMBR). The CMBR is \nlight that is directly coming to us from a time when the universe was \nabout 100,000 years old and at a temperature of about 3000 K. This\n``recombination'' epoch is significant because protons and electrons\ncombine to form hydrogen atoms at about 3000 K. After recombination, \nthe universe contains electrically neutral atoms and, since light \ndoes not scatter off neutral atoms, it can travel freely to us. Before \nrecombination, however, \nthe matter in the universe is electrically charged and light scatters\nstrongly. During this period, light propagates as if it were in a \nfog and so light from the pre-recombination universe cannot reach us. \nThe CMBR is the earliest light we could possibly see and it is very\nsignificant that we have actually seen this light \n(see Fig. \\ref{cmbspec}) \\cite{firas}.\n\n\\begin{figure}[tbp]\n\\caption{ \\label{lssobs}\nThe points in the wedges show the distribution of galaxies in a slice of \nthe sky as observed by the\nLas Campanas Redshift Survey. The survey covers three strips \nof the sky in the Northern hemisphere and another three strips \nin the Southern hemisphere. The larger angular width of the Northern \nhemisphere strips is shown on top of the figure ($10^h$ to $16^h$ of\nRight Ascension). The smaller angular widths of the strips is about a\nfew degrees and the Declination of each of the strips is specified on the \nside of the figure. The radial distance to a galaxy is measured in terms\nof a velocity corresponding to the observed redshift of the galactic \nlight.\n}\n\\epsfxsize = \\hsize \\epsfbox{lcrs.ps}\n\\end{figure}\n\nThe CMBR is extremely uniform in all directions. The uniformity\nis only spoilt by \ntiny fluctuations of about 1 part in $10^5$. In other words, the\ntemperature of the CMBR is $T=2.7$ K, no matter in which direction\nyou choose to look but there are fluctuations $\\delta T$ in this\ntemperature:\n$$\n\\left ( {{\\delta T}\\over T} \\right )_{rms}  \\simeq 10^{-5} \\ .\n$$\nFurther, due to the growth in the number of observational experiments,\nit is now becoming possible to say something about the map of\n$\\delta T$ over the sky. The observations determine the\ntemperature fluctuation on the sky at different angular scales and so\none has quantities related to the spherical harmonics of $\\delta T$.\nThe usual procedure in a calculation of the anisotropy is to decompose \nthe temperature fluctuations on the sky \n(coordinates $\\theta$ and $\\phi$) in spherical harmonics:\n$$\n{{\\delta T} \\over T} (\\theta , \\phi )\n= \\sum_{l=0}^\\infty \\sum_{m=-l}^{m=+l} a_{lm} Y_{lm} (\\theta , \\phi )\n$$\nand then calculate \n$$\nc_l = < | a_{lm}^2 | > ~ ,\n$$\nwhere the angular brackets denote an ensemble average. Then it is\nconventional to find\n$$\n<Q> \\equiv \\left [ {l(l+1)c_l}\\over {2\\pi} \\right ]^{1\/2} T_{cmbr}\n$$\nas a function of $l$,\nwhere $T_{cmbr}$ is the CMBR temperature measured in $\\mu$K.\nThe calculated $<Q>$'s are then compared with observations\n(see Fig. \\ref{cmbcls}) \\cite{cobeandmore}.\n\n\\begin{figure}[tbp]\n\\caption{ \\label{cmbspec}\nThe temperature of the CMBR in degrees Kelvin as measured at various \nfrequencies in GHz.\n(FIRAS was a satellite borne experiment to measure the spectrum of the\nCMBR.)\n}\n\\epsfxsize = \\hsize \\epsfbox{speccmbr.ps}\n\\end{figure}\n\nThe fluctuations of the temperature of the CMBR can be produced in\ntwo ways. The first is if the matter that the photons last scattered\noff (the ``last scattering surface'') was not quite uniform, and the\nsecond is if there are objects between the last scattering surface\nand the present that disturb the photons. In the first case, the CMBR\nprovides a very definite determination of the state of the 100,000 year\nold universe and in the second case, it provides a probe of the universe\nbetween now and the last scattering surface. By considering the details\nof the fluctuations of the CMBR, we hope to be able to derive both the\nstate of the early universe and the intervening influences.\n\n\\begin{figure}[tbp]\n\\caption{ \\label{cmbcls}\nThe observed distribution of $<Q>$ - a quantity related to the anisotropy\nof the CMBR in the $l^{th}$ multipole moment (see text) - together with \nerror bars. The\ncurve is the prediction of an inflationary model. (The $<Q>$ in this\nplot is normalized with an extra factor of $(5\/12)^{1\/2}$ as compared \nto the definition in the text for historical reasons.)\n}\n\\epsfxsize = \\hsize \\epsfbox{clcmbr2.ps}\n\\end{figure}\n\n\\section{Fossils from the Early Universe}\n\nBased on our knowledge of particle physics, the\ngradual cooling of the universe must have been\npunctuated by sharp phase transitions. This is\nsimilar to the violent climatic changes on earth\nthat would have affected the otherwise gradual \nevolution of life. Further, just as we seek fossils\nof the early forms of life, we can seek fossils from \nthe early universe in the form of topological defects.\nIn fact, topological defects are our main hope of\ndirectly studying the very early universe.\n\nThe current belief that the electromagnetic, weak and \nstrong forces unified at about $10^{16}$ GeV implies\na cosmological GUT phase transition at a temperature\nof $10^{29}$ K at the young age of $10^{-35}$ s. Then\nwe are led to consider the formation of topological\ndefects corresponding to this scale. These defects\ncould be magnetic monopoles, strings or domain walls.\n\nMagnetic monopoles formed at the GUT scale would \ndilute with the expansion of the universe while keeping\ntheir number fixed. This means that the energy density in\nmonopoles goes down as $a^{-3}$ where $a$ is the scale factor \nof the universe. However, the dominant energy in the \nearly universe is radiation. The energy density of radiation\nnot only gets diluted by the expansion but the energy of\neach radiation quanta also gets red-shifted. Therefore\nthe energy density in radiation falls off as $a^{-4}$. This\nmeans that the energy in monopoles becomes more important\nas the universe expands. Following this argument by a more\ncareful and detailed analysis, Zel'dovich and Khlopov \\cite{yzmk},\nand John Preskill \\cite{jp} found \nthat GUT monopoles would start dominating the universe very early \nand would overclose the universe ({\\it i.e.} the energy density \nin monopoles would exceed the critical density and the universe \nwould recollapse in a very short time). This is clearly not the case.\n\nAround 1980, the monopole overabundance problem led to a tension \nbetween a central belief in particle physics - that of grand \nunification - and cosmology. For consistency, either grand unification \nhad to succumb, or, cosmology needed revision. The breakthrough was\nachieved when Alan Guth realized \\cite{ag} that an exponential inflationary \nperiod in cosmology, during which the energy density in monopoles is \ndiluted to acceptable levels, would alleviate the tension. \nThe following years have seen a number of other solutions \nto the monopole problem but inflationary cosmology has survived because \nit also offers solutions to a number of other cosmological puzzles.\n(See Andrei Linde's book on inflationary cosmology \\cite{linde} for an\naccount of the field.) \n\nDomain walls formed at the GUT scale, like magnetic monopoles, would \nbe a cosmological disaster. If we had one domain wall of mass per unit\narea equal to $\\sigma$ in our visible universe (size $t$), the total\nenergy contained in it would be of order $\\sigma t^2$. And the energy \nin all the other matter would be of order $\\rho t^3 \\sim t\/G$ where\n$\\rho \\sim 1\/ Gt^2$ is the energy density in matter and $G$ is\nNewton's gravitational constant. So the ratio of domain wall energy\nto other forms of energy is $G\\sigma t$. By inserting the value\nof $\\sigma$ ($10^{52}$ gms\/sq-cm) and $G$, we find that domain walls\nstart dominating the universe very early and would lead to a universe\nunlike ours. This rules out the formation of GUT domain walls. \n(Though, if the GUT model leads to an inflationary universe,\nGUT domain walls would be acceptable.)\n\nCosmic strings formed at the GUT scale are more benign than magnetic\nmonopoles and domain walls. To see this, however, is considerably\nmore difficult. The problem has been studied over the years\nusing intensive computer simulations by\nthree groups: Andy Albrecht and Neil Turok, \nDave Bennett and Francois Bouchet, and \nBruce Allen and Paul Shellard (see \\cite{proceed} for reviews). \nAnalytic tools to study the problem have been devised by\nTom Kibble, Dave Bennett, Ed Copeland and others (references may be\nfound in \\cite{mhtk,avps}). Most recently, Mark Hindmarsh, inspired by \nideas from condensed matter physics, has devised a technique to\nstudy the evolution of domain walls in an expanding universe \\cite{mh}. \nThis seems like a \npromising approach to study the evolution of cosmic strings too, though\nthe technique has not yet been applied to this problem.\n\nThe results on cosmic string evolution, first\nestablished by Bennett and Bouchet, were that the network\nevolves in such a way that the long strings shed very small loops and\nthe energy density in strings remains a fixed\nsmall fraction of the energy density in the universe. This leads\nto the possibility that cosmic strings may have been produced at the\nGUT scale and could be ``out there'' for us to discover. In addition,\ncosmic strings would have influenced the light and matter around them\nand so it may be possible to detect this influence by careful observations\nof the present universe. In particular, cosmic strings may have left\ntheir imprint on the CMBR and the large-scale structure. \nThe flip side of the coin is that if the effect \nof GUT scale cosmic strings on the CMBR \nand on large-scale structure formation disagrees with observation,\nwe would be able to say that they do not exist and thus gain some\nimportant information about particle physics at very high energies. \n\n\\begin{figure}[tbp]\n\\caption{ \\label{lens1}\nThe filled circles show the location of several hypothetical unlensed\nsources in the presence of a foreground string segment that was generated\nby computer simulation. The sharp kinks in the string\nare partially due to the fact that what is shown is the projection of the\nstring onto a plane and, on small scales, due to the simulation grid used\nto generate the string.  The inner box (dotted line) is 25''$\\times$25''.\n}\n\\epsfxsize = \\hsize \\epsfbox{figure.2.ps}\n\\end{figure}\n\nThere are two known ways to hunt for cosmic strings. The first is by\nrealizing that a cosmic string that is illuminated on the backside by \na light source would act as a lens for the source since the string\ncurves the intervening spacetime. So a string would cause multiple \nimages of a background quasar or galaxy. The observation of such an\nevent would not only tell us that there are cosmic strings in the universe\nbut it would also tell us where the string is currently located. \nWith Andrew de Laix and Lawrence Krauss, I have recently been investigating \nthis scheme for a cosmic string hunt \\cite{adlktv}. \nIn Fig. \\ref{lens1} the location of a \nstring with several background sources is shown. The string causes the light \nfrom the sources to bend and the field appears as shown in Fig. \\ref{lens2}. \nIn this hunt for strings, there is an uncertainty in the details of the \nlensing pattern since the shape of cosmic strings is not precisely\nknown. However, the \nlimiting factor is the small probability for looking in the right direction \nfor observing a string lensing event. Ongoing and planned surveys, however, \nwill be covering roughly a quarter of the sky and should find GUT cosmic \nstrings if they are there.\n\n\\begin{figure}[tbp]\n\\caption{ \\label{lens2}\nThe appearance of the field of sources in the 25''$\\times$25'' size\nbox shown in the previous figure due to gravitational lensing by the\nstring. The stringy appearance of the lensed sources seems evident.\nThe challenge in real surveys would be to pick out the stringy nature\nof the signal in a field of other astronomical objects.