multiple topological structures of chaotic attractors ruling the emission of a driven laser we analyze the chaotic emission of an erbium doped fiber laser with sine wave pump power modulation for different modulation frequencies and modulation indexes for each working condition considered the template which summarizes the corresponding chaotic attractor has been determined by means of topological analysis techniques the variety of templates obtained is much wider than in previous works apart from horseshoes reverse horseshoes or jellyroll structures with different global torsions two more kinds of structures have been found particularly one is especially notable as its folding mechanism does not fit the rolling scheme valid for all templates found in former experimental studies apart from strengthening the usefulness of templates as convenient objects for characterization of chaotic attractors the interest of these results lies in the identification of folding mechanisms for chaos generation not found in nature up to now

on multiscale entropy analysis for physiological data abstract we perform an analysis of cardiac data using multiscale entropy as proposed in costa et al multiscale entropy analysis of complex physiological time series phys rev lett 89 2002 068102 we reproduce the signatures of the multiscale entropy for the three cases of young healthy hearts atrial fibrillation and congestive heart failure we show that one has to be cautious how to interpret these signatures in terms of the underlying dynamics in particular we show that different dynamical systems can exhibit the same signatures depending on the sampling time and that similar systems may have different signatures depending on the time scales involved besides the total amount of data we identify the sampling time the correlation time and the period of possible nonlinear oscillations as important time scales which have to be involved in a detailed analysis of the signatures of multiscale entropies we illustrate our ideas with the lorenz equation as a simple deterministic chaotic system

symbolic dynamics and characterization of complexity abstract symbolic dynamics deals with robust properties of dynamics without digging into numbers for one dimensional maps there have been some new results periodic window theorem generalized composition rule construction of median words without using harmonics and antiharmonics incorporation of discrete symmetry to analyze symmetry breaking and symmetry restoration etc this new understanding has been applied to ordinary differential equations in particular the systematics of stable periodic solutions in the lorenz model has been shown to be given basically by symbolic dynamics of the cubic map symbolic dynamics may be used to extract invariant characteristics from time series its relation with grammatical complexity will also be commented on

multiscale analysis of financial time series by renyi distribution entropy abstract it is of great interests to analyze the complexity or dependence of time series of the daily stock closing price in different regions entropy is typically a fundamental technique to explore the complexity of time series the well studied irregularity measure sample entropy sampen and a more recently proposed complexity measure distribution entropy disten are treated as classification features in our study based on renyi entropy which has been recently suggested as a measure of complexity in nonlinear systems we propose multiscale renyi distribution entropy mrde to research the complexity of nonlinear time series over multiple time scales considering the validity and accuracy renyi entropy obtains more information than the shannon entropy for time series mingled with much noise like financial time series which is mainly reflected in the choice of parameters for frequent events and rare events both the simulated signals logistic map and gaussian noise and the financial time series with three different stock markets are calculated and compared using an innovative method in different situations for different stock markets the entropy decreases as the scale factor increases except when the parameter is the maximum in the special case meanwhile we illustrate the necessity and advantage of renyi distribution entropy rde method by comparing rde results with the sampen results about stability and consistency on synthetic data and financial time series it is applied to measure the dependence by detecting the length of the time series and special algorithm parameters

a model of neural control of the heart rate a simple model of heart rate regulation is proposed using the assumption that the nervous system regulates the generation of pulses of the pacemaker previous values of intervals between heart beats rr intervals are used for this regulation which is described by a nonlinear feedback loop with time delay the conductance of the excitation in the heart is phenomenologically described by one nonlinear function recovery curve the model reproduces time series of rr intervals different known patterns of heart rate variability are observed depending on the type of control and the parameter values