\n}\n\\epsfxsize = \\hsize \\epsfbox{figure.3.ps}\n\\end{figure}\n\nA second way to search for strings is to seek their imprint on the\nCMBR. Just as a string distorts the\nimages of background sources, Kaiser and Stebbins \\cite{nkas}\nshowed that moving strings would change the energy of photons\nthat pass by. Since the CMBR\nis background illumination for cosmic strings, it should have\ntemperature fluctuations induced by strings. Ongoing experiments are\ndetermining the CMBR fluctuations very\ncarefully and theorists have been calculating the influence of strings\non the background. The theoretical predictions depend on various other\nfactors (such as the energy density in the universe). At the moment,\nthe simplest cosmological model with strings does not appear to be\nconsistent with the observations \\cite{ballenetal,aalbrechtetal}.\nIn another 5 years, with more and better data and with further\ncharacterization of the string network, we should be able to say with\ngreater certainty if the observed anisotropy in the CMBR can be due to\nGUT cosmic strings.\n\nIn addition, as first pointed out by Zeldovich \\cite{yz} \nand by Alex Vilenkin \\cite{avlss}, it is possible to consider the influence \nof cosmic strings on the formation of structure (galaxies {\\it etc.}) in the \nuniverse. Over the years, \nour understanding of the influence of cosmic strings \non the matter in the universe has evolved. At first it was believed that \ncosmic string {\\em loops} would be centers around which galaxies would\nform. Later the potential importance of long (infinite) strings for structure \nformation was realized by Silk and Vilenkin \\cite{jsav}, \nStebbins {\\it et. al.} \\cite{asetal} and by me \\cite{tvlss}. \nThis realization\ngained force once it was found that the string loops were too small to be\nof much importance in the formation of large-scale structure \\cite{tvav}.\nThe implications of cosmic strings for structure formation continue to\nbe worked out with great vigour by researchers like Albrecht, Allen, \nBrandenberger, Shellard, Stebbins, and others. \n\nIt should be added that there are great cosmological, astrophysical and \ntheoretical uncertainties in the research on formation of large-scale structure\nby strings ({\\it eg.} see the paper by Rees \\cite{rees}). \nHowever, if the calculated distribution of large-scale structure due to\nstrings roughly agrees with the observed distribution \n(Fig. \\ref{lssobs}), this would provide hope for the existence of strings. \nA disagreement would provide evidence against string seeded structure \nformation and hence a constraint on GUT models in particle physics. \n\nAs first pointed out by Hogan and Rees \\cite{hoganrees}, there\nis yet another constraining observation that cosmic strings must\nsatisfy - this is the observed limit on a cosmological background\nof gravitational waves. Since cosmic strings generate gravitational\nradiation, their energy density has to be low enough such that their\ngravitational radiation remain within limits imposed by the timing\nof the millisecond pulsar \\cite{jtaylor}. \n(A gravitational wave background would introduce noise in the \nmillisecond pulsar timing \nbeyond that what is observed and accounted for.) An estimate of the \ngravitational radiation from strings depends sensitively on the \nstructure of the string network. Based on the current understanding of\nthe network, the gravitational wave constraints are evaded\nby GUT strings, though by a small margin.\n\nIn an effort spearheaded by Turok \\cite{turok}, \nSpergel \\cite{spergel} and Durrer \\cite{durrer}, cosmologists \nhave also examined the influence of texture \nand other global defects on the CMBR and large-scale structure. Once again,\nanalysis of the simplest theoretical models indicates that GUT scale\nglobal defects by themselves cannot simultaneously explain\nlarge-scale structure formation and the anisotropy of the CMBR.\n\nThe interest in the GUT phase transition comes from the underlying\nunification philosophy, the apparent convergence of the known \ncoupling constants (Fig. \\ref{coupcons}), \nand, the cosmological relevance of the GUT\nenergy scale. (The GUT energy scale seems suitable for laying out\nthe seeds of density inhomogeneities that will later grow to become \ngalaxies.) However, our knowledge of particle physics is not yet \ncomplete enough that we can say that the electroweak and GUT phase\ntransitions were the only unifying cosmological phase transitions. \nIndeed, there are several particle physics models in which phase\ntransitions would have occurred between the electroweak and GUT\nepochs. Defects produced at these epochs may not have been responsible\nfor galaxy formation but it would be invaluable to know if they\nexist in the universe. Since these defects would be lighter, it\nis unlikely that they will be seen due to their gravitational interactions. \nInstead, to hunt them, one has to rely on their particle physics\ninteractions which can lead to electromagnetic radiation and cosmic\nrays, an effort actively pursued by Bhattacharjee and others \n\\cite{pbgsds,tkay,avcosmicrays}.\n\n\\section{Down on Earth}\n\nThe fact that cosmological phase transitions and condensed matter \nphase transitions are described by the same physical principles, \nallows us to consider performing ``cosmological experiments''\nin the lab. These are experiments in condensed matter systems that \nare motivated by cosmology. \nThis idea was first suggested by Zurek \\cite{zurek}. For example,\ncondensed matter physicists have studied topological defects for \na long time and have been interested in their microphysical properties \nand also in how the system of defects relaxes\nwith time thus leading to the completion of the phase\ntransition. However, until now, they were not\ninterested in the number of defects, or in the size\ndistribution of vortices (strings) that are formed during\na phase transition. Both these quantities are of crucial interest\nto cosmologists since the number and distribution of\ndefects determines their astronomical and astrophysical\nrelevance. Hence an experiment that studies the \ndistribution of vortices produced during a phase transition\nwould be called a ``cosmological experiment'' \\footnote{The exchange of \nideas between cosmologists and condensed matter physicists was greatly \nfacilitated by a six month long program and a NATO workshop on topological \ndefects held at the Isaac Newton Institute during 1994. The workshop\nlectures can be found in the proceedings edited by Davis and\nBrandenberger \\cite{robertanne}.}.\n\nOver the last several years, a number of condensed matter\nexperiments of a cosmological flavour have been performed. First\nwas the experiment in nematic liquid crystals by \nChuang {\\it et. al.} where the authors studied the relaxation \nof a network of strings \\cite{chuang}. This was later followed\nby efforts to study the formation of defects in liquid crystals\nby Srivastava and collaborators \\cite{ams}. \nThe attention then turned to phase \ntransitions closer to expected particle physics phase transitions \nand experiments studying the formation of vortices in $^4$He\nwere carried out by Peter McClintock and his group in Lancaster\n\\cite{he4}. Most recently, there have been a number of ingenious \nexperiments in $^3$He conducted in Grenoble, Helsinki and\nLancaster \\cite{grenoble,helsinki}\nthat have also studied the formation of strings. \n(These are described in the article by A. Gill \\cite{agill}.) \n\nA leading personality in the effort to connect particle physics and\ncosmology with $^3$He is Grisha Volovik. The point he has tried to\nconvey to the physics community is that there are strong similarities \nbetween the basic structure of particle physics and $^3$He \\cite{gvtv}. \nSo one can imagine simulating particle physics processes of cosmological \ninterest in $^3$He provided one is careful to ask the right questions. \nFor example, as has been done with astounding success, it is possible to \nsimulate the cosmological formation of strings in $^3$He since the formation\nof topological defects is not peculiar to details of the cosmological \nenvironment. At the same time, it seems that it may not be\npossible to simulate the cosmological evolution of strings in condensed\nmatter systems since that depends on the \nHubble expansion and the absence of strong\ndissipative processes, both of which are cosmological conditions and\nhard to find in the lab setting.\nHere I will describe another process that has been studied in $^3$He\n\\cite{baryo3he}\nand which is of great interest to cosmologists - this is the generation\nof matter, also called ``baryogenesis''.\n\nIn the absence of an external magnetic field, \n$^3$He is known to have two superfluid phases which are\ncalled the A and B phases. At high temperature, \n$^3$He is invariant under rotations of the Cooper pair\nspin (S), orbital angular momentum (L), and, also, phase \nrotations of the wave-function that lead to the\nconservation of particle number (N). So the (continuous)\nsymmetry group of $^3$He is:\n$$\nG = SO(3)_S \\times SO(3)_L \\times U(1)_N \\ .\n$$\nThe spontaneous symmetry\nbreaking pattern for the transition into the A phase is:\n\\be\nG \\rightarrow SO(2)_{S_3} \\times U(1)_Q \\ ,\n\\end{equation}\nwhere \n$$\nQ = L_3 - {N \\over 2}\n$$\nNote that in this symmetry breaking, the $SO(3)_S$ group\nbreaks to $SO(2)_{S_3}$ while \nthe remaining symmetry breaking pattern appears to be exactly \nthat of the electroweak model. There are, however, subtle\ndifferences in certain discrete symmetries in the two\nmodels that are important in determining the topology \nof the vacuum manifold and hence, the topological defects.\n(Nonetheless, a direct analog of the non-topological electroweak \nstring is present in $^3$He \\cite{gvtv}.) Another difference is \nthat $^3$He does not contain fundamental gauge fields other than the \nelectromagnetic fields.  \nIn the electroweak model, however, such gauge fields exist and \nare important.  It is useful to be aware of these subtle differences \nbecause it allows us to meaningfully compare $^3$He experiments with\nparticle theory expectations.\n\nThe common elements in $^3$He and (hypothetical) particle theory \nis the presence of non-trivial topology. Therefore processes\nsuch as the formation of topological defects can be studied in\n$^3$He and the results translated to the particle physics world.\nIn addition, $^3$He contains quasiparticles that correspond to the \nfundamental particles (leptons and quarks) in particle theory. \nThese quasiparticles interact with the order parameter of $^3$He \njust as the fundamental fermions interact with the electroweak gauge \nfields. So $^3$He does contain``effective'' gauge fields besides the \nordinary electromagnetic fields.\nThis similarity is very valuable since the behaviour of fermions\nin fixed background gauge fields can be simulated \nby the interaction of quasiparticles in the background of some\norder parameter configuration in $^3$He. \nIndeed this is precisely what is needed\nto simulate the violation of baryon number in $^3$He.\n\nIn vacuum the energy of a free fermion is given by:\n$$\nE = \\pm \\sqrt{{\\bf p} ^{~ 2} + m^2 } \\ ,\n$$\nwhere $\\bf p$ is the momentum and $m$ is the mass of the\nfermion ($c$ has been set to 1). Therefore to create a fermion\nand antifermion pair from the vacuum requires at least an\nenergy equal to $2m$.\nIn the presence of certain scalar and gauge field configurations,\nhowever, the dispersion relation for fermions can display a\n``zero mode'' (see Fig. \\ref{zeromode}). \nThis can be seen by solving the Dirac equation in the non-trivial\nscalar and gauge field background. If the Dirac equation has a\nsolution with zero energy eigenvalue then this solution is the\nzero mode. (Alternatively, the existence of the zero mode follows \nfrom certain mathematical ``index theorems'' which I will not\ndescribe here.) For example, there can be a zero mode in the background \nof a string lying along the $z-$axis. Effectively this says that\nthere are fermions trapped on the string that behave as if they are \nmassless as long as they stay on the string. But these trapped fermions\ncan travel along the string and their dispersion relation in the\none dimensional space of the string is:\n$$\nE = \\pm ~ p_z   \\ .\n$$\n\n\\begin{figure}[tbp]\n\\caption{\\label{zeromode}\n{\\bf a)}\nThe energy versus the momentum of fermions along the $^3$He vortex\n(assumed to lie along the $z-$axis). The crucial feature in this\nspectrum is the presence of a zero mode (the $n=0$ line) which\ncrosses from the negative energy to the positive energy region. The\napplication of an electric field lifts the level of\nthe Dirac sea along the $n=0$ mode and particles move from the\nvacuum ($E<0$) to the physical world ($E>0$).\nThis is the anomalous production of fermions from the vacuum.\n{\\bf b)} The spectrum of $u$ and $d$ quarks on strings in the\nelectroweak model. As the level of the Dirac sea rises (or falls),\n$u$ and $d$ quarks are produced (or destroyed) in such a way that\nthe total electric charge, $q$, remains conserved but the total\nbaryon number $B$ is violated. ($C$ is the charge under an\noperation called ``particle conjugation''.)\n}\n\\epsfxsize = \\hsize \\epsfbox{zeromode.ps}\n\\epsfxsize = \\hsize \\epsfbox{udonstring.ps}\n\\end{figure}\n\nThe existence of zero modes leads to the anomalous creation\nof fermions (Fig. \\ref{zeromode}). An intuitive picture\nis that, if an electric field is applied along the string,\nthe Dirac sea can rise as a whole and particles from the Dirac\nsea can get pushed into positive vacuum states along the zero mode. \nThis can happen in string-like configurations in the electroweak model \nof particle physics and also along vortices in $^3$He. The anomalous \ncreation of fermions in the electroweak model leads to the creation of\nmatter (baryons) over antimatter (antibaryons) or vice versa,\nwhile the anomalous creation of quasiparticles in $^3$He \nleads to the violation of total quasiparticle momentum and is\nobserved as an excess force on moving vortices. \nIn a cosmological scenario, such processes\ntogether with other suitable conditions such as thermal non-equilibrium\nand CP violation can lead to the creation of matter in the universe\n(baryogenesis). \n\nWhen we apply an electric field ${\\bf E}$ along a string that\ncarries magnetic field ${\\bf B}$, the\nrate of production of fermions of charge $q$ is:\n\\be\n\\dot n = {{q^2} \\over {4\\pi^2}} {\\bf E}\\cdot {\\bf B} \\ ,\n\\label{ndot}\n\\end{equation}\nwhere $n$ is the number density of fermions.\n(The electric field itself can be induced via Faraday's law if \nthe string moves across an ambient magnetic field.) \n\nThe anomaly equation (\\ref{ndot}) is applicable to both the electroweak\nmodel and $^3$He. The possibility of anomalous generation of baryon number \nalong strings was discussed by Witten \\cite{ew}. \nIn the electroweak case, a non-Abelian generalization\nof (\\ref{ndot}) leads to the possibility of anomalous baryon charge\non electroweak string knots (see Fig. \\ref{link}) as I showed in \ncollaboration with George Field \\cite{tvgf}, and, Jaume Garriga \\cite{jgtv}. \nIn $^3$He, the \nanomaly equation leads to quasiparticle production. The measurable quantity, \nhowever, is the momentum, $\\bf P$, carried off by the anomalously created \nquasiparticles:  \n$$\n\\partial_t {\\bf P} = {{1} \\over {2\\pi^2}} \n          \\int d^3 x (p_F {\\bf {\\hat l}}) {\\bf E} \\cdot {\\bf B} \\ ,\n$$\nwhere, $p_F$ is the Fermi momentum and ${\\bf {\\hat l}}$ is the orientation\nof the Cooper pair angular momentum.\n\nIn the Cooper pair plus quasiparticle system, momentum is obviously\nexactly conserved. In the absence of the anomaly, the momentum in\nthe Cooper pairs and quasiparticles is separately conserved. Due to\nthe anomaly, however, momentum is transferred from the $^3$He vacuum\n(Cooper pairs) to the quasiparticles and vice versa. This transfer\nof momentum leads to an extra force on moving vortices:\n\\begin{equation}\n{\\bf F} = \\partial_t {\\bf P} = \\pi \\hbar N C_0 \n           {\\bf {\\hat z}} \\times ({\\bf v}_n - {\\bf v}_L ) \\ ,\n\\label{extraforce}\n\\end{equation}\nwhere $N$ is the winding of the vortex, the coefficient $C_0$ is\na temperature dependent coefficient,\nthe vortex lies in the ${\\bf {\\hat z}}$ direction, and\n${\\bf v}_L - {\\bf v}_n$ is the vortex line velocity with respect to\nthe normal fluid. \n\nThe Manchester group, led by Henry Hall and John Hook, \nused a clever experimental setup \nin which an array of vortices was created by rotating a sample of $^3$He. \nA diaphragm placed within the sample had two orthogonal modes of oscillation\nwhich could be driven electrically and also detected. Oscillations in one of \nthe modes was used to create the relative velocity ${\\bf v}_L - {\\bf v}_n$.\nThe extra force on the vortices given by eq. (\\ref{extraforce}) produces\nforces perpendicular to the driven mode of oscillation and hence couples \nto the other oscillation mode of the diaphragm. The oscillations in this\northogonal mode can then be measured. This leads to the measurement of \nquantities related to the coefficient $C_0$ at different temperatures. \nThe results confirm the anomalous production of quasiparticles \non the vortex.\n\n\\begin{figure}[tbp]\n\\caption{\\label{link}\nA knotted configuration of electroweak strings that has associated\nbaryon number. Here ``baryon number'' is defined in terms of\nparticles trapped on the string and this is somewhat different from\nthe usual meaning which is defined in terms of particles in the vacuum.\n}\n\\epsfxsize = \\hsize \\epsfbox{link.ps}\n\\end{figure}\n\nThe observation of ``momentogenesis'' in $^3$He confirms \n``baryogenesis'' in the electroweak model. The experiments, however, do \nnot say anything about the cosmological process of baryogenesis since these\ndepend on various other cosmological factors such as departures from\nthermal equilibrium and CP violation.\n\n\n\\section{Outlook}\n\nIn the study of the early universe, the last several decades \nhave seen a remarkable confluence of ideas originating in \nvastly different branches of physics. Who could have imagined \nthe possibility of fossils from the early universe and that \none day we would be ``digging'' for them? That the mysteries\nof the atom could be revealed by astronomical observations, while \nthe secrets of the big bang may be locked in particle accelerators? \nIt requires an even further stretch of imagination to contemplate\nsimulating the early universe in a vial of helium.\nYet this is the current state of early universe cosmology\nand we can be sure of many equally surprising developments\nin the years to come.\n\n\\smallskip\n\n\\noindent {\\it Acknowledgements:} I am grateful to Andrew de Laix \nfor the figures showing gravitational lensing by strings, to Martin \nWhite for providing the CMBR figures, and to the Department of Energy \n(USA) for research support. \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Introduction}\nThis paper is concerned with the efficient simulation of parameter-dependent partial differential equations (PDEs), with a parameter varying in a given set $\\mathcal{G}$. For complex physical systems, computational costs can be huge. It may happen, for instance in the context of parameter optimization or real time simulations in an industrial context, that the same problem needs to be solved for several parameter values.\n\nIn such cases, different model order reductions (MOR) like the reduced basis methods have been proposed (see eg~\\cite{rb, hesthaven2016certified}) based on POD or greedy selection of the reduced basis, the reduced basis elements being computed accurately enough through a high fidelity code. In these approaches, the efficient implementation of the reduced method, leading to reductions in the computational time, requires to be able to deeply enter into the high fidelity code, in order to compute offline, a key ingredient which saves the implementation costs online. This can be tedious, even impossible when the code has been bought, as it is often the case in an industrial context. The Non Intrusive Reduced Basis methods (NIRB)~\\cite{madaychakir,NIRB1} has been proposed in this framework. This method is useful to reduce computational costs of parametric-dependent PDEs in a non intrusive way.  Unlike other MOR, the NIRB method does not require to modify the solver code and hence does not depend on the numerical approach underlying the code. \nThis method, based on two grids, one fine where high fidelity computations are done offline and one coarse which is used online, has been introduced in~\\cite{madaychakir,NIRB1}. It was presented and analysed in the case where the high fidelity code is based on a finite element solver. In these papers, an optimal error estimate is recovered and illustrated with numerical simulations. The method can be extended to other classical discretizations but the key ingredient is a better approximation rate in the $L^2$ norm than in the energy norm, thanks to the Aubin-Nitsche's  trick that is easy for variational approximations. In addition, the degrees of freedom in FVM don't have the same status as in FEM and the transfer of information from one grid to another must be adapted. \nThe aim of this paper is to propose the adaptation of the NIRB method to FV and to propose the numerical analysis able to recover the classical error estimate with Finite Volume (FV) schemes. \n\n\\subsection*{The Non Intrusive Reduced Basis Method.}\n\nLet $\\Omega$ be an open bounded domain in $\\mathbb{R}^d$ with $d\\leq 3$.\nThe NIRB method in the context of a high fidelity solver of finite element or finite volume types involves two partitioned meshes, one fine mesh  $\\mathcal{M}_h$ and one coarse mesh $\\mathcal{M}_H$, where $h$ and $H$ are the respective sizes of the meshes and $h<\\! < H$. The size $h$ (res. $H$) is defined as $h=\\underset{K \\in \\mathcal{M}_h}{\\max\\ } h_{K}$ (resp. $H=\\underset{K \\in \\mathcal{M}_H}{\\max\\ } H_{K}$) where the diameter $h_K$ (or $H_K$) of any element $K$ in a mesh is equal to $\\underset{x,y \\in K}{\\sup \\ } |x-y|$. \\\\ \n\nAs it is classical in other reduced basis methods, the NIRB method is based on the assumption (assumed or actually checked) that the manifold of all solutions $\\mathcal{S} = \\{u(\\mu), \\mu \\in \\mathcal{G} \\}$ has a small Kolmogorov n-width $\\varepsilon(n)$~\\cite{kolmo}. This leads to the fact that very few well chosen solutions are sufficient to approximate well any element in~$\\mathcal{S}$. These well chosen elements are called the snapshots. In this frame, the method\n is based on two steps :\none offline step and one online. The ``offline'' part is costly in time because the snapshots must be generated with a  high fidelity code on the fine mesh $\\mathcal{M}_h$. The  ``online'' step \nis performed on the coarse mesh $\\mathcal{M}_H$, and thus much less expensive than a high fidelity computation. This algorithm remains effective as the offline part is performed only once and in advance and also independently from the online stage. The online stage can then be done as many times  as desired.\n\n   \n\\begin{itemize}\n    \\item In the offline part, several snapshots are  computed on the fine mesh for different well chosen parameters in the parameter set $\\mathcal{G}$ with the (fine and costly) solver. The best way to determine the required parameters is through a greedy procedure~\\cite{veroy2003posteriori, barrault2004empirical, greedy} if available or through an SVD approach. \n    \\item The online part consists in computing a coarse solution with the same solver for some (new) parameter $\\mu \\in \\mathcal{G}$ and then $L^2$-project this (coarse) solution on the (fine) reduced basis. This results in an improved approximation, in the sense that we may retrieve almost fine error estimates with a much lower computational cost. \n\\end{itemize}\n\\subsection*{Motivation and earlier works.}\nSeveral papers have underlined the efficiency of the NIRB method in the finite element context, illustrated both with numerical results presenting error plots and the online part compurational time~\\cite{madaychakir,NIRB1, NIRB2,NIRB3}. \nHowever, to the best of our knowledge, works with Finite Volume (FV) schemes have not yet been studied with a non intrusive approach~\\cite{iliev2013two,stabile2017pod,haasdonk2008reduced,TPFABR,Casenave2014}, and they are often preferred to finite element methods in an industrial context. Thanks to recent works on super-convergence~\\cite{supconv2}, and with some technical subtleties, we are now able to generalize the two-grids method which is non intrusive to FV methods and propose the numerical analysis of this method.