from template analysis to generating partitions ii characterization of the symbolic encodings we give numerical evidence of the validity of a previously described algorithm for constructing symbolic encodings of chaotic attractors from a template analysis we verify that the different solutions that can be found are dynamically equivalent and that our approach yields results that are consistent with those obtained from methods based on homoclinic tangencies this is further confirmed by verifying directly that the computed partitions are generating to a high degree of accuracy and that they can be used to estimate precisely the metric entropy it is also shown that the correct number of symbols needed to describe the dynamics is naturally provided and that a compact parameterization of a partition can easily be determined which makes our algorithm suitable for applications such as real time encoding

lyapunov exponents from observed time series we examine the question of accurately determining lyapunov exponents for a time series we find that it is advantageous to use local mappings with higher order taylor series rather than linear maps as done earlier we demonstrate this procedure for the ikeda map and the lorenz system we present methods for identifying spurious exponents by analyzing data set singularities and by determining the lyapunov direction vectors the behavior of spurious exponents in the presence of noise is also investigated and found to be different from that of the true exponents

hardy s inequality and the l1 norm of exponential sums in this paper we generalize hardy s inequality 3 for measures of analytic type and obtain a proof of the littlewood conjecture 4 for the l norm of exponential sums as a simple consequence let t be the circle group z the additive group of integers and m t the customary convolution algebra of borel measures on t for pe e m t and n g z put m e ino dp o denote by h t the classical space of all measures e e m t such that j n 0 for all n 0 and let c denote the complex numbers we now state hardy s inequality for measures of analytic type theorem 1 if m c h t then

braid analysis of a bouncing ball i identify the template organizing the chaotic dynamics of a bounding ball system i further show how to estimate the topological parameter values of the system directly from a time series a process i call topological time series analysis two distinct methods to determine the topological parameters are illustrated and compared the pruning front procedure and a braid analysis both procedures lead to compatible results

on the symbolic dynamics of the henon map the authors study a method for finding all periodic orbits of the dissipative henon map for the canonical values a 1 4 b 0 3 this method is found to give indeed the good symbolic dynamics proposed earlier by two of them but it does not work as proposed for larger values of b this is seen in two ways although the method is thus not universally valid it can be used to study problems related to periodic orbits they find that for the usual values a 1 4 b 0 3 expressions for average quantities are not easily obtained precisely from periodic orbits in agreement with recent claims

entropy pattern entropy and related methods for the analysis of data on the time intervals between heartbeats from 24 h electrocardiograms sequences of the time intervals between heartbeats char22 medically termed rr intervals char22 extracted from 24 h electrocardiogram recordings are examined as three dimensional return map images the recordings were made in humans by means of the medically widely used portable electrocardiograph holter system a time window measured in the number of heartbeats is used and different types of behavior are classified bifurcations between the types of dynamics of the heart are noted and a form of intermittency is found an alternative quantitative measure char22 a form pattern entropy of the return map image char22 is defined that characterizes the dynamics of the rr interval sequence it is shown that this is a measure of the degree of ordering of the rr interval sequence and as such it is a good novel medical diagnostic tool for analyzing heart rate variability which distinguishes between illness and health where other diagnostics fail

numerical solution of differential algebraic equations for constrained mechanical motion the two most popular formulations of the equations of constrained mechanical motion thedescriptor andstate space forms each have severe practical limitations in this paper we discuss and relate some proposed reformulations of the equations which have improved numerical properties

estimates for the norms of products of sines and cosines abstract in this paper we prove asymptotic formulas for the l p norms of p n k 1 n 1 e i k and q n k 1 n 1 e i k these products can be expressed using k 1 n sin k 2 and k 1 n cos k 2 respectively we prove an estimate for p n at a point near where its maximum occurs finally we give an asymptotic formula for the maximum of the fourier coefficients of q n

cumulative tsallis entropy based on multi scale permuted distribution of financial time series abstract a method based on multi scale permuted distribution cumulative tsallis entropy mpdcte is proposed in this paper to measure the complexity and dissimilarity between sequences it avoids the influence of permutation on spatial distribution and uses the calculation of spatial distance matrix in distributed entropy to effectively measure the complexity of time series we apply the mpdcte method to the simulation data verify the effectiveness of the method discuss the influence of the parameters and compare it with the traditional entropy metric the results show that this method is insensitive to parameter changes and has a low dependence on data length the dependencies all show superiority finally it was applied to the real financial market and we selected 9 stocks in the world to analyze the mpdcte method clearly divided the stocks by analyzing the sequence which is consistent with the phenomenon of the financial market

homoclinic bifurcations for the henon map abstract chaotic dynamics can be effectively studied by continuation from an anti integrable limit we use this limit to assign global symbols to orbits and use continuation from the limit to study their bifurcations we find a bound on the parameter range for which the henon map exhibits a complete binary horseshoe as well as a subshift of finite type we classify homoclinic bifurcations and study those for the area preserving case in detail simple forcing relations between homoclinic orbits are established we show that a symmetry of the map gives rise to constraints on certain sequences of homoclinic bifurcations our numerical studies also identify the bifurcations that bound intervals on which the topological entropy is apparently constant

how random is your heart beat we measure the content of random uncorrelated noise in heart rate variability using a general method of noise level estimation using a coarse grained entropy we show that usually except for atrial fibrillation the level of such noise is within 5 15 of the variance of the data and that the variability due to the linearly correlated processes is dominant in all cases analyzed but atrial fibrillation the nonlinear deterministic content of heart rate variability remains significant and may not be ignored

comment on multiscale entropy analysis of complex physiologic time series a comment on the letter by madalena costa ary l goldberger and c k peng phys rev lett 89 068102 2002 the authors of the letter offer a reply

intensity approximation of random fluctuation in complex systems abstract the markov and non markov processes in complex systems are examined with the help of dynamical information shannon entropy method here we consider the essential role of two mutually independent channels of entropy involving creation of correlation and annihilation of correlation the developed method has been used to analyze the intensity fluctuation of the complex systems of various nature in psychology to analyze numerical and pattern short time human memory to study the effect of stress on the parameters of the dynamical taping test and in cardiology to analyze the random dynamics of rr intervals in human ecg s and to diagnose various diseases of human cardiovascular systems the received results show that the application of intensity approximation allows to improve essentially the diagnostics of parameters in the evolution of human dynamic states

estimating a generating partition from observed time series symbolic shadowing we propose a deterministic algorithm for approximating a generating partition from a time series using tessellations using data generated by henon and ikeda maps we demonstrate that the proposed method produces partitions that uniquely encode all the periodic points up to some order and provide good estimates of the metric and topological entropies the algorithm gives useful results even with a short noisy time series

knotted periodic orbits in suspensions of smale s horseshoe extended families and bifurcation sequences abstract we consider knotted periodic orbits in the natural suspension of smale s horseshoe map as constructed by holmes and williams 1985 and holmes 1986 we identify quartets and octets of periodic orbits which are isotopic knots and indicate how they are related via the sequences of bifurcations in which they are created in two parameter families such as the henon map

estimating generating partitions of chaotic systems by unstable periodic orbits an outstanding problem in chaotic dynamics is to specify generating partitions for symbolic dynamics in dimensions larger than 1 it has been known that the infinite number of unstable periodic orbits embedded in the chaotic invariant set provides sufficient information for estimating the generating partition here we present a general dimension independent and efficient approach for this task based on optimizing a set of proximity functions defined with respect to periodic orbits our algorithm allows us to obtain the approximate location of the generating partition for the ikeda hammel jones moloney map