\n\n\\begin{center}\n\\begin{tikzpicture}[every text node part\/.style={align=center}]\n     \n      \n     \\node[draw=black,thick,rectangle,rounded corners=3pt,fill=orange!40] (snap)at(0,4.5){   \\textcolor{black}{ Snapshots:} \\\\  \\textcolor{black}{\\{$u_h(\\mu_1),\\ldots,u_h(\\mu_{N})$\\}} \\\\ \\textcolor{black}{computed on a fine mesh $\\mathcal{M}_h$}};\n        \n\n    \\draw[color=orange] (-5,2) rectangle (5,6);\n    \\draw (0,5.7) node[] {\\textbf{Offline}};\n     \n \\node[draw,rectangle,color=blue!80,minimum width=10cm,minimum height=4cm](aaaa)at(0,-0.5){};\n    \n \\draw (0,1.2) node[] {\\textbf{Online}};\n        \\node[draw,circle,rounded corners=2pt,fill=black!25] (solver)at(-5,1.6){\\quad   Solver   \\quad};\n        \\node[draw,rectangle,rounded corners=3pt,fill=orange!40] (basis)at(0,2.8){Orthonormal basis: $(\\Phi_i^h)_{i=1,\\ldots,N}$};\n        \\node[draw=yellow,thick,rectangle,rounded corners=3pt,fill=blue!30](coarsesol)at(0,0.3){Coarse solution: $u_H(\\mu)$ computed on $\\mathcal{M}_H$};\n    \n         \\node[draw,rectangle,rounded corners=3pt,fill=blue!25] (app)at(0,-1.8){\\textcolor{red}{NIRB approximation: $u_{hH}^N(\\mu)$} };\n         \n          \\draw[->,draw=orange,fill=blue,line width=0.3mm] (snap) -- (basis) node[midway,left]{\\textcolor{orange}{Greedy}};\n         \\draw[->,draw=orange,fill=blue,line width=0.3mm] (basis) -- (aaaa){};\n   \n          \\draw[->,draw=black,fill=blue,line width=0.3mm] (solver) -- (snap) node[midway,left]{};\n         \\draw[->,draw=black,fill=blue,line width=0.3mm] (solver) -- (coarsesol) node[midway,left]{};\n  \\draw[->,draw=blue!80,fill=blue,line width=0.3mm] (coarsesol) -- (app) node[midway,left]{\\textcolor{blue!80}{$L^2$-projection: set $u_{hH}^N(\\mu) =$}\\\\ \\textcolor{blue!80}{ $\\overset{N}{\\underset{i=1}{\\sum}}(u_H(\\mu),\\Phi_i^h)\\Phi_i^h$}};\n  \n   \n\n    \n\n\\end{tikzpicture}\n\n  \\end{center}\n\n\\subsection*{Main results.}\n\nIn the context of $P_1$-FEM solvers, the works~\\cite{madaychakir,NIRB1} retrieve an estimate error of the order of $\\mathcal{O}(h+H^2)$ in the energy norm using the Aubin-Nitsche's Lemma \\cite{FE} for the coarse grids solution  (for a reduced basis dimension large enough). With FV schemes, no equivalent of the Aubin-Nitsche's lemma is available, instead, we consider the class of Hybrid Mimetic Mixed methods (HMM) schemes for elliptic equations and use a super-convergence property proven in (~\\cite{supconv1,supconv2,tpfascheme}).\\\\\nLet us consider the following linear second-order parameter dependent problem as our model problem:\n\\begin{subnumcases}\n     \\strut - \\ \\textrm{div} (A(\\mu) \\nabla u)=f \\textrm{ in } \\Omega,& \\label{ellip1}\\\\\n    \\label{ellip2} u = 0 \\textrm{ on } \\partial \\Omega, &\n\\end{subnumcases}\nwhere  $f\\in L^2(\\Omega)$,  $\\mu$ is  a parameter in a set $\\mathcal{G}$, and for any $\\mu \\in \\mathcal{G}$,  $A(.; \\mu):\\Omega \\to\n\\mathbb{R}^{d\\times d}$ is measurable, bounded, uniformly elliptic, and $A(\\textbf{x}; \\mu)$ is symmetric for a.e. $\\textbf{x} \\in \\Omega$.\n\n\n\nUnder general hypotheses, it is well known that this problem has a unique solution.\n\n\\noindent The usual weak formulation for problem \\eqref{ellip1}-\\eqref{ellip2} reads:\\\\\nFind $u \\in H_0^1(\\Omega)$ such that,\n\\begin{equation}\n    \\forall v \\in H_0^1(\\Omega), \\quad a(u,v;\\mu)=(f,v),\n    \\label{varellip}\n\\end{equation}\nwhere \\begin{equation*}\n    a(w,v;\\mu)=\\int_{\\Omega}A(\\textbf{x};\\mu)\\nabla w(\\textbf{x})\\cdot \\nabla v (\\textbf{x})\\ d\\textbf{x},\\quad \\forall w, v\\in H_0^1(\\Omega).\n\\end{equation*}\n\nThe main result of this paper is the following estimate:\n\n\\begin{thrm} [NIRB error estimate for hMFD solvers]\n       \\label{th11}\n   Let $u_{hH}^N(\\mu)$ be the reduced solution projected on the fine mesh and generated with the hMFD solver with the unknowns defined on $\\textbf{x}_k=\\overline{\\textbf{x}}_K$ (the cell centers of mass), and $u(\\mu)$ be the exact solution of \\eqref{varellip} under an $H^2$ regularity assumption \\eqref{h2reg} (which will be stated later), then the following estimate holds \\\\\n\\begin{equation}\n    \\norm{u(\\mu) - u_{hH}^N(\\mu)}_{\\mathcal{D}}\\leq \\varepsilon(N) +C_1 h + C_2(N) H^2, \n    \\label{estimationNIRB}\n\\end{equation}\nwhere $C_1$ and $C_2$ are constants independent of $h$ and $H$,$C_2$ depends on $N$, the number of functions in the basis, and $\\norm{\\cdot}_{\\mathcal{D}}$ is the discrete norm introduced in section \\ref{sect2}, and $\\varepsilon$ depends of the Kolmogorov n-width. If $H$ is such as $H^2\\sim h$, and $\\varepsilon(N)$ small enough, it results in an error estimate in $\\mathcal{O}(h)$.\n\\end{thrm}\nNote that if $H$ is chosen such as $H^2 \\sim h$ and $\\varepsilon(N)$ small enough, it results in an error estimate in $\\mathcal{O}(h)$. \n\n\\subsection*{Outline  of  the  paper.} The  rest  of  this  paper  is  organized  as  follows. In section \\ref{sect2} we describe the mathematical context. In section \\ref{sect3} we recall the two-grids method. Section \\ref{sect4} is devoted to the proof of theorem \\ref{th11} with the hybrid-Mimetic Finite Difference scheme (hMFD). Section \\ref{sect5} generalizes theorem \\ref{th11} to other schemes, such as the Two Point Flux Approximation (TPFA). In the last section, the implementation is discussed and we illustrate the estimate with several numerical results on the NIRB method. \n\n\\section{Mathematical Background}\n\\label{sect2}\n\\subsection{The Hybrid Mimetic Finite Difference method (hMFD)}\n\nIn this section, we recall the hybrid-Mimetic Finite Difference method (hMFD)~\\cite{hmfd} and all the notations that will be necessary for the analysis of NIRB method in this finite volume context. \n\nThis scheme uses interface values and fluxes as unknowns. The hMFD scheme, which is part of the family of Hybrid Mimetic Mixed methods (HMM) (\\cite{cindy,hmm2,jd,hmm3,supconv1}), is a finite volume method despite its name. Indeed hMFD scheme relies on both a flux balance equation and on a local conservativity of numerical fluxes. HMM also includes mixed finite volume schemes (MFV) \\cite{mfv} and hybrid finite volume schemes (HFV), a hybrid version of the SUSHI scheme~\\cite{hfv}. This scheme is built on a general mesh, namely a polytopal mesh, which is a star-shaped mesh regarding the unknowns of the cells. \\\\\n\n\\noindent Describing the hMFD method requires to introduce the Gradient Discretisation (GD) method~\\cite{cindy}, which is a general framework for the definition and the convergence analysis of many numerical methods (finite element, finite volume, mimetic finite difference methods, etc).\\\\\nThe GD schemes involve a discete space, a reconstruction operator and a gradient operator, which taken together are called a Gradient Discretisation. Selecting the gradient discretisation mostly depends on the boundary conditions (BCs). We now introduce the definition of GD for Dirichlet BCs as in~\\cite{cindy} and the GD scheme associated to our model problem. \\\\\n\n\n\n\\begin{dfntn}(Gradient Discretisation) For homogeneous\nDirichlet BCs, a gradient discretisation $\\mathcal{D}$ is a triplet $(X_{\\mathcal{D},0},\\Pi_{\\mathcal{D}},\\nabla_{\\mathcal{D}})$, where the space of degrees of\nfreedom $X_{\\mathcal{D},0}$ is a discrete version of the continuous space $H_0^1(\\Omega)$.\n\\begin{itemize}\n    \\item $\\Pi_{\\mathcal{D}}:X_{\\mathcal{D},0}\\to L^2(\\Omega)$ is a function reconstruction operator that relates an element of $X_{\\mathcal{D},0}$ to a function in $L^2(\\Omega)$.\n\\item $\\nabla_{\\mathcal{D}}:X_{\\mathcal{D},0}\\to L^2(\\Omega)^d$ is a gradient reconstruction in $L^2(\\Omega)$ from the degrees of freedom. It must be chosen such that $\\norm{\\cdot}_{\\mathcal{D}}=\\norm{\\nabla_{\\mathcal{D}} \\cdot}_{{L^2(\\Omega)}^d}$ is a norm on $X_{\\mathcal{D},0}$.\n\\end{itemize}\n\\end{dfntn}\n\\noindent In what follows, we will refer to $\\Pi_\\mathcal{D}^H$ or $\\Pi_\\mathcal{D}^h$ depending on the mesh considered and for the gradient reconstruction too (respectively $\\nabla_\\mathcal{D}^H$ or $\\nabla_\\mathcal{D}^h$).\\\\\n\n\\begin{dfntn} (Gradient discretisation scheme) For the variational form \\eqref{varellip}, the related gradient discretisation scheme with the new operators is defined by:\\\\\nFind $u_{\\mathcal{D}} \\in X_{\\mathcal{D},0}$ such that, $\\forall v_\\mathcal{D} \\in X_{\\mathcal{D},0},$\n\\begin{equation}\n     \\int_{\\Omega}A(\\mu) \\nabla_{\\mathcal{D}} u_{\\mathcal{D}} \\cdot \\nabla_{\\mathcal{D}} v_\\mathcal{D} \\ d\\textbf{x}=\\int_{\\Omega}f\\  \\Pi_{\\mathcal{D}} v_\\mathcal{D} \\ d\\textbf{x}.\n    \\label{variat}\n\\end{equation}\n\\end{dfntn}\nWe will use two general polytopal meshes (Definition 7.2~\\cite{cindy}) which are admissible meshes for the hMFD scheme.\n\n\\begin{dfntn} (Polytopal mesh) Let $\\Omega$ be a bounded polytopal open subset of $\\mathbb{R}^d (d\\geq 1).$ A polytopal mesh of $\\Omega$ is a quadruplet $\\mathcal{T} = (\\mathcal{M}, \\mathcal{F},\\mathcal{P}, \\mathcal{V})$, where:\n\\begin{enumerate}\n    \\item $\\mathcal{M}$  is a finite family of non-empty connected polytopal open disjoint subsets $\\Omega$ (the cells) such that $\\overline{\\Omega}=\\underset{K\\in\\mathcal{M}}{\\cup}\\overline{K}.$ \n    For any $K\\in \\mathcal{M}$,\\ $|K|>0$ is the measure of $K$ and $h_K$ denotes the diameter of $K$.\n\\item $\\mathcal{F} =  \\mathcal{F}_{int}\\cup  \\mathcal{F}_{ext}$ is a finite family of disjoint subsets of $\\overline{\\Omega}$ (the edges of the mesh in 2D), such that any $\\sigma \\in \\mathcal{F}_{int}$ is contained in $\\Omega$ and any $\\sigma \\in \\mathcal{F}_{ext}$ is contained in $\\partial \\Omega$. Each $\\sigma \\in \\mathcal{F}$ is assumed to be a nonempty open subset of a hyperplane of $\\mathbb{R}^d$, with a positive $(d-1)$-dimensional measure $|\\sigma|$. Furthermore, for all $K \\in \\mathcal{M}$, there exists a subset\n$\\mathcal{F}_K$ of $\\mathcal{F}$ such that $\\partial K = \\underset{\\sigma \\in \\mathcal{F}_K}{\\cup}\\overline{\\sigma}$. \n We assume that for all $\\sigma \\in \\mathcal{F}, \\mathcal{M}_{\\sigma}=\\{K\\in \\mathcal{M}:\\ \\sigma \\in \\mathcal{F}_K\\}$ has exactly one element and $\\sigma \\subset \\partial \\Omega$ or $\\mathcal{M}_{\\sigma}$ has two elements and $\\sigma \\subset\\Omega$.\nThe center of mass is $\\overline{\\textbf{x}}_{\\sigma}$, and, for $K \\in \\mathcal{M}$ and $\\sigma \\in \\mathcal{F}_K$, $\\textbf{n}_{K,\\sigma}$ is the (constant) unit vector normal to $\\sigma$ outward to $K$.\n\\item $\\mathcal{P}$ is a family of points of $\\Omega$ indexed by $\\mathcal{M}$ and $\\mathcal{F}$, denoted by   $\\mathcal{P}=((\\textbf{x}_K)_{K\\in \\mathcal{M}},(\\textbf{x}_{\\sigma})_{\\sigma \\in \\mathcal{F}} )$, such that for all $K \\in \\mathcal{M},\\ \\textbf{x}_K \\in K$ and for all $\\sigma \\in \\mathcal{F},\\  \\textbf{x}_{\\sigma} \\in \\sigma$. We then denote by $d_{K,\\sigma}$ the signed orthogonal distance between $\\textbf{x}_K$ and $\\sigma \\in \\mathcal{F}_K$, that is:\\\\\n$d_{K,\\sigma } = (\\textbf{x} - \\textbf{x}_K) \\cdot \\textbf{n}_{K,\\sigma}$, for all $\\textbf{x} \\in \\sigma. $\nWe then assume that each cell $K \\in \\mathcal{M}$ is strictly star-shaped with respect to $\\textbf{x}_K$, that is $d_{K,\\sigma} > 0$ for all $\\sigma \\in \\mathcal{F}_K$. This implies that for all $\\textbf{x}  \\in K$, the line segment $[\\textbf{x}_K, \\textbf{x}]$\nis included in $K$. We denote $\\overline{\\textbf{x}}_K $ the center of mass of $K$ and by $\\overline{\\textbf{x}}_{\\sigma} $ the one of $\\sigma$.\nFor all $K \\in \\mathcal{M}$ and $\\sigma \\in \\mathcal{F}_K$, we denote by $D_{K,\\sigma}$ the cone with vertex $\\textbf{x}_K$ and basis $\\sigma$, that is $D_{K,\\sigma}= \\{t\\textbf{x}_K + (1 - t)\\mathbf{y}, t \\in (0, 1), \\mathbf{y} \\in \\sigma \\}$. \n\\item $\\mathcal{V}$ is a set of points (the vertices of the mesh). For $K \\in \\mathcal{M}$, the set of\nvertices of $K$, i.e. the vertices contained in $\\overline{K}$, is denoted $\\mathcal{V}_K$. Similarly,\nthe set of vertices of $\\sigma \\in F$ is $\\mathcal{V}_\\sigma$.\n\\end{enumerate}\n\\end{dfntn}\n\n\\begin{figure}[H]\n\\begin{center}\n\\begin{pspicture}(-1,-1)(6,6)\n  \\psline(0,0)(2,-0.75)(5,2.75)(3,4.5)(0.5,5)(0,0)\n  \\psline[linestyle=dashed](0,0)(1.8,1.7)\n\n  \\psline[linestyle=dashed](0.5,5)(1.8,1.7)\n  \\psline[linestyle=dashed](3,4.5)(1.8,1.7)\n  \\psline[linestyle=dashed](5,2.75)(1.8,1.7)\n  \\psline[linestyle=dashed](2,-0.75)(1.8,1.7)\n\\psdots[dotstyle=+,dotscale=2.5,dotangle=45](1.8,1.7)\n\\uput[d](1.7,1.45){$\\mathbf{x}_K$}\n\\pspolygon[opacity=0.3,fillstyle=solid,fillcolor=yellow](1.8,1.7)(2,-0.75)(5,2.75)\n\\uput[d](5.1,2.75){$\\sigma$}\n\\uput[d](2.1,4.){$K$}\n\\psline[linestyle=dashed]{<->}(1.8,1.7)(3.15,0.67)\n\\uput[d](2.6,1.1){$d_{K,\\sigma}$}\n\\uput[d](3.3,1.8){$D_{K,\\sigma}$}\n\\psline{->}(4.15,1.7)(4.6,1.4)\n\\uput[d](4.65,2.1){$\\mathbf{n}_{K,\\sigma}$}\n\\end{pspicture}\n\\end{center}\n\\caption{A cell $K$ of a polytopal 2D mesh}\n\\end{figure}\n\nThe regularity factor for the mesh is\n\\begin{equation}\n  \\theta=\\underset{\\sigma \\in \\mathcal{F}_{int},\\mathcal{M}_{\\sigma=\\{K,K'\\}}}{\\max }\n  \\frac{d_{K,\\sigma}}{d_{K',\\sigma}}+\\underset{K\\in \\mathcal{M}}{\\max}(\\underset{\\sigma\\in \\mathcal{F}_K}{\\max}\\frac{h_K}{d_{K,\\sigma}}+\\textrm{Card}(\\mathcal{F}_K)).