reconstruction of dynamical and geometrical properties of chaotic attractors from threshold crossing interspike intervals we reconstruct the largest lyapunov exponent and fractal dimension of a chaotic attractor using thresholdcrossing interspike intervals alone we show that in certain cases one may reconstruct from this data a set looking very similar to the initial attractor we also give an explanation of this possibility based on the concept of instantaneous frequency s1063 651x 98 51207 x pacs number s 05 45 1b the problem of extracting information about a dynamical system from experimental data is of great importance for all those who study the systems of natural origin that may be thought of as black boxes the traditional way of obtaining experimental data is to fix the sampling stepdt and store the values of some state variable s t corresponding to the time moments idt the fundamental works 1 state that the delay method applied to such time series often allows one to reconstruct a set being topologically equivalent to the initial system attractor on the base of these works a series of methods have been developed allowing one to evaluate attractor dimension 2 and the largest lyapunov characteristic exponent lce 3 to detect unstable equilibrium points 4 and unstable periodic orbits 5 and finally to create in some cases global models 6 or different predictors 7 another way of getting experimental data which is often preferred by biologists is to generate spike train data 8 one imposes a certain condition on the value of state variable s t and records the interspike intervals isi ti between the time moments for which the condition is fulfilled there are two general methods for recording spike train data the first one is to generate a spike when s t crosses some threshold level u in one direction and measure isi as an alternative one may measure the time intervals between the local maxima of s t fig 1 a the values of ti can be treated as the times of trajectory return to the surface defined by the condition imposed to the initial s t the condition s t 5u means the definition of a plane in the phase space while the condition s t 50 s t being a realization x j t o f a dynamical system dx w dt5f

knotted periodic orbits in suspensions of smale s horseshoe period multiplying and cabled knots abstract following earlier work on knotted periodic orbits in a suspension of smale s 1 horseshoe diffeomorphism holmes and williams 2 we define a notion of iterated horseshoe knots we show that approximately half the horseshoe knots are cabled and that among these cablings are infinitely many non isotopic iterated torus knots we obtain uniqueness results for certain families of resonant torus knots related to cascades of period doubling and period multiplying bifurcations which occur as a horseshoe is created by passing through a family of diffeomorphisms such as the henon maps we give formulae from which linking numbers can be computed and provide closed form expressions for the type numbers of certain resonant iterated torus knots

structure in the bifurcation diagram of the duffing oscillator we identify four levels of structure in the bifurcation diagram of the two well periodically driven duffing oscillator plotted as a function of increasing control parameter t the period of the driving term the superstructure or bifurcation peninsula repeats periodically as t increases by ensuremath sim 2 ensuremath pi beginning and ending with symmetric period one orbits whose local torsions differ by 2 within each bifurcation peninsula there is a systematic window structure the primary window structure is due to newhouse and newhouse like orbits fine structure is due to a farey sequence of well ordered orbits between the primary windows hyperfine structure consists of very narrow windows associated with non well ordered orbits we construct a template for the duffing oscillator a two dimensional return map and a one dimensional return map which describes the systematics of orbit creation and annihilation all structures are identified by topological indices our predictions are based on and compatible with numerical computations

efficient numerical solution of constrained multibody dynamics systems abstract a numerical algorithm for conducting coupled system dynamical simulation is presented the interconnected system comprising numerous modules is treated as a constrained multibody dynamics system of particular focus is the efficient solution of coupled system simulation without sacrificing the independence of the separate dynamical modules the proposed algorithm maggi s equations with perturbed iteration mepi emanates from numerical methods for differential algebraic equations separate treatment of the constraint equations from the resolution of subsystem dynamical responses marks mepi s main characteristic

phase relationships between two or more interacting processes from one dimensional time series i basic theory a general approach is developed for the detection of phase relationships between two or more different oscillatory processes interacting within a single system using one dimensional time series only it is based on the introduction of angles and radii of return times maps and on studying the dynamics of the angles an explicit unique relationship is derived between angles and the conventional phase difference introduced earlier for bivariate data it is valid under conditions of weak forcing this correspondence is confirmed numerically for a nonstationary process in a forced van der pol system a model describing the angles behavior for a dynamical system under weak quasiperiodic forcing with an arbitrary number of independent frequencies is derived to establish from experimental data whether or not two or more interacting processes are synchronized is an old and important problem in the absence of noise the existence of synchronization between two processes in interacting limit cycle oscillators was originally taken to imply that their basic frequencies of oscillation are related as integer numbers i e are rationally connected 1 2 and that their instantaneous phases are permanently locked this definition suggests that the detection of whether or not synchronization exists can be established by computation of the ratio of basic frequencies in the fourier spectrum of the signal from one of the sub systems involved even in this simplest case however the finite observation time and the discreteness of the digitiza tion steps used in practice will make it appear that all fre quencies are rationally connected thereby complicating the reliable estimation of their ratio furthermore the noise that is invariably present in all real macroscopic systems means that only effective synchronization 3 can normally take place meaning that the phases can remain locked only dur ing finite time intervals and that the basic frequencies may no longer be rationally connected 4 serious difficulties may also arise due to the nonstationarity of experimental data in view of these problems modern techniques for estab lishing the presence or absence of synchronization are based on the assumption that the behavior of each subsystem can be considered separately and that their individual time series can be compared by a variety of techniques e g by compu tation of the phase difference between them this approach has been justified theoretically 3 6 and is widely used to detect synchronization not only in periodic noisy but also in

average entropy measurement of disorder for cardiac rr interval signals abstract we introduce average entropy ae as a disorder measure of heart rate time series the ae measurement reflects both the probability distribution of all discrete values in a time series and the randomness of the series the ae analysis is able to differentiate the congestive heart failure chf the healthy and the atrial fibrillation af subjects for both short and long heart rate time series the differentiation accuracy of ae is higher than 90 in addition since the meaning of disorder is easily confused with complexity we explore the relation between them by analyzing heart rate time series specifically we demonstrate that the complexity measured by entropy of entropy eoe versus the disorder measured by ae plot of heart rate time series exhibits a distinct inverted u relation

complex temporal patterns in molecular dynamics a direct measure of the phase space exploration by the trajectory at macroscopic time scales computer simulated trajectories of bulk water molecules form complex spatiotemporal structures at the picosecond time scale this intrinsic complexity which underlies the formation of molecular structures at longer time scales has been quantified using a measure of statistical complexity the method estimates the information contained in the molecular trajectory by detecting and quantifying temporal patterns present in the simulated data velocity time series two types of temporal patterns are found the first defined by the short time correlations corresponding to the velocity autocorrelation decay times 0 1 ps remains asymptotically stable for time intervals longer than several tens of nanoseconds the second is caused by previously unknown longer time correlations found at longer than the nanoseconds time scales leading to a value of statistical complexity that slowly increases with time a direct measure based on the notion of statistical complexity that describes how the trajectory explores the phase space and independent from the particular molecular signal used as the observed time series is introduced

structure of generating partitions for two dimensional maps we discuss the non uniqueness of the generating partitions of two dimensional maps and we give new explanations for ambiguities that are to be expected in the construction of generating partitions if a parameter is varied both points are discussed with the help of the connecting lines of homoclinic tangencies in the entire phase space i e also outside the attractor that can be calculated very easily with a new algorithm introduced recently

diversity analysis based on ordered patterns abstract diversity index based on entropy such as emlen index e i is suggested as a highly effective algorithm to detect complexity in nonlinear systems in this paper a method considering the amplitude information called weighted emlen index w e i is proposed to extract the amplitude information and track the signal changes and then we expand e i w e i methods to multiple time scales thus multiscale emlen index m s e i and multiscale weighted emlen index m s w e i will be used to discuss the complexity of traffic time series under different scales results show that there are differences between the complexity of weekend traffic time series and workday time series which gives a better classification of traffic systems for making prediction

horseshoe templates with global torsion in a driven laser dimensional system with many longitudinal modes oscillating simultaneously hence in sec ii we describe the experimental system and check that topological analysis is applicable by estimating the lyapunov dimension of the attractors then sec iii is devoted to the extraction of upo from experimental data and the computation of their topological invariants we show there how careful signal processing can greatly help in extracting topological information from noisy time series of finite length and precision finally in sec iv we determine the templates of the different attractors compare them and relate them to the values of the modulation frequency in particular we show that these templates have a nonzero global torsion and that this torsion increases by one full turn when the period of modulation increases by 2p v r this observation is in close agreement with the theoretical studies of modulated nonlinear oscillators by gilmore and mccallum 10