\\label{reg}\n\\end{equation}\n\nIn what follows, we will consider two polytopal meshes. The fine mesh will be denoted $\\mathcal{T}^h= (\\mathcal{M}^h,\\mathcal{F}^h,\\mathcal{P}^h,\\mathcal{V}^h)$ and $\\mathcal{T}^H= (\\mathcal{M}^H,\\mathcal{F}^H,\\mathcal{P}^H,\\mathcal{V}^H)$ will be referred to as the coarse mesh.\\\\\n\n\\noindent All HMM schemes require to choose one point inside each mesh cell $\\textbf{x}_K$, and in the case the center of mass $\\overline{\\textbf{x}}_K$ is chosen, then the scheme corresponds to hMFD and superconvergence is well known~\\cite{supconv1,supconv2,hmm2}. Until section \\ref{sect5}, we will consider $\\textbf{x}_K=\\overline{\\textbf{x}}_K$. \\\\\n\n\\begin{dfntn} (Hybrid Mimetic Mixed gradient discretisation (HMM-GD)) \\\\\nFor hMFD scheme, we use the following GD (Definition 13.1.1~\\cite{cindy}):\n\\begin{enumerate}\n    \\item Let $X_{\\mathcal{D},0}= \\{v=((v_K)_{K\\in \\mathcal{M}}, (v_{\\sigma})_{\\sigma \\in \\mathcal{F}}): v_K\\in \\mathbb{R},v_{\\sigma}\\in \\mathbb{R},v_{\\sigma}=0 \\textrm{ if } \\sigma \\in \\mathcal{F}_{ext}\\},$\n    \\item $\\Pi_\\mathcal{D}:X_{\\mathcal{D},0}\\to L^2(\\Omega)$ is the following piecewise constant reconstruction on the mesh:\\\\\n$\\forall v \\in X_{\\mathcal{D},0}, \\forall K \\in \\mathcal{M},$\n\\begin{equation}\n  \\Pi_{\\mathcal{D}}v(\\textbf{x})=v_K \\textrm{ on }K.\\label{PiD}\n\\end{equation}\n\\item $\\nabla_{\\mathcal{D}}:X_{\\mathcal{D},0}\\to L^2(\\Omega)^d$ reconstructs piecewise constant gradients on the cones $(D_{K,\\sigma})_{K\\in\\mathcal{M},\\sigma \\in \\mathcal{F}_K}$:\\\\\n$\\forall v \\in X_{\\mathcal{D},0}, \\forall K \\in \\mathcal{M}, \\forall \\sigma \\in \\mathcal{F},\\\\$\n\\begin{equation}\n    \\nabla_{\\mathcal{D}}v(\\textbf{x})=\\nabla_{K}v+\\frac{\\sqrt{d}}{d_{K,\\sigma}}[\\mathcal{L}_K R_K(v)]_{\\sigma}\\ \\textbf{n}_{K,\\sigma} \\textrm{ on } D_{K,\\sigma},\\label{NablaD}\n\\end{equation}\nwhere:\n\\begin{itemize}\n    \\item $ \\nabla_{K}v=\\frac{1}{|K|}\\sum_{\\sigma\\in \\mathcal{F}_K}|\\sigma| v_{\\sigma} \\textbf{n}_{K,\\sigma}$,\n \\item $R_K:X_{\\mathcal{D},0}\\to \\mathbb{R}^{\\mathcal{F}_K} $ is given by $ R_K(v)=(R_{K,\\sigma(v)}))_{\\sigma \\in \\mathcal{F}_K}$ with $ R_{K,\\sigma}(v)=v_{\\sigma}-v_K-\\nabla_{K}v\\cdot(\\overline{\\textbf{x}}_{\\sigma}-\\textbf{x}_K)$,\n \\item $\\mathcal{L}_K$ is an isomorphism of the space $Im(R_K)$.\n \\end{itemize}\n\\end{enumerate}\n\\end{dfntn}\n As explained in the introduction of this chapter, hMFD,\nHFV and MFV schemes are three different presentations of the same method. With the notations above, any HMM method for the weak form \\eqref{varellip} can be written (Equation 2.25~\\cite{jd}):\\\\\nFind $u_\\mathcal{T}(\\mu) \\in X_{\\mathcal{D},0}$ such that, for all $v_\\mathcal{T} \\in X_{\\mathcal{D},0},$\n\\begin{equation*}\n   \\mu  \\underset{K\\in\\mathcal{M}}{\\sum}|K| A_K(\\mu)\\nabla_Ku_\\mathcal{T}\\cdot \\nabla_K v_\\mathcal{T}+\\underset{K\\in\\mathcal{M}}{\\sum} R_K(v_\\mathcal{T})^T\\mathbb{B}_KR_K(u_\\mathcal{T})=\\underset{K\\in\\mathcal{M}}{\\sum}v_K \\int_Kf(\\textbf{x}) \\ d\\textbf{x},\n    \\label{hmm}\n\\end{equation*}\nwhere $A_K(\\mu)$ is the $L_2$ projection of $A(\\mu)$ on $K$ and $\\mathbb{B}_K = ((\\mathbb{B}_K)_{\\sigma,\\sigma'})_{\\sigma,\\sigma' \\in \\mathcal{F}_K}$ is a symmetric positive definite matrix.\\\\\nFor a certain choice of isomorphism $\\mathcal{L}_K:\\Im(R_K)\\to \\Im(R_K)$, the HMM scheme \\eqref{hmm} is identical to GDs \\eqref{variat} (see Theorem 13.7~\\cite{cindy}).\\\\\n\n\n\\noindent We now introduce the super-convergence property which will be used in the proof of theorem \\ref{th11}, but first we need the following $H^2$ regularity assumption (which holds if $A$ is Lipschitz continuous and $\\Omega$ is convex):\\\\\n\n\\noindent Let $f \\in L^2(\\Omega),$ the solution $u(\\mu)$ to \\eqref{varellip} belongs to $H^2(\\Omega),$ and\n\\begin{equation}\n    \\norm{u(\\mu)}_{H^2(\\Omega)}+\\norm{A(\\mu)\\nabla u(\\mu)}_{H^1(\\Omega)^d} \\leq C \\norm{f}_{L^2(\\Omega)},\n    \\label{h2reg}\n\\end{equation}\nwith $C$ depending only on $\\Omega$ and $A$.\\\\\n\nWe define $\\pi_{\\mathcal{M}^h}:L^2(\\Omega)\\to L^2(\\Omega)$ as the orthogonal projection on the piecewise constant functions on $\\mathcal{M}^h$ that is\n\\begin{equation*}\n  \\forall \\Psi \\in L^2(\\Omega), \\quad \\forall K \\in \\mathcal{M}^h,\\quad \\pi_{\\mathcal{M}^h}\\Psi=\\frac{1}{|K|}\\int_K \\Psi(\\textbf{x})\\  d\\textbf{x} \\textrm{ on }K. \n\\end{equation*}\n\n\\begin{thrm}[Super-convergence for hMFD schemes, Theorem 4.7~\\cite{supconv2}]\nLet $d\\leq 3$, $f\\in H^1(\\Omega)$, and $u(\\mu)$ be the solution of $\\eqref{varellip}$ under assumption \\eqref{h2reg}. Let $\\mathcal{T}_h$ be a polytopal mesh, and $\\mathcal{D}$ be an HMM gradient discretisation on $\\mathcal{T}_h$ with the unknowns defined on $\\textbf{x}_K$, and let $u_{h}(\\mu)$ be the solution of the corresponding GD. Recall that $\\overline{\\textbf{x}}_K$ is the center of mass of $K$ and we are in the case where $\\textbf{x}_K=\\overline{\\textbf{x}}_K$. Then, considering $u_{\\mathcal{P}}(\\mu)$ as the piecewise constant function on $\\mathcal{M}_h$ equal to $u(\\overline{\\textbf{x}}_K;\\mu)$ on $K\\in \\mathcal{M}$, there exists $C>0$ not depending on $h$ such that\n\n\\begin{equation}\n   \\norm{\\Pi_{\\mathcal{D}}^hu_{h}(\\mu)-u_{\\mathcal{P}}(\\mu)}_{L^2(\\Omega)}\\leq C(\\norm{f}_{H^1(\\Omega)}+\\norm{u}_{H^2(\\Omega)})h^2.\n    \\label{superconvhmm}\n\\end{equation}\n\\end{thrm}\n\nTo recover \\eqref{superconvhmm} in the case $\\textbf{x}_K=\\overline{\\textbf{x}}_K$, we used the Lemma 7.5 of~\\cite{supconv2} on the approximation of $H_2$ functions by affine functions  to obtain\n\\begin{equation*}\n  \\norm{\\pi_{\\mathcal{M}^h}u(\\mu)-u_{\\mathcal{P}}(\\mu)}_{L^2(\\omega)}\\leq Ch^2\\norm{u}_{H^2(\\Omega)}.\n  \\end{equation*}\n\n\\begin{rmrk}\n\n We consider here $\\norm{\\cdot}_\\mathcal{D}$ as the discrete semi norm of $H^1$ so as not to make notations too cumbersome. The usual discrete semi-norm for $H^1$ is defined by\n\\begin{equation}\n    \\forall v \\in \\mathcal{T},\\ |v|_{\\mathcal{T},2}^2= \\underset{K \\in \\mathcal{M}}{\\sum} \\underset{\\sigma \\in \\mathcal{F}_K}{\\sum} |\\sigma| d_{K,\\sigma}\\left\\rvert\\frac{v_{\\sigma}-v_K}{d_{K,\\sigma}}\\right\\rvert^2. \\label{discretenorme}\n\\end{equation}\nUnder some conditions on the regularity of the mesh, this norm and $\\norm{\\nabla_\\mathcal{D} \\cdot}_{L^2(\\Omega)^d}$ are equivalent (Lemma 13.11~\\cite{cindy}).\n\\end{rmrk}\n\nIn the next section, we recall the offline and the online parts of the two-grids algorithm. \n\n\\section{The Non Intrusive Reduced Basis method (NIRB)}\n\\label{sect3}\nThis section recalls the main steps of the two-grids method algorithm~\\cite{madaychakir, NIRB1}.\\\\ \n\n\\noindent Let $u_h(\\mu)$ refer to the hMFD solution on a fine polytopal mesh $\\mathcal{T}_h$, with cells $\\mathcal{M}_h$ and respectively $u_H(\\mu)$ the one on a coarse mesh $\\mathcal{T}_H$, with the cells $\\mathcal{M}_H$.\\\\\n\n\\noindent We briefly recall the NIRB method. Points 1 and 2 are in the offline part, and the others are done online. \n\\begin{enumerate}\n    \\item Several snapshots $\\{u_{h}(\\mu_i)\\}_{i \\in \\{1,\\dots N\\}}$ are computed with the hMFD scheme $\\eqref{variat}$, where $\\mu_i\\in \\mathcal{G}\\quad \\forall i=1,\\cdots,N$. The space generated by the snapshots is named $X_h^N = Span\\{ u_h(\\mu_1 ), \\dots ,  u_h(\\mu_N)\\}$.\n    \\item We generate the basis functions $(\\Phi_i^h)_{i=1,\\cdots,N}$ with the following steps:\n    \\begin{itemize}\n    \\item A Gram-Schmidt procedure is used, which involves $L^2$ orthonormalization of the reconstruction functions. \n    \\item This procedure is also completed by the following eigenvalue problem:\n    \\begin{numcases}\n      \\strut \\textrm{Find } \\Phi^h \\in X_h^N, \\textrm{ and } \\lambda \\in \\mathbb{R} \\textrm{ such that: }    \\nonumber\\\\\n        \\forall v \\in X_h^N, \\int_{\\Omega} \\nabla_\\mathcal{D}^h \\Phi^h \\cdot \\nabla_\\mathcal{D}^h v \\ d\\textbf{x}= \\lambda \\int_{\\Omega} \\Pi_\\mathcal{D}^h \\Phi^h \\cdot \\Pi_\\mathcal{D}^h v \\ d\\textbf{x}, \\label{orthoTPFA} \n    \\end{numcases} \n    \n    where $\\nabla_\\mathcal{D}^h$ and $\\Pi_\\mathcal{D}^h$ are respectively the discrete gradient and the discrete reconstruction operators as in the definition of the HMM GD (\\eqref{PiD}, \\eqref{NablaD}). We get an increasing sequence of eigenvalues $\\lambda_i$, and orthogonal eigenfunctions $(\\Pi_\\mathcal{D}^h\\Phi_i^h)_{i=1,\\cdots,N}$, orthonormalized in $L^2(\\Omega)$ and orthogonalized in $H^1(\\Omega)$, such that ($\\Phi_i^h)_{i=1,\\cdots,N}$ defines a new basis of the space $X_h^N$.\n    \\end{itemize}\n    \\item We solve the hMFD problem \\eqref{variat} on the coarse mesh $\\mathcal{T}_H$ for a new parameter $\\mu \\in \\mathcal{G}$. Let us denote by $u_H(\\mu)$ the solution.\n    \\item We then introduce $\\alpha_i^H(\\mu)=\\int_{\\Omega}\\Pi_\\mathcal{D}^H u_H(\\mu) \\cdot \\Pi_\\mathcal{D}^h \\Phi_i^h\\ d\\textbf{x}$. The approximation used in the two-grids method is $u_{Hh}^N(\\mu)=\\overset{N}{\\underset{i=1}{\\sum}}\\alpha_i^H(\\mu) \\Pi_\\mathcal{D}^h \\Phi_i^h.$\n    \\end{enumerate}\n    In the next section, we detail how to obtain the classical finite elements estimate in $\\mathcal{O}(h)$ on the NIRB algorithm, when the snapshots are computed with the hMFD GD using a polytopal mesh.\n    \\section{NIRB error estimate}\n    \\label{sect4}\n   In this section, we consider $\\textbf{x}_K=\\overline{\\textbf{x}}_K$ which is the case with the hMFD scheme. Some other cases will be detailed in section \\ref{sect5}. \\\\\nWe now continue with the proof of theorem \\ref{th11}. \n\n\\begin{proof}\nIn this proof, we will denote $A \\lesssim B$ for $A \\leq  CB$ with $C$ not depending on $h$ or $H$.\n\nWe use the triangle inequality on $\\norm{u(\\mu)-u_{Hh}^N(\\mu)}_{\\mathcal{D}}$ to get\n\\begin{align}\n\\norm{u(\\mu)-u_{Hh}^N(\\mu)}_{\\mathcal{D}}&\\leq \\norm{u(\\mu)-\\Pi_\\mathcal{D}^h u_{h}(\\mu)}_{\\mathcal{D}}+\\norm{\\Pi_\\mathcal{D}^hu_h(\\mu)-u_{hh}^N(\\mu)}_{\\mathcal{D}}+ \\norm{u_{hh}^N(\\mu)-u_{Hh}^N(\\mu)}_{\\mathcal{D}} \\nonumber \\\\\n&=:T_1+T_2+T_3, \n\\label{triangleinequality}\n\\end{align}\nwhere $u_{hh}^N(\\mu)=\\overset{N}{\\underset{i=1}{\\sum}}\\alpha_i^h(\\mu) \\Pi_\\mathcal{D}^h \\Phi_i^h,$ and $\\alpha_i^h(\\mu)=\\int_{\\Omega}\\Pi_\\mathcal{D}^hu_h(\\mu) \\cdot \\Pi_\\mathcal{D}^h \\Phi_i^h\\ d\\textbf{x}$.\\\\\n\\begin{itemize}\n    \\item The first term $T_1$ can be estimated using a classical result for finite volume schemes (Consequence of Proposition 13.16~\\cite{cindy}) such that: \n\\begin{equation}\n    \\norm{u(\\mu)-\\Pi_\\mathcal{D}^h u_{h}(\\mu)}_{\\mathcal{D}} \\lesssim  h\\norm{u}_{H^2(\\Omega)}.\n    \\label{VFestim}\n\\end{equation}\n\\item The best achievable error in the uniform sense of a fine solution projected into $X_h^N$ relies on the notion of Kolmogorov n-width (Theorem 20.1~\\cite{nummod}). \nIf $\\mathcal{K}$ is a compact set in a Banach space $V$, the Kolmogorov n-width of $\\mathcal{K}$ is \\begin{equation}\n    d_n(\\mathcal{K})=\\underset{\\textrm{ dim} (V_n)\\leq n}{\\inf}\\quad \\underset{v\\in \\mathcal{K}}{\\sup}\\quad \\underset{w\\in V_n}{\\min} \\norm{v-w}_{V}.\n\\end{equation} \nHere we suppose the set of all the reconstructions of the solutions $\\mathcal{S}=\\{\\Pi_\\mathcal{D}^h u_h(\\mu),\\mu \\in \\mathcal{G}\\}$ has a low complexity which means for an accuracy $\\varepsilon=\\varepsilon(N)$ related to the Kolmogorov n-width of the manifold $\\mathcal{S}$, there exists a set of parameters $\\{\\mu_1,\\dots,\\mu_N\\} \\in \\mathcal{G}$, such that~\\cite{cohen,madaychakir,NIRB1,greedy}\n\\begin{equation}\n  T_2=\\norm{\\Pi_\\mathcal{D}^h u_h(\\mu)-\\overset{N}{\\underset{i=1}{\\sum}}\\alpha_i^h(\\mu)\\Pi_\\mathcal{D}^h \\Phi_i^h}_{\\mathcal{D}} \\leq \\varepsilon(N).\\label{kolmotpfa}\n  \\end{equation}\n\n\\item Consider the term $T_3$ now. We will need the following proposition where the property of super-convergence for the hMFD scheme \\eqref{superconvhmm} is used.\\\\\n\n\\begin{prpstn}\nLet $u_H(\\mu)$ be the solution of the hMFD on a polytopal mesh $\\mathcal{T}_H$ with the unknowns on $\\textbf{x}_K=\\overline{\\textbf{x}}_K$. Denote by $u(\\mu)$ the exact solution of equation  $\\eqref{varellip}$, and let $(\\Phi_i^h)_{i=1,\\cdots,N}$ be the basis functions of the NIRB algorithm, then there exists a constant $C=C(N)>0$ not depending on $H$ or $h$,and depending on $N$ such that\n\\begin{equation}\n    \\left\\rvert\\int_{\\Omega}(u(\\mu)-\\Pi_\\mathcal{D}^H u_H(\\mu))\\cdot \\Pi_\\mathcal{D}^h \\Phi_i^h \\ d\\textbf{x} \\right\\rvert\\lesssim ((\\norm{\\Phi_i}_{L^{\\infty}(\\Omega)}+C(N))\\norm{u}_{H^2(\\Omega)}+\\norm{f}_{H^1(\\Omega)})H^2. \\label{EstimH2prop}\n\\end{equation}\n\\end{prpstn}\n\\begin{proof}\nSince $\\mathcal{M}_H$ is a partition of $\\Omega$,  \n\\begin{align}\n    \\int_{\\Omega}\\Pi_\\mathcal{D}^H u_H(\\mu) \\cdot \\Pi_\\mathcal{D}^h \\Phi_i^h \\ d\\textbf{x} &=\\underset{K\\in \\mathcal{M}_H}{\\sum} \\int_K \\Pi_\\mathcal{D}^H u_H(\\mu)  \\cdot \\Pi_\\mathcal{D}^h \\Phi_{i}^h \\ d\\textbf{x}. \\label{avecH}\n\\end{align}\nTo begin with, let $\\Pi_0^H:\\mathcal{C}(\\Omega)\\to L^\\infty(\\Omega)$ be the piecewise constant projection operator on $\\mathcal{M}_H$ such that:\n\\begin{equation}\n     \\Pi_0^H \\Phi(\\textbf{x})=\\Psi(\\textbf{x}_K),\\quad \\textrm{ on } K ,\\quad \\forall K \\in \\mathcal{M}_H,\\quad  \\forall \\Psi \\in \\mathcal{C}(\\Omega).\\label{pi0}\\\\\n\\end{equation}\nWe use the triangle inequality on the left part of the inequality \\eqref{EstimH2prop} and therefore,  \n\\begin{align}\n\\left\\lvert \\int_{\\Omega}(u(\\mu)-\\Pi_\\mathcal{D}^H u_H(\\mu))\\cdot \\Pi_\\mathcal{D}^h \\Phi_i^h \\ d\\textbf{x} \\right\\rvert &\\leq \\left\\rvert \\int_{\\Omega} (u(\\mu)-\\Pi_0^H u(\\mu))\\cdot \\Pi_\\mathcal{D}^h \\Phi_i^h \\ d\\textbf{x} \\right\\rvert + \\left\\rvert \\int_{\\Omega} (\\Pi_0^H u(\\mu)-\\Pi_\\mathcal{D}^H u_H(\\mu))\\cdot\\Pi_\\mathcal{D}^h \\Phi_i^h \\ d\\textbf{x} \\right\\rvert,\\nonumber \\\\\n&=:T_{3,1} + T_{3,2}. \\label{decompose1}\n\\end{align}\n\n\\begin{itemize}\n\n    \\item We first consider the term $T_{3,1}$, but beforehand the estimate of $T_{3,1}$ requires the use of a further operator which we now introduce. \n  Each cell $K\\in \\mathcal{M}_H$ is star-shaped with respect to a ball $B_K$ centered in $\\textbf{x}_K$ of radius $\\rho=\\underset{\\sigma \\in \\mathcal{F}_K}{\\textrm{min }} d_{K,\\sigma}$ (Lemma B.1~\\cite{cindy}). We then use an averaged Taylor polynomial as in~\\cite{FE} but simplified. Let us consider the following polynomial of $u(\\mu)$ averaged over $B_K$:\\\\\n\\begin{equation}\n    Q_K u(\\textbf{x};\\mu)=\\frac{1}{|B_K|}\\int_{B_K} [u(\\mathbf{y};\\mu)+D^1 u(\\mathbf{y};\\mu)(\\textbf{x}-\\mathbf{y})] \\ d\\mathbf{y}.\\\\\n    \\label{defQ}\n\\end{equation}\nThis polynomial is of degree less or equal to $1$ in $\\textbf{x}$.\\\\\nLet us introduce $\\Pi_1^H:H^1(\\Omega) \\cap \\mathcal{C}(\\Omega)\\to \\mathbb{R}$, the piecewise affine projection operator on $\\mathcal{M}_H$ such that:\n\\begin{equation}\n     \\Pi_1^H \\Psi=Q_K \\Psi(\\textbf{x}), \\quad \\textrm{ on } K, \\quad \\forall K \\in \\mathcal{M}_H, \\quad \\forall \\Psi \\in H^1(\\Omega)\\cap \\mathcal{C}.\\label{pi01}\\\\\n\\end{equation}\nWith the triangle inequality, we obtain \\begin{align}\n    T_{3,1} &\\leq \\left\\rvert\\int_{\\Omega} (u(\\mu)-\\Pi_1^Hu(\\mu))\\cdot \\Pi_\\mathcal{D}^h \\Phi_i^h \\ d\\textbf{x} \\right\\rvert + \\left\\rvert\\int_{\\Omega} (\\Pi_1^Hu(\\mu)-\\Pi_0^H u(\\mu))\\cdot\\Pi_\\mathcal{D}^h \\Phi_i^h \\ d\\textbf{x} \\right\\rvert,\\nonumber \\\\\n    &=: T_{3,1,1} + T_{3,1,2}. \\label{decompose2}\n\\end{align}\n \n  \\begin{itemize}\n       \\item \n    Using the Cauchy-Schwarz inequality,\n\\begin{align}\n    T_{3,1,1} & \\leq\\int_{\\Omega} \\left\\rvert (u(\\mu)-\\Pi_1^Hu(\\mu))\\cdot \\Pi_\\mathcal{D}^h \\Phi_i^h \\right\\rvert\\ d\\textbf{x} , \\nonumber \\\\ \n    &\\leq \\norm{u(\\mu)-\\Pi_1^Hu(\\mu)}_{L^2(\\Omega)}\\norm{\\Pi_{\\mathcal{D}}^h \\Phi_i^h}_{L^2(\\Omega)},  \\nonumber \\\\ \n    \\label{331e} &\\leq \\norm{u(\\mu)-\\Pi_1^Hu(\\mu)}_{L^2(\\Omega)}, \\textrm{ since $\\Pi_\\mathcal{D}^h\\Phi_i^h\\quad  \\forall i=1,\\cdots,N$ are normalized in } L^2.\n    \\end{align}\n    \nLet $K\\in\\mathcal{M}_H$. As in Proposition 4.3.2~\\cite{FE},\n\\begin{equation}\n\\underset{\\textbf{x}\\in \\overline{K}}{\\sup\\ } |u(\\textbf{x};\\mu)-Q_K u(\\textbf{x};\\mu)|\\lesssim H_K^{2-\\frac{d}{2}} |u(\\mu)|_{H^2(K)}.\n\\label{supestim}\n\\end{equation}\nSince $K\\subset B(\\textbf{x},H)$ for all $\\textbf{x} \\in K$, \n\\begin{align}\n    |K|&\\leq |B(\\textbf{x}_K,H)|=|B(0,1)|H_K^d. \\label{K}\n\\end{align}\nThus, with the inequalities \\eqref{K} and \\eqref{supestim}, we get  \\begin{equation}\n    \\underset{\\textbf{x}\\in \\overline{K}}{\\sup\\ } |u(\\textbf{x};\\mu)-Q_K u(\\textbf{x};\\mu)|\\lesssim H_K^{2}|K|^{-\\frac{1}{2}} |u(\\mu)|_{H^2(K)},\n\\end{equation}\ntaking the square and integrating over $K$, we obtain\n\\begin{equation}\n    \\int_K |u(\\mu)-\\Pi_1^H u(\\mu)|^2 \\ d\\textbf{x} \\lesssim H_K^{4} |u(\\mu)|_{H^2(K)}^2,\n\\end{equation}\nand summing over $K$ yields \n\\begin{equation}\n    \\norm{u(\\mu)-\\Pi_1^H u(\\mu)}_{L^2(\\Omega)}\\lesssim H^2 |u(\\mu)|_{H^2(\\Omega)}.\n    \\label{bramblehilbert}\n\\end{equation}\nThe inequality \\eqref{bramblehilbert}, combined with \\eqref{331e}, entails that\n\\begin{equation}\n    T_{3,1,1}  \\lesssim H^2 |u(\\mu)|_{H^2(\\Omega)}.\n    \\label{311estim}\n\\end{equation}\n\n\\item The term $T_{3,1,2}$ can be estimated using a continuous reconstruction of $\\Phi_i^h$, denoted by $\\Phi_i$ .\\\\\nWith the triangle inequality,\n\\begin{align}\n  \\left\\rvert \\int_{\\Omega} (\\Pi_1^H u(\\mu)-\\Pi_0^H u (\\mu))\\cdot \\Pi_{\\mathcal{D}}^h \\Phi_i^h \\ d\\textbf{x}\\right\\rvert&\\leq \\left\\rvert \\int_{\\Omega}(\\Pi_1^Hu(\\mu)-\\Pi_0^H u(\\mu))( \\Pi_{\\mathcal{D}}^h\\Phi_i^h-\\Pi_0^H\\Phi_i) \\ d\\textbf{x} \\right\\rvert \\nonumber \\\\\n  &+ \\left\\rvert \\int_{\\Omega} (\\Pi_1^Hu(\\mu)-\\Pi_0^H u(\\mu))\\cdot\\Pi_0^H\\Phi_i) \\ d\\textbf{x} \\right\\rvert. \\label{inegalit\u00e9triangulairesurT312}\n  \\end{align}\n\nSince $\\textbf{x}_K$ is the center of mass, $\\int_{K} \\textbf{x} \\ d\\textbf{x} =|K|\\textbf{x}_K$. Therefore,\n\\begin{equation}\n    \\int_{K}Q_K u(\\textbf{x};\\mu)\\ d\\textbf{x} = |K| Q_K u (\\textbf{x}_K; \\mu).\n    \\label{centerofmass}\n\\end{equation}\nFrom the inequality \\eqref{supestim}, \n\\begin{align}\n    |Q_K u(\\textbf{x}_K;\\mu)-u(\\textbf{x}_K;\\mu)|\\lesssim H_K^{2-\\frac{d}{2}} |u(\\mu)|_{H^2(K)}.\n    \\label{estimsurxk}\n\\end{align}\n\nThus, since $\\Pi_0^H\\Phi_i$ is constant on each cell $ K \\in \\mathcal{M}_H$, and $|K|\\lesssim H_K^d$ \\eqref{K},\n\\begin{align}\n    \\left\\rvert \\int_{\\Omega} (\\Pi_1^H u(\\mu)-\\Pi_0^Hu(\\mu)) \\cdot \\Pi_0^H\\Phi_i \\ d\\textbf{x} \\right\\rvert &= \\left\\rvert \\underset{K\\in \\mathcal{M}_H}{\\sum}  \\int_K (Q_K u(\\textbf{x};\\mu)-u(\\textbf{x}_K;\\mu)) \\cdot \\Pi_0^H \\Phi_i \\ d\\textbf{x}  \\right\\rvert, \\nonumber \\\\\n    &\\leq \\underset{K\\in \\mathcal{M}_H}{\\sum}  \\left\\rvert  \\Phi_i(x_K) \\int_{K} Q_Ku(\\textbf{x};\\mu) -u(\\textbf{x}_K;\\mu)  \\ d\\textbf{x}  \\right\\rvert, \\nonumber \\\\\n    &\\leq \\underset{K\\in \\mathcal{M}_H}{\\sum} |K| \\left\\rvert  \\Phi_i(x_K) (Q_Ku(\\textbf{x}_K;\\mu) -u(\\textbf{x}_K;\\mu))  \\right\\rvert,   \\textrm{ from \\eqref{centerofmass}},\\nonumber \\\\\n    &\\leq  \\norm{\\Phi_i}_{L^{\\infty}(\\Omega)}\\underset{K\\in \\mathcal{M}_H}{\\sum}  |K|\\left\\rvert Q_Ku(\\textbf{x}_K;\\mu)  - u(\\textbf{x}_K;\\mu) \\right\\rvert, \\nonumber  \\\\\n&\\lesssim \\norm{\\Phi_i}_{L^{\\infty}(\\Omega)} \\underset{K\\in \\mathcal{M}_H}{\\sum}  |K|H_K^{2-\\frac{d}{2}}|u(\\mu)|_{H^2(K)} \\textrm{ from \\eqref{estimsurxk}}, \\nonumber \\\\\n&\\lesssim \\norm{\\Phi_i}_{L^{\\infty}(\\Omega)} \\underset{K\\in \\mathcal{M}_H}{\\sum}  H_K^{2+\\frac{d}{2}}|u(\\mu)|_{H^2(K)}. \\label{test}\n\\end{align}\nSince Card$(\\mathcal{M}_H)\\simeq H^{-d} $, using the Cauchy-Schwarz inequality, the inequality \\eqref{test} becomes \n\n\\begin{align}\n   \\left\\rvert \\int_{\\Omega} (\\Pi_1^H u(\\mu)-\\Pi_0^Hu(\\mu)) \\cdot \\Pi_0^H\\Phi_i \\ d\\textbf{x} \\right\\rvert &\\lesssim \\norm{\\Phi_i}_{L^{\\infty}}H^{2} (\\underset{K\\in \\mathcal{M}_H}{\\sum} |u(\\mu)|_{H^2(K)}^2)^{\\frac{1}{2}}, \\nonumber \\\\\n    &=\\norm{\\Phi_i}_{L^{\\infty}}|u(\\mu)|_{H^2(\\Omega)}H^2 \\label{inter},\n\\end{align}\n \nwhich implies that there exists a constant $\\widetilde{C_1}>0$ not depending on $h$ or $H$ such that  \\eqref{inegalit\u00e9triangulairesurT312} becomes \n \n  \n  \\begin{equation}\n      T_{3,1,2} \\leq  \\int_{\\Omega} \\left\\rvert(\\Pi_1^H u(\\mu)-\\Pi_0^H u(\\mu))( \\Pi_{\\mathcal{D}}^h\\Phi_i^h-\\Pi_0^H\\Phi_i) \\right\\rvert d\\textbf{x}  + \\widetilde{C_1}\\norm{\\Phi_i}_{L^{\\infty}}|u(\\mu)|_{H^2(\\Omega)}H^2.\n      \\label{derniereestimsurT312}\n  \\end{equation}\n  \nFrom the Cauchy-Schwarz inequality and the inequality \\eqref{derniereestimsurT312},\n\\begin{align}\n        T_{3,1,2}\n        &\\leq \\norm{\\Pi_1^Hu(\\mu)-\\Pi_0^H u(\\mu)}_{L^2(\\Omega)}\\norm{\\Pi_{\\mathcal{D}}^h \\Phi_i^h-\\Pi_0^H\\Phi_i}_{L^2(\\Omega)} + \\widetilde{C_1}\\norm{\\Phi_i}_{L^{\\infty}}|u(\\mu)|_{H^2(\\Omega)}H^2.        \\label{enoH2}\n\\end{align}\nFrom Bramble-Hilbert's Lemma (see~\\cite{FE}), we deduce that\n\\begin{equation}\n    \\norm{u(\\mu)-\\Pi_0^H u(\\mu)}_{L^2(\\Omega)}\\lesssim H \\norm{u(\\mu)}_{H^2(\\Omega)}.\\label{pi0eq}\n\\end{equation}\nFor the first term in the right-hand side of \\eqref{enoH2}, from \\eqref{bramblehilbert}-\\eqref{pi0eq} and the triangle inequality,  \n\\begin{align}\n    \\norm{\\Pi_1^H u(\\mu)- \\Pi_0^H u (\\mu)}_{L^2(\\Omega)} &\\leq \\norm{\\Pi_1^H u(\\mu)- u (\\mu)}_{L^2(\\Omega)} + \\norm{u(\\mu)- \\Pi_0^H u (\\mu)}_{L^2(\\Omega)}, \\nonumber \\\\\n    &\\lesssim H\\norm{u}_{H^2(\\Omega)}, \\textrm{ neglecting the estimate in $H^2$}, \\label{1erterme}\n\\end{align} \nand the inequality \\eqref{pi0eq} and the classical finite volume estimate as for \\eqref{VFestim} ($\\Pi_{\\mathcal{D}}^h \\phi_i^h$ being a linear combination of the family $(\\Pi_{\\mathcal{D}}^hu_j^h)_{j=1}^N,\\ \\forall i=1,\\cdots,N$) implies that there exists $\\widetilde{C_2}=\\widetilde{C_2}(N)>0$ not depending of $H$ or $h$ but depending on $N$ such that\n    \\begin{align}\n      \\norm{\\Pi_{\\mathcal{D}}^h \\Phi_i^h-\\Pi_0^H \\Phi_i}_{L^2(\\Omega)} & \\leq \\norm{\\Pi_{\\mathcal{D}}^h \\Phi_i^h-\\Phi_i}_{L^2(\\Omega)}+\\norm{\\Phi_i-\\Pi^H_0 \\Phi_i}_{L^2(\\Omega)}, \\nonumber \\\\\n      &\\leq \\widetilde{C_2}(N) H, \\textrm{ neglecting the estimate in $h$}. \\label{2emeterme}\n\\end{align}\nFrom \\eqref{1erterme}-\\eqref{2emeterme}, we deduce that each $L^2$ term is in $\\mathcal{O}(H)$ in the product of the right-hand side of \\eqref{enoH2}. Hence the equation \\eqref{inegalit\u00e9triangulairesurT312} yields to \n\\begin{equation}\n\\label{premier2terme}\n     T_{3,1,2}=\\left\\rvert \\int_{\\Omega} (\\Pi_1^H u(\\mu)-\\Pi_0^H u(\\mu))\\cdot \\Pi_{\\mathcal{D}}^h \\Phi_i^h \\ d\\textbf{x} \\right\\rvert \\lesssim (\\widetilde{C_1}\\norm{\\Phi_i}_{L^{\\infty}(\\Omega)}+\\widetilde{C_2}(N)) \\norm{u}_{H^2(\\Omega)}H^2.\n\\end{equation}\n   \\end{itemize} \n\n  \n\\item We now proceed with the estimate on $T_{3,2}:$\\\\\nWith the super-convergence property on the hMFD scheme \\eqref{superconvhmm}, and with the normalization of $\\Pi_{\\mathcal{D}}^h \\Phi_i^h $ in $L^2(\\Omega)$\n\\begin{align}\n    \\left\\rvert \\int_{\\Omega} (\\Pi_{\\mathcal{D}}^H u_H(\\mu)-\\Pi_0^H u(\\mu)) \\cdot \\Pi_{\\mathcal{D}}^h \\Phi_i^h \\ d\\textbf{x} \\right\\rvert & \\leq  \\int_{\\Omega} \\left \\rvert (\\Pi_{\\mathcal{D}}^H u_H(\\mu)-\\Pi_0^H u(\\mu)) \\cdot \\Pi_{\\mathcal{D}}^h \\Phi_i^h\\right\\rvert \\ d\\textbf{x} , \\nonumber \\\\\n    &\\leq \\norm{\\Pi_{\\mathcal{D}}^H u_H(\\mu)-\\Pi_0^H u(\\mu)}_{L^2(\\Omega)}\\norm{\\Pi_{\\mathcal{D}}^h \\Phi_i^h}_{L^2(\\Omega)}, \\nonumber \\\\\n    &\\lesssim (\\norm{f}_{H^1(\\Omega)} +\\norm{u}_{H^2(\\Omega)})H^2. \\label{term2bis}\n\\end{align}\n\\end{itemize}\n\n\nCombining the estimates \\eqref{311estim}-\\eqref{premier2terme}-\\eqref{term2bis} with the inequalities \\eqref{decompose1}-\\eqref{decompose2}, this results in the inequality \\eqref{EstimH2prop}.\n\\end{proof}\n\n\\noindent We now consider the third term $T_3=\\norm{u_{hh}^N(\\mu)-u_{Hh}^N(\\mu)}_{\\mathcal{D}}$.