a time integration algorithm for flexible mechanism dynamics the dae α method this paper introduces a new family of second order methods for solving the index 2 differential algebraic equations daes of motion for flexible mechanism dynamics these methods which extend the methods for odes of structural dynamics to daes possess numerical dissipation that can be controlled by the user convergence and stability analysis is given and verifies that the dae methods introduce no additional oscillations and preserve the stability of the underlying ode system convergence of the newton iteration which can be a source of difficulty in solving nonlinear oscillatory systems with large stepsizes is achieved via a coordinate split modification to the newton iteration numerical results illustrate the effectiveness of the new methods for simulation of flexible mechanisms

predicting chaotic time series with wavelet networks a new technique wavelet network is introduced to predict chaotic time series by using this technique firstly we make accurate short term predictions of the time series from chaotic attractors secondly we make accurate predictions of the values and bifurcation structures of the time series from dynamical systems whose parameter values are changing with time finally we predict chaotic attractors by making long term predictions based on remarkably few data points where the correlation dimensions of predicted attractors are calculated and are found to be almost identical to those of actual attractors

homoclinic tangencies and non normal jacobians effects of noise in nonhyperbolic chaotic systems abstract a general study of the effects of noise interacting with the deterministic dynamics of nonhyperbolic chaotic 2 d maps and 3 flows is presented noise in these systems can have dramatic effects on the invariant measure of the system in regions of homoclinic tangencies perturbations are transported away from the neighborhood of the attractor leading to deformations which can be one to two orders of magnitude larger than the noise level a qualitative understanding is attained by a study of the universal properties of the unstable and stable foliation of the phase space in the vicinity of homoclinic tangencies through the investigation of nontrivial structures of the tangent space at homoclinic tangencies related to the non normal jacobians we obtain a quantitative description of these noise induced deformations of the attractor local expansion rates which are much larger than the maximal lyapunov exponent can be used as a measure of the system s structural instability under perturbations we exemplify our general results on the henon ikeda and duffing system a new and effective algorithm to calculate the homoclinic tangencies in the entire phase space based on the results of this paper is presented

growth of the sudler product of sines at the golden rotation number abstract we study the growth at the golden rotation number 5 1 2 of the function sequence p n r 1 n 2 sin r this sequence has been variously studied elsewhere as a skew product of sines birkhoff sum q pochhammer symbol on the unit circle and restricted euler function in particular we study the fibonacci decimation of the sequence p n namely the sub sequence q n r 1 f n 2 sin r for fibonacci numbers f n and prove that this renormalisation subsequence converges to a constant from this we show rigorously that the growth of p n is bounded by power laws this provides the theoretical basis to explain recent experimental results reported by knill and tangerman 2011 10

automatic integration of euler lagrange equations with constraints abstract numerical difficulties in the integration of euler lagrange and similar equations are discussed a technique for reducing their index from three to two is introduced and it is shown that variable order variable step bdf methods converge for these index two problems the practical application of this reduction in a numerical setting is examined

classification of strange attractors by integers we show how to characterize a strange attractor by a set of integers these are extracted from the chaotic time series data by first reconstructing the low period orbits and then determining the template or knot holder which supports all periodic orbits embedded in the strange attractor and the strange attractor itself the template is identified by a set of integers which therefore characterize the strange attractor this identification is explicitly demonstrated for the pirogon using a relatively small data set 5000 points

on the topology of the henon map topological invariants of the henon map are investigated by means of the pruning front first a long sequence of primary homoclinic tangencies is computed confirming the monotonicity of the front an algorithm to extract forbidden sequences is then introduced and discussed forbidden sequences of increasing lengths are used to construct a hierarchy of regular grammars represented by directed graphs which approximate the exact grammar arbitrarily well the topological entropy is estimated as the largest eigenvalue of their adjacency matrix it exhibits an exponential convergence towards the asymptotic value with an exponent in agreement with a previous conjecture based on the growth rate of the number of forbidden words