\n\\begin{align}\n    T_3&= \\norm{\\overset{N}{\\underset{i=1}{\\sum}}\\alpha_i^h(\\mu) \\Pi_{\\mathcal{D}}^h \\Phi_i^h-\\overset{N}{\\underset{i=1}{\\sum}}\\alpha_i^H(\\mu) \\Pi_{\\mathcal{D}}^h \\Phi_i^h}_{\\mathcal{D}}, \\nonumber \\\\ \n    &\\leq \\overset{N}{\\underset{i=1}{\\sum}} \\left\\rvert \\alpha_i^h(\\mu)-\\alpha_i^H(\\mu)\\right\\rvert \\norm{\\Pi_{\\mathcal{D}}^h \\Phi_i^h}_{\\mathcal{D}}, \\nonumber \\\\\n     &= \\overset{N}{\\underset{i=1}{\\sum}} \\left\\rvert(\\Pi_{\\mathcal{D}}^h u_h(\\mu)-\\Pi_{\\mathcal{D}}^H u_H(\\mu),\\Pi_{\\mathcal{D}}^h \\Phi_i^h)_{L^2}\\right\\rvert \\norm{\\Pi_{\\mathcal{D}}^h \\Phi_i^h}_{\\mathcal{D}}. \\label{on sensert}\n\\end{align}\nFrom \\eqref{orthoTPFA}, we get that \n\\begin{align}\n   \\norm{\\Pi_\\mathcal{D}^h\\Phi_i^h}_{\\mathcal{D}}^2\\ &=\\ \\int_{\\Omega} |\\nabla_\\mathcal{D} \\Phi_i^h|^2 \\ d\\textbf{x} \\ = \\ \\lambda_i \\norm{\\Pi_\\mathcal{D} \\Phi_i^h}_{L^2(\\Omega)}^2 \\leq  \\underset{i=1,\\cdots,N}{\\max}(\\lambda_i)= \\lambda_N.\\label{lambdamaxortho}\n\\end{align}\nTherefore we obtain from \\eqref{on sensert} and \\eqref{lambdamaxortho},\n\\begin{equation}\n    T_3\\leq \\sqrt{\\lambda_N} \\overset{N}{\\underset{i=1}{\\sum}} \\left\\rvert (\\Pi_{\\mathcal{D}}^h u_h(\\mu)-\\Pi_{\\mathcal{D}}^H u_H(\\mu),\\Pi_{\\mathcal{D}}^h \\Phi_i^h)_{L^2}\\right\\rvert. \\label{on sensert bis}\\\\\n\\end{equation}\nUsing the triangle inequality in the right-hand side of \\eqref{on sensert bis}, \n\\begin{equation}\n    T_3\\leq \\sqrt{\\lambda_N} \\overset{N}{\\underset{i=1}{\\sum}} \\left\\rvert(\\Pi_{\\mathcal{D}}^h u_h(\\mu)-u(\\mu),\\Pi_\\mathcal{D}^h \\Phi_i^h)\\right\\rvert+\\left\\rvert(u(\\mu)-\\Pi_{\\mathcal{D}}^H u_H(\\mu),\\Pi_{\\mathcal{D}}^h \\Phi_i^h)\\right\\rvert. \\label{on sensert bisbis}\\\\\n\\end{equation}\nFrom Proposition 1, with the estimate \\eqref{EstimH2prop} applied to $\\mathcal{M}_h$ and $\\mathcal{M}_H$, neglecting the estimate in $\\mathcal{O}(h^2)$\n\\begin{equation}\n    T_3 \\lesssim \\sqrt{\\lambda_N}N((\\norm{\\Phi_i}_{L^{\\infty}(\\Omega)}+C(N))\\norm{u}_{H^2(\\Omega)}+\\norm{f}_{H^1(\\Omega)}) H^2.\n    \\label{T3}\n\\end{equation}\n\\end{itemize}\nThe conclusion follows combining the estimates on $T_1, T_2$ and $T_3$ (estimates \\eqref{VFestim},\\eqref{kolmotpfa} and \\eqref{T3}).\n\\begin{align}\n\\norm{u(\\mu)-u_{Hh}^N(\\mu)}_\\mathcal{D} &= \\norm{u(\\mu)-\\sum_{i=1}^N  \\alpha_i^H(\\mu) \\Pi_{\\mathcal{D}}^h \\Phi^h_i}_{\\mathcal{D}}, \\nonumber \\\\\n& \\leq \\varepsilon(N) + C_1 h + C_2(N) H^2 \\sim \\mathcal{O}(h) \\textrm{ if }h \\sim H^2.\n\\end{align}\n\\end{proof}\n\\section{Results on other FV schemes}\n\\label{sect5}\nIn this section, we consider the case where $\\textbf{x}_K$ is not the center of mass, as it is the case for some FV schemes. Therefore the left hand side of the inequality \\eqref{test} cannot be estimated using equation \\eqref{centerofmass}.\nThe unknowns $\\textbf{x}_K$ are not necessarily the centers of mass of the cells neither with HMM methods nor with the Two-Point Flux Approximation (TPFA) scheme~\\cite{boyer,tpfascheme}. Under the following superadmissibility condition \\begin{equation}\n    \\forall K\\in \\mathcal{M}_H, \\ \\sigma \\in \\mathcal{F}_K: \\ \\textbf{n}_{K,\\sigma}=\\frac{\\overline{\\textbf{x}}_{\\sigma}-\\textbf{x}_K}{d_{K,\\sigma}},\n\\end{equation} the TPFA scheme is a member of the the HMM family schemes (~\\cite{cindy} section 13.3 ,\\ ~\\cite{jd} section 5.3) with the choice $\\mathcal{L}_K=Id$. This leads to take $\\textbf{x}_K$ as the circumcenters of the cells with 2D  triangular meshes. Theorem 1.1 holds in 2D on uniform rectangles with TPFA since the superadmissibility condition is satisfied in this case where $\\textbf{x}_K$ is the centre of mass of the cells. The TPFA scheme is rather simple to implement, and therefore we will present in the last section  numerical results with a TPFA solver.\n We will use the definition of a local grouping of the cells as in~\\cite{supconv2} (Definition 5.1).\nWe will extend the Theorem 1.1 in the case where such groupings of cells exist. \n\n\\begin{dfntn} (Local grouping of the cells). Let $\\mathcal{T}_H$ be a polytopal mesh of $\\Omega$. A local grouping of the cells of $\\mathcal{T}_H$ is a partition $\\mathfrak{G}$ of $\\mathcal{M}_H$, such that for each $G\\in \\mathfrak{G}$, letting $U_G:= \\underset{K \\in G}{\\cup} K$, there exists a ball $B_G\\subset U_G$ such that $U_G$ is star-shaped with respect to $B_G$. This implies that for all $\\textbf{x} \\in U_G$ and all $\\mathbf{y} \\in B_G$, the line segment $[\\textbf{x},\\mathbf{y}]$ is included in $U_G$. We then define the regularity factor of $\\mathfrak{G}$ \\begin{equation}\n    \\mu_G:= \\quad \\underset{G\\in \\mathfrak{G}}{\\max }\\quad \\textrm{Card}(G)\\quad +\\quad \\underset{G\\in \\mathfrak{G}}{\\max}\\quad \\underset{K\\in G}{\\max}\\quad \\frac{H_K}{\\textrm{diam}(B_G)},\n\\end{equation}\n\nand, with $e_K=\\overline{\\textbf{x}}_K-\\textbf{x}_K$, and\n\\begin{equation}\n  e_{G}:=\\frac{1}{|U_G|}\\ \\underset{K\\in G}{\\sum}\\ |K|\\ \\mathbf{e}_K ,\\quad \\forall G \\in \\mathfrak{G},\n\\end{equation}\n\n\\begin{equation}\n    e_{\\mathfrak{G}}:= \\underset{G \\in \\mathfrak{G}}{\\max} \\quad \\left\\rvert \\mathbf{e}_G \\right\\rvert.\n\\end{equation}\n\\end{dfntn}\n\\noindent\nNote that we are interested in situations where $|\\mathbf{e}_{G}|=\\left\\rvert\\frac{1}{|U_G|}\\sum_{K\\in G} |K|\\ \\mathbf{e}_K\\right\\rvert$ is much smaller than $|\\mathbf{e}_K|\\quad \\forall K\\in G$.\nThe aim of this section is to estimate the left hand side of the inequality $\\eqref{test}$ in $\\mathcal{O}(H^2)$ using a local grouping of the cells. The rest of the proof remains unchanged. \n\nWe will need the following Theorem of super-convergence for HMM schemes with local grouping (Theorem 5.4~\\cite{supconv2}).\n\n\\begin{thrm}[Super-convergence for HMM schemes with local grouping (Theorem 5.4 ~\\cite{supconv2})]\nLet $f\\in H^1(\\Omega)$, and $u(\\mu)$ be the solution of $\\eqref{varellip}$ under assumption \\eqref{h2reg}. Let $\\mathcal{T}_h$ be a polytopal mesh, and $\\mathcal{D}$ be an HMM gradient discretisation on $\\mathcal{T}_h$ and $e_{\\mathfrak{G}}$ be a local grouping, and let $u_{h}(\\mu)$ be the solution of the corresponding GD. Then, considering $u_{\\mathcal{P}}(\\mu)$ as the piecewise constant function on $\\mathcal{M}_h$ equal to $u(\\textbf{x}_K;\\mu)$ on $K\\in \\mathcal{M}$, there exists $C$ not depending on $H$ or $h$ such that \n\\begin{equation}\n    \\norm{\\Pi_{\\mathcal{D}}^hu_{h}(\\mu)-u_{\\mathcal{P}}(\\mu)}_{L^2(\\Omega)}\\leq C \\norm{f}_{H^1(\\Omega)}(h^2+e_{\\mathfrak{G}}) .\n    \\label{superconvhmm2}\n\\end{equation}\n\\end{thrm}\n\\begin{thrm}[NIRB error estimate with local grouping] Let $u_{hH}^N(\\mu)$ be the reduced solution projected on the fine mesh and generated with the hMFD solver with the unknowns defined on $\\textbf{x}_k$ such that $e_{\\mathfrak{G}}$ is in $\\mathcal{O}(H^2)$ on the coarse mesh, and $u(\\mu)$ be the exact solution of \\eqref{varellip} under assumption \\eqref{h2reg}, then the following estimate holds \\\\\n\\begin{equation}\n    \\norm{u(\\mu) - u_{hH}^N(\\mu)}_{\\mathcal{D}}\\leq \\varepsilon(N) +C_1 h + C_2(N) H^2, \n   \\label{estimationNIRBBB}\n\\end{equation}\nwhere $C_1$ and $C_2$ are constants independent of $h$ and $H$,$C_2$ depends on $N$, the number of functions in the basis, and $\\norm{\\cdot}_{\\mathcal{D}}$ is the discrete norm introduced in section \\ref{sect2}, and $\\varepsilon$ depends of the Kolmogorov n-width. If $H$ is such as $H^2\\sim h$, and $\\varepsilon(N)$ small enough, it results in an error estimate in $\\mathcal{O}(h)$.\n\n\\end{thrm}\n\\begin{proof}\nIn this proof, we will still denote $A\\lesssim B $ for $A \\leq C B$ with $C$ not depending on $h$ or $H$. The reconstruction $\\Phi_i$ of $\\Phi_i^h$ must belong to $W^{1,\\infty}.$\nAs in the previous section, with the equation \\eqref{centerofmass},\n\\begin{align}\n    \\left\\rvert \\int_{\\Omega} (\\Pi_1^H u(\\mu)-\\Pi_0^Hu(\\mu)) \\cdot \\Pi_0^H\\Phi_i \\ d\\textbf{x} \\right\\rvert &= \\left\\rvert \\underset{K\\in \\mathcal{M}_H}{\\sum}  \\int_K (Q_K u(\\textbf{x};\\mu)-u(\\textbf{x}_K;\\mu)) \\cdot \\Pi_0^H \\Phi_i \\ d\\textbf{x}  \\right\\rvert, \\nonumber \\\\\n    &=  \\left\\rvert \\underset{K\\in \\mathcal{M}_H}{\\sum} \\Phi_i(\\textbf{x}_K) |K| \\Big[Q_Ku(\\overline{\\textbf{x}}_K;\\mu) -u(\\textbf{x}_K;\\mu)\\Big] \\right\\rvert, \\nonumber \\\\\n    &\\leq \\left\\rvert \\underset{K\\in \\mathcal{M}_H}{\\sum} \\Phi_i(\\textbf{x}_K) |K| \\Big[Q_Ku(\\overline{\\textbf{x}}_K;\\mu)  - Q_Ku(\\textbf{x}_K;\\mu)\\Big] \\right\\rvert \\nonumber \\\\&+  \\norm{\\Phi_i}_{L^{\\infty}(\\Omega)} \\underset{K\\in \\mathcal{M}_H}{\\sum}  \\left\\rvert Q_Ku(\\textbf{x}_K;\\mu) -u(\\textbf{x}_K;\\mu) \\right\\rvert \\textrm{ from the triangle inequality}. \n    \\label{test2}\n\\end{align}\n\nAs in the previous section \\eqref{inter},\n\\begin{equation}\n    \\norm{\\Phi_i}_{L^{\\infty}(\\Omega)}\\underset{K\\in \\mathcal{M}_H}{\\sum} |K| \\left\\rvert Q_Ku(\\textbf{x}_K;\\mu)-u(\\textbf{x}_K;\\mu)\\right\\rvert \\lesssim \\norm{\\Phi_i}_{L^{\\infty}(\\Omega)}\\norm{u}_{H^2(\\Omega)}H^2 \\label{inter22}.\n\\end{equation}\nThus, the inequality \\eqref{test2} yields\n\\begin{equation}\n  \\left\\rvert \\int_{\\Omega} (\\Pi_1^H u(\\mu)-\\Pi_0^Hu(\\mu)) \\cdot \\Pi_0^H\\Phi_i \\ d\\textbf{x} \\right\\rvert \\lesssim\\left\\rvert \\underset{K\\in \\mathcal{M}_H}{\\sum} \\Phi_i(\\textbf{x}_K) |K| \\Big[Q_Ku(\\overline{\\textbf{x}}_K;\\mu)  - Q_Ku(\\textbf{x}_K;\\mu)\\Big] \\right\\rvert + \\norm{\\Phi_i}_{L^{\\infty}(\\Omega)}\\norm{u}_{H^2(\\Omega)}H^2.\n  \\label{test3}\n\\end{equation}\n\nWith the triangle inequality, the first term in \\eqref{test3} becomes\n\n\\begin{align}\n  \\left\\rvert \\underset{K\\in \\mathcal{M}_H}{\\sum} \\Phi_i(\\textbf{x}_K) |K| \\Big[Q_Ku(\\overline{\\textbf{x}}_K;\\mu)  - Q_Ku(\\textbf{x}_K;\\mu)\\Big] \\right\\rvert &\\lesssim \\left\\rvert \\underset{K\\in \\mathcal{M}_H}{\\sum} \\Big[\\Phi_i(\\textbf{x}_G)+(\\Phi_i(\\textbf{x}_K)-\\Phi_i(\\textbf{x}_G))\\Big] |K| \\Big[Q_Ku(\\overline{\\textbf{x}}_K;\\mu)  - Q_Ku(\\textbf{x}_K;\\mu)\\Big] \\right\\rvert, \\nonumber \\\\\n  &\\lesssim \\left\\rvert \\underset{K\\in \\mathcal{M}_H}{\\sum} \\Phi_i(\\textbf{x}_G) |K| \\Big[Q_Ku(\\overline{\\textbf{x}}_K;\\mu)  - Q_Ku(\\textbf{x}_K;\\mu)\\Big] \\right\\rvert \\nonumber \\\\& +  \\norm{\\nabla \\Phi_i}_{L^{\\infty}(\\Omega)}  \\underset{K\\in \\mathcal{M}_H}{\\sum} H_K  |K|  \\left\\rvert Q_Ku(\\overline{\\textbf{x}}_K;\\mu)  - Q_Ku(\\textbf{x}_K;\\mu) \\right\\rvert  \\textrm{ since diam($U_G$) $\\leq \\mu_G H_K$ }.  \\label{test4}\n\\end{align}\n\nUsing the decomposition of the mesh in patches $U_G$ and with the definition of $Q_K$, the first term of \\eqref{test4} gives\n\\begin{align}\n  \\left\\rvert \\underset{K\\in \\mathcal{M}_H}{\\sum} \\Phi_i(\\textbf{x}_G) |K| \\Big[Q_Ku(\\overline{\\textbf{x}}_K;\\mu)  - Q_Ku(\\textbf{x}_K;\\mu)\\Big] \\right\\rvert &\\leq \\left\\rvert  \\underset{G\\in \\mathfrak{G}}{\\sum} \\underset{K\\in G}{\\sum} \\Phi_i(\\textbf{x}_G) \\frac{|K|}{|B_K|} \\int_{B_K}D^1u(\\mathbf{y})\\cdot \\mathbf{e}_K \\ d\\mathbf{y} \\right\\rvert, \\nonumber \\\\\n  &\\leq \\underset{G\\in \\mathfrak{G}}{\\sum} \\norm{\\Phi_i}_{L^{\\infty}(G)}\\left\\rvert \\underset{K\\in G}{\\sum} \\Big( \\frac{1}{|B_K|} \\int_{B_K}D^1u(\\mathbf{y}) \\ d\\mathbf{y} \\Big) |K|\\ \\mathbf{e}_K  \\right\\rvert.\\label{test8}\n\\end{align}\n\nUsing the definition of $Q_K$ \\eqref{defQ}, the second term in \\eqref{test4} yields\n\\begin{align}\n  \\norm{\\nabla \\Phi_i}_{L^{\\infty}(\\Omega)}  \\underset{K\\in \\mathcal{M}_H}{\\sum} H_K |K|  \\left\\rvert Q_Ku(\\overline{\\textbf{x}}_K;\\mu)  - Q_Ku(\\textbf{x}_K;\\mu) \\right\\rvert &=  \\norm{\\nabla \\Phi_i}_{L^{\\infty}(\\Omega)} \\underset{K\\in \\mathcal{M}_H}{\\sum} H_K  \\frac{|K|}{|B_K|}  \\left\\rvert \\int_{B_K} D^1u(\\mathbf{y})\\cdot \\mathbf{e}_K \\ d\\mathbf{y} \\right\\rvert, \\nonumber \\\\\n  &\\lesssim \\norm{\\nabla \\Phi_i}_{L^{\\infty}(\\Omega)} \\underset{K\\in \\mathcal{M}_H}{\\sum} H_K^2 \\norm{\\nabla u}_{L^1(B_K)}, \\textrm{ since $|B_K|\\geq \\theta_H^{-1} |K|$ \\eqref{reg}}, \\nonumber \\\\\n  & \\leq H^2\\norm{\\nabla \\Phi_i}_{L^{\\infty}(\\Omega)} \\norm{\\nabla u}_{L^1(\\Omega)}.