asymptotic behaviour of the sudler product of sines for quadratic irrationals abstract we study the asymptotic behaviour of the sequence of sine products p n r 1 n 2 sin r for real quadratic irrationals in particular we study the subsequence q n r 1 q n 2 sin r where q n is the nth best approximation denominator of and show that this subsequence converges to a periodic sequence whose period equals that of the continued fraction expansion of this verifies a conjecture recently posed by verschueren and mestel 2016 15

knotted periodic orbits in suspensions of annulus maps we consider a class of suspensions of diffeomorphisms of the annulus as flows in the orientable 3 manifold t 2 x i using results of birman williams topology 22 47 82 1983 contemp math 20 1 60 1983 we construct a knotholder or template that carries the set of periodic orbits of the flow we define rotation numbers and show that any orbit of period q and rotation number p q can be arranged as a positive braid on p strands this yields existence and uniqueness results for families of resonant torus knots p braids that are p q torus knots of period q p which correspond to order preserving birkhoff periodic orbits of the diffeomorphism we show that all other q periodic p braids have higher genus and we establish bounds on the genera of such knots we obtain existence and uniqueness results for a number of other non resonant torus knots including non order preserving q s q torus knots of rotation number 1

reshape scale method a novel multi scale entropic analysis approach abstract the reshape scale rs method was introduced in this article as a novel approach to perform multi scale transition of sample entropy this method was able to quantify the orderliness in the signal by determining the distance over which the subsequent data points can remain affiliated to one another entropic half life enhl was introduced to characterize such an affiliation the method was tested for 1 f processes for different values furthermore the dependency of the multi scale entropy analysis developed by costa et al 2002 6 to the probability density function and the standard deviation of autoregressive signals was studied and discussed

stability of computational methods for constrained dynamics systems many methods have been proposed for numerically integrating the differential algebraic systems arising from the euler lagrange equations for constrained motion these are based on various problem formulations and discretizations we offer a critical evaluation of these methods from the standpoint of stability nconsidering a linear model we first give conditions under which the differential algebraic problem is well conditioned this involves the concept of an essential underlying ode we review a variety of reformulations which have been proposed in the literature and show that most of them preserve the well conditioning of the original problem then we consider stiff and nonstiff discretizations of such reformulated models in some cases the same implicit discretization may behave in a very different way when applied to different problem formulations acting as a stiff integrator on some formulations and as a nonstiff integrator on others we present the approach of projected invariants as a method for yielding problem reformulations which are desirable in this sense

efficient detection of periodic orbits in chaotic systems by stabilizing transformations an algorithm for detecting periodic orbits in chaotic systems r l davidchack and y c lai phys rev e 3 60 1999 pp 6172 6175 which combines the set of stabilizing transformations proposed by schmelcher and diakonos phys rev lett 78 1997 pp 4733 4736 with a modified semi implicit euler iterative scheme and seeding with periodic orbits of neighboring periods has been shown to be highly efficient when applied to low dimensional systems the difficulty in applying the algorithm to higher dimensional systems is mainly due to the fact that the number of the stabilizing transformations grows extremely fast with increasing system dimension here we analyze the properties of stabilizing transformations and propose an alternative approach for constructing a smaller set of transformations the performance of the new approach is illustrated on the four dimensional kicked double rotor map and the six dimensional system of three coupled henon maps

from template analysis to generating partitions i periodic orbits knots and symbolic encodings abstract we present a detailed algorithm to construct symbolic encodings for chaotic attractors of three dimensional flows it is based on a topological analysis of unstable periodic orbits embedded in the attractor and follows the approach proposed by lefranc et al phys rev lett 73 1994 1364 for each orbit the symbolic names that are consistent with its knot theoretic invariants and with the topological structure of the attractor are first obtained using template analysis this information and the locations of the periodic orbits in the section plane are then used to construct a generating partition by means of triangulations we provide numerical evidence of the validity of this method by applying it successfully to sets of more than 1500 periodic orbits extracted from numerical simulations and obtain partitions whose border is localized with a precision of 0 01 a distinctive advantage of this approach is that the solution is progressively refined using higher period orbits which makes it robust to noise and suitable for analyzing experimental time series furthermore the resulting encodings are by construction consistent in the corresponding limits with those rigorously known for both one dimensional and hyperbolic maps