\\label{test6}\n\\end{align}\n\n\nThus \\eqref{test4} becomes\n\\begin{align}\n \\left\\rvert \\underset{K\\in \\mathcal{M}_H}{\\sum} \\Phi_i(\\textbf{x}_K) |K| \\Big[ Q_Ku(\\overline{\\textbf{x}}_K;\\mu)  - Q_Ku(\\textbf{x}_K;\\mu) \\Big] \\right\\rvert &\\lesssim \\underset{G\\in \\mathfrak{G}}{\\sum} \\norm{\\Phi_i}_{L^{\\infty}(G)}\\left\\rvert \\underset{K\\in G}{\\sum} \\Big( \\frac{1}{|B_K|} \\int_{B_K}D^1u(\\mathbf{y}) \\ d\\mathbf{y} \\Big) |K|\\ \\mathbf{e}_K  \\right\\rvert \\nonumber\\\\ &+ H^2\\norm{\\nabla \\Phi_i}_{L^{\\infty}(\\Omega)} \\norm{\\nabla u}_{L^1(\\Omega)}. \\label{test7}\n\\end{align}\n\n\nNow, the Lemma 7.6. in ~\\cite{supconv2} is going to be used three times on the first term the right hand side of \\eqref{test7}. This lemma reads: \\\\\nLet $U,\\ V$ and $O$ be open sets of $\\mathbb{R}^d$ such that, for all $(\\textbf{x},\\mathbf{y})\\in U \\times V, \\ [\\textbf{x},\\mathbf{y}] \\subset O$. There exists $C$ only depending on $d$ such that, for all $\\Phi \\in W^{1,1}(O)$,\n\\begin{equation}\n\\left\\rvert \\frac{1}{|U|}\\int_U \\Phi(\\textbf{x}) \\ d\\textbf{x} - \\frac{1}{|V|}\\int_V \\Phi(\\textbf{x}) \\ d\\textbf{x} \\right\\rvert \\leq C \\frac{\\textrm{diam}(O)^{d+1}}{|U||V|}\\int_O |\\nabla \\Phi(\\textbf{x})|\\ d\\textbf{x}.\n\\end{equation}\n\n\\noindent We will use it successively with $[U,V,O]=[B_K,K,U_G]$, $[U,V,O]=[K,B_G,U_G]$, and $[U,V,O]=[B_G,U_G,U_G]$.\\\\\n\n\\noindent We use the triangle inequality on \\eqref{test8},\n\n\\begin{align}\n  \\underset{G\\in \\mathfrak{G}}{\\sum} \\norm{\\Phi_i}_{L^{\\infty}(G)} \\left| \\underset{K\\in G}{\\sum} \\Big( \\frac{1}{|B_K|} \\int_{B_K} D^1 u(\\mathbf{y}) \\ d\\mathbf{y} \\Big) |K| \\ \\mathbf{e}_K  \\right| &\\leq  \\underset{G\\in \\mathfrak{G}}{\\sum} \\norm{\\Phi_i}_{L^{\\infty}(G)} \\Big| \\underset{K\\in G}{\\sum} \\Big( \\left| \\frac{1}{|B_K|} \\int_{B_K}D^1u(\\mathbf{y}) \\ d\\mathbf{y} - \\ \\frac{1}{|K|}  \\int_{K} D^1 u(\\mathbf{y};\\mu)\\ d\\mathbf{y} \\right| \\nonumber \\\\\n    &+ \\ \\left| \\frac{1}{|K|}  \\int_{K} D^1 u(\\mathbf{y};\\mu) \\ d\\mathbf{y} \\ - \\ \\frac{1}{|B_G|}  \\int_{B_G} D^1 u(\\mathbf{y};\\mu) \\ d\\mathbf{y} \\right| \\nonumber  \\\\\n    &+  \\left| \\frac{1}{|B_G|}  \\int_{B_G} D^1 u(\\mathbf{y};\\mu)\\ d\\mathbf{y} \\ - \\ \\frac{1}{|U_G|}  \\int_{U_G} D^1 u(\\mathbf{y};\\mu)\\ d\\mathbf{y} \\right|  \\nonumber \\\\\n    &+     \\frac{1}{|U_G|}  \\int_{U_G} D^1 u(\\mathbf{y};\\mu)\\ d\\mathbf{y} \\Big)\\ |K| \\ \\mathbf{e}_K \\Big| . \\label{test10}\n    \\end{align}\n    and we get\n    \n    \\begin{align}\n  \\underset{G\\in \\mathfrak{G}}{\\sum} \\norm{\\Phi_i}_{L^{\\infty}(G)} \\left| \\underset{K\\in G}{\\sum} \\Big( \\frac{1}{|B_K|} \\int_{B_K} D^1 u(\\mathbf{y}) \\ d\\mathbf{y} \\Big) |K| \\ \\mathbf{e}_K  \\right| &\\lesssim  \\underset{G\\in \\mathfrak{G}}{\\sum} \\norm{\\Phi_i}_{L^{\\infty}(G)} \\Big| \\underset{K\\in G}{\\sum} \\Big( \\norm{u}_{W^{2,1}(U_G)}\\textrm{diam}(U_G)^d \\Big[  \\frac{\\textrm{diam}(U_G)}{|B_K||K|} \\nonumber \\\\\n    &+ \\frac{\\textrm{diam}(U_G)}{|B_G||K|}+\\frac{\\textrm{diam}(U_G)}{|U_G||B_G|} \\Big] +  \\frac{1}{|U_G|}  \\int_{U_G} D^1 u(\\mathbf{y};\\mu) \\ d\\mathbf{y} \\Big) \\ |K| \\ \\mathbf{e}_K \\Big| .  \\label{test11}\n    \\end{align}\n    \n\n\\noindent With the regularity factor $\\theta_{H}$ (see the previous definition of a polytopal mesh \\eqref{reg}), $|K|\\leq |B(0,1)|H_K^d \\lesssim |B_K|\\theta_H^{d}. $ Since $\\textrm{Card}(G)$ is bounded by $\\mu_G$, $\\textrm{diam}(U_G)\\leq \\mu_G H_K$. Thus, \n$\\textrm{diam}(U_G)^d\\leq \\mu_G^d H_K^d,$ and $\\frac{\\textrm{diam}(U_G)}{|B_K|}\\leq C$, $\\ |B_G|\\geq \\mu_G^{-d} \\textrm{diam}(U_G)^d,\\ |B_G|\\gtrsim \\mu_G^{-d}H_K^d\\gtrsim \\mu_G^{-d}|K|,$ and $|U_G|\\geq \\textrm{diam}(U_G)^d.$\\\\\nTherefore \\eqref{test11} becomes \n    \\begin{align}\n  \\underset{G\\in \\mathfrak{G}}{\\sum} \\norm{\\Phi_i}_{L^{\\infty}(G)} \\left| \\underset{K\\in G}{\\sum} \\Big( \\frac{1}{|B_K|} \\int_{B_K} D^1 u(\\mathbf{y}) \\ d\\mathbf{y} \\Big) |K| \\mathbf{e}_K  \\right| &\\lesssim  \\underset{G\\in \\mathfrak{G}}{\\sum} \\norm{\\Phi_i}_{L^{\\infty}(G)} \\Big| \\underset{K\\in G}{\\sum} \\Big( \\norm{u}_{W^{2,1}(U_G)}\\frac{\\textrm{diam}(U_G)}{|K|} \\nonumber \\\\\n    &+  \\frac{1}{|U_G|}  \\int_{U_G} D^1 u(\\mathbf{y};\\mu) \\ d\\mathbf{y} \\Big) \\ |K| \\ \\mathbf{e}_K \\Big|.  \\label{test12}\n    \\end{align}\n\n\nSince $\\textrm{diam}(U_G)\\leq  \\mu_G H_K$ and $|\\mathbf{e}_K|\\leq H_K$,\n\\begin{align}\n  \\underset{G\\in \\mathfrak{G}}{\\sum} \\norm{\\Phi_i}_{L^{\\infty}(G)} \\left| \\underset{K\\in G}{\\sum} \\Big( \\frac{1}{|B_K|} \\int_{B_K} D^1 u(\\mathbf{y}) \\ d\\mathbf{y} \\Big) |K|\\ \\mathbf{e}_K  \\right| &\\lesssim  \\underset{G\\in \\mathfrak{G}}{\\sum} \\norm{\\Phi_i}_{L^{\\infty}(G)} \\Big[ \\underset{K\\in G}{\\sum} H_K^2 \\norm{u}_{W^{2,1}(U_G)} \\nonumber \\\\\n    &+ \\left\\rvert \\frac{1}{|U_G|} \\underset{K\\in G}{\\sum} \\int_{U_G} D^1 u(\\mathbf{y};\\mu)\\ d\\mathbf{y}  |K| \\mathbf{e}_K  \\right\\rvert \\Big] . \\label{test13}\n\\end{align}\n\n\nThen,\n\\begin{align}\n \\underset{G\\in \\mathfrak{G}}{\\sum} \\norm{\\Phi_i}_{L^{\\infty}(G)} \\left| \\underset{K\\in G}{\\sum} \\Big( \\frac{1}{|B_K|} \\int_{B_K} D^1 u(\\mathbf{y}) \\ d\\mathbf{y} \\Big) |K| \\mathbf{e}_K  \\right|  &\\lesssim  \\underset{G\\in \\mathfrak{G}}{\\sum} \\norm{\\Phi_i}_{L^{\\infty}(G)} \\underset{K\\in G}{\\sum} H_K^2 \\norm{u}_{W^{2,1}(U_G)} \\nonumber \\\\\n    &+   \\underset{G\\in \\mathfrak{G}}{\\sum} \\norm{\\Phi_i}_{L^{\\infty}(G)}\\left\\rvert \\frac{1}{|U_G|} \\underset{K\\in G}{\\sum}  |K|\\ \\mathbf{e}_K  \\right\\rvert \\left\\rvert \\int_{U_G} D^1 u(\\mathbf{y};\\mu)\\ d\\mathbf{y}  \\right\\rvert, \\label{test14}\n\\end{align}\n\n\nwhich implies, since $\\textrm{Card}(G)\\leq \\mu_G$,\n\\begin{align}\n\\underset{G\\in \\mathfrak{G}}{\\sum} \\norm{\\Phi_i}_{L^{\\infty}(G)} \\left| \\underset{K\\in G}{\\sum} \\Big( \\frac{1}{|B_K|} \\int_{B_K} D^1 u(\\mathbf{y}) \\ d\\mathbf{y} \\Big) |K| \\mathbf{e}_K  \\right|  &\\lesssim  \\underset{G\\in \\mathfrak{G}}{\\sum} \\norm{\\Phi_i}_{L^{\\infty}(G)} H^2 \\norm{u}_{W^{2,1}(U_G)} \\nonumber \\\\\n    &+   \\underset{G\\in \\mathfrak{G}}{\\sum} \\norm{\\Phi_i}_{L^{\\infty}(G)}\\left\\rvert \\frac{1}{|U_G|} \\underset{K\\in G}{\\sum}  |K| \\ \\mathbf{e}_K  \\right\\rvert \\norm{u}_{W^{1,1}(U_G)}. \\label{test15}\n\\end{align}\n\nand finally,\n\\begin{equation}\n \\underset{G\\in \\mathfrak{G}}{\\sum} \\norm{\\Phi_i}_{L^{\\infty}(G)}\\left\\rvert \\underset{K\\in G}{\\sum} \\frac{1}{|B_K|} \\int_{B_K}D^1u(\\mathbf{y}) \\ d\\mathbf{y} |K|\\ \\mathbf{e}_K  \\right\\rvert \\leq  \\norm{\\Phi_i}_{L^{\\infty}(\\Omega)} \\norm{u}_{W^{2,1}(\\Omega)} H^2 \n    +   \\norm{\\Phi_i}_{L^{\\infty}(\\Omega)} \\underset{G \\in \\mathfrak{G}}{\\max\\ } \\norm{u}_{W^{1,1}(\\Omega)} \\left\\rvert \\frac{1}{|U_G|} \\underset{K\\in G}{\\sum}  |K| \\ \\mathbf{e}_K  \\right\\rvert. \\label{test16}\n\\end{equation}\n\nThis results  using \\eqref{test2}, \\eqref{inter22}, \\eqref{test4}, \\eqref{test6}, and \\eqref{test16} in\n\\begin{equation}\n \\left\\rvert \\int_{\\Omega} (\\Pi_1^H u(\\mu)-\\Pi_0^Hu(\\mu)) \\cdot \\Pi_0^H\\Phi_i \\ d\\textbf{x} \\right\\rvert \\lesssim (\\norm{\\Phi_i}_{W^{1,\\infty}(\\Omega)}\\norm{u}_{W^{2,1}(\\Omega)}+\\norm{u}_{H^2(\\Omega)}\\norm{\\Phi_i}_{L^{\\infty}(\\Omega)})H^2+(\\norm{\\Phi_i}_{L^{\\infty}(\\Omega)}\\norm{u}_{W^{1,1}(\\Omega)})e_{\\mathfrak{G}}. \n\\end{equation}\nIf $e_{\\mathfrak{G}}=\\underset{G \\in \\mathfrak{G}}{\\max\\ }  \\left\\rvert \\bigg(\\frac{1}{|U_G|}  \\underset{K \\in G}{\\sum} |K|\\ \\mathbf{e}_K\\bigg)\\right\\rvert$ is in $\\mathcal{O}(H^2)$ then the estimate of $ \\left\\rvert \\int_{\\Omega} (\\Pi_1^H u(\\mu)-\\Pi_0^Hu(\\mu)) \\cdot \\Pi_0^H\\Phi_i \\ d\\textbf{x} \\right\\rvert$ is in $\\mathcal{O}(H^2).$\nThis concludes the proof since the rest is similar to the one of Theorem \\ref{th11}. Note that for the estimate of $T_{3,2}$ \\eqref{term2bis}, the equation \\eqref{superconvhmm2} from the Theorem of super-convergence with local grouping is used instead of \\eqref{superconvhmm}.\n\\end{proof}\n\\section{Some details on the implementation and numerical results}\nWe consider two simple cases in 2D for the numerical results based with the TPFA scheme. Both results are computed on the unit square. We use an harmonic averaging of the diffusion coefficient(~\\cite{jd} section 5.3). Our variable parameter is $\\mu\\in \\mathbb{R}^4=(\\mu_1,\\mu_2,\\mu_3,\\mu_4)$. For both cases, the size of mesh $h$ is defined as the maximum length of the edges. The diffusion coefficient we consider here is $A(\\mu)=(2\\mu_1+\\mu_2 \\sin(x+y)\\cos(xy))$ and $f=(\\mu_3(1-y)+\\mu_4x(1-x))$.\nWe choose random coefficients for the snapshots with $N=5$ and our solution is defined with $\\mu_1=0.99,\\ \\mu_2=0.8,\\ \\mu_3=0.2,\\ \\mu_4=0.78$. For the exact solution, we consider the TPFA solution on a finer mesh (Figures \\ref{fig:rectangle}, \\ref{fig:triangle}). For the computation of the norm, we use the discrete semi-norm as in the remark of the section \\ref{sect2} \\eqref{discretenorme}. NIRB results are compared to the classical finite volume error  (Figures \\ref{fig:resurect}, \\ref{fig:resutriangle}). We measure the following relative error\n\\begin{equation}\n  \\frac{\\norm{u(\\mu) - u_{Hh}^N(\\mu)}_{\\mathcal{T},2}}{\\norm{u(\\mu)}_{\\mathcal{T},2}}.\n  \\end{equation}\n\n\n\n\\paragraph{Uniform grid}\n\nThe first case presents results on a rectangular uniform grid where $\\textbf{x}_K $ is the center of mass of the cell.\n\n\\vspace{-0.8cm}\n\\begin{figure}[H]\n  \\centering\n  \\includegraphics[scale=0.13]{coarse_square_exacte_solution.eps}\\qquad\n\\includegraphics[scale=1.5]{fine_square_exacte_solution2.eps}\n\\caption{coarse and fine solution with the uniform grid}\\label{fig:rectangle}\n\\end{figure}\n\n\\begin{figure}[H]\n    \\centering\n    \\includegraphics[scale=0.3]{case_rectangle.eps}\n   \\caption{Numerical result on the uniform grid}\n  \\label{fig:resurect}\n\\end{figure}\n\n\\paragraph{Triangular mesh}\n\nThe second case is defined on a triangular mesh where $\\textbf{x}_K $ are the circumcenter of the cells, such that $e_{\\mathfrak{G}}$ is in $\\mathcal{O}(H^2)$. \\\\\n\n\\vspace{-0.5cm}\n\\begin{figure}[H]\n  \\centering\n\\includegraphics[scale=0.13]{coarse_triangle_exacte_solution.eps}\\qquad\n\\includegraphics[scale=1.5]{fine_triangle_exacte_solution2.eps}\n\\caption{coarse and fine solution with the triangular mesh}\\label{fig:triangle}\n\\end{figure}\n\n\n\\begin{figure}[H]\n    \\centering\n    \\includegraphics[scale=0.3]{triangle_case.eps}\n    \\caption{Numerical result on the triangular mesh}\n    \\label{fig:resutriangle}\n  \n\\end{figure}\n\n\\paragraph{Discussion on the implementation}\n\nWe implemented the TPFA scheme on Scilab and retrieved several solutions for the NIRB algorithm on Python to highlight the black box side of the solver. \n\n\\begin{itemize}\n\\item Implementation of TPFA\\\\\n  The TPFA on $\\mathcal{T}_h$ reads:\n  Find $u_h=(u_K)_{K \\in \\mathcal{M}}$ such that:\\\\\n  \\begin{equation}\n    \\forall K\\in \\mathcal{M}_h,\\underset{\\sigma \\in \\mathcal{F}_K\\cap \\mathcal{F}_{int}}{\\sum} \\tau_{\\sigma}(u_K-u_L) + \\underset{\\sigma \\in \\mathcal{F}_K\\cap \\mathcal{F}_{ext}}{\\sum} \\tau_{\\sigma} u_K= \\int_K f(\\textbf{x}) d\\textbf{x},\n     \\end{equation}\n    where the harmonic average $\\tau_{\\sigma}=|\\sigma|\\frac{A(\\textbf{x}_L;\\mu)A(\\textbf{x}_K;\\mu)}{A(\\textbf{x}_L;\\mu)\\times d_{L,\\sigma}+ A(\\textbf{x}_K;\\mu)\\times d_{K,\\sigma}}$ on $\\mathcal{F}_{int}$, and $\\tau_{\\sigma}=|\\sigma|\\frac{A(\\textbf{x}_K;\\mu)}{d_{K,\\sigma}}$ on $\\mathcal{F}_{ext}$.\n   \n   To assemble the matrices $A$ of the TPFA scheme, we iterate on each edge, and we add the harmonic average $\\tau_{\\sigma}$ on each cell, and for $b$ we add the term $|D_{K,\\sigma}|\\times f(x_K)$.\n\n \\vspace{1cm}\n  \\item Time execution (min,sec)\n\n    \\begin{tabular}{ |c ||c| c| }\n      \\hline\n      & NIRB Online & FV solver \\\\\n      \\hline\n      uniform grid & 00:06 & 01:48 \\\\\n      \\hline\n      triangular mesh & 00:05 & 01:15 \\\\\n      \\hline\n    \\end{tabular}\n    \n\\end{itemize}\n \\vspace{1cm}\n \\begin{rmrk}\n   Note that for the discontinuous diffusion coefficient $A$, with the TPFA scheme, we recovered numerically the same estimate as in the Lipschitz continuous case, when we use the harmonic mean even if the proof no longer works.\n   \n\\end{rmrk}\n\n\n \\paragraph{Acknowledgements}\n This work is supported by the FUI MOR\\_DICUS. We would like to give special thanks to Nora A\u00efssiouene at the LJLL for her precious help\n\n\\newpage\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}