computational geometry an introduction from the reviews this book offers a coherent treatment at the graduate textbook level of the field that has come to be known in the last decade or so as computational geometry the book is well organized and lucidly written a timely contribution by two founders of the field it clearly demonstrates that computational geometry in the plane is now a fairly well understood branch of computer science and mathematics it also points the way to the solution of the more challenging problems in dimensions higher than two

interspike interval attractors from chaotically driven neuron models abstract sequences of intervals between firing times interspike intervals isis from single neuron models with chaotic forcing are investigated we analyze how the dynamical properties of the chaotic input determine those of the output isi sequence and assess how various biophysical parameters affect this input output relationship the attractors constructed from delay embeddings of isis and of the chaotic input are compared from the points of view of geometry and nonlinear forecastability for the three integrate and fire if models investigated the similarity between these attractors is high only when the mean firing rate is high and when firings occur over a large range of the input signal when these conditions are satisfied isis are related to input values at which firings occur by a simple map and their distribution can be derived from that of input signal values attractor reconstruction is found to be more sensitive to mean firing rate than to isi distribution at low firing rates for which the input is under sampled and isis become larger than the short term prediction horizon or when the dynamics do not allow a smooth invertible mapping between signal and isis information is lost our results relevant to all dynamical systems generating events show that information about continuous time dynamics is difficult to retrieve at low event rates and that information about inputs can be isolated only under restricted conditions

characterization of strange attractors a new measure of strange attractors is introduced which offers a practical algorithm to determine their character from the time series of a single observable the relation of this new measure to fractal dimension and information theoretic entropy is discussed

extracting unstable periodic orbits from chaotic time series data a general nonlinear method to extract unstable periodic orbits from chaotic time series is proposed by utilizing the estimated local dynamics along a trajectory we devise a transformation of the time series data such that the transformed data are concentrated on the periodic orbits thus one can extract unstable periodic orbits from a chaotic time series by simply looking for peaks in a finite grid approximation of the distribution function of the transformed data our method is demonstrated using data from both numerical and experimental examples including neuronal ensemble data from mammalian brain slices the statistical significance of the results in the presence of noise is assessed using surrogate data s1063 651x 97 15205 9

event and time scale characteristics of heart rate dynamics the cardiac system shows various scale dynamic activities from secondly to yearly therefore multiple time scale characteristics of heart dynamics have received much attention for understanding and distinguishing healthy and pathological cardiac systems in this paper we expand the multiple time scale analysis into event and time scales to investigate scale characteristics in healthy and pathologic cardiac systems to do this we define a measure based on symbolic dynamics which calculates complexity at each time and event scale called the unit time block entropy utbe this measure allows a reliable comparison of experimental data through matching the number of words and the total measurement time at the same time for all rr interval sequences which are composed of the time durations between consecutive r waves of electrocardiograms we apply the utbe to the healthy heart rate hr group and pathological hr groups and find that the rr interval acceleration is more effective than the rr interval in distinguishing each group and we also find that the normal and pathological hr groups are clearly distinguished in some specific event and time scale regions

knotted periodic orbits in suspensions of smale s horseshoe torus knots and bifurcation sequences we construct a suspension of smale s horseshoe diffeomorphism of the two dimensional disc as a flow in an orientable three manifold such a suspension is natural in the sense that it occurs frequently in periodically forced nonlinear oscillators such as the duffing equation from this suspension we construct a knot holder or template a branched two manifold with a semiflow in such a way that the periodic orbits are isotopic to those in the full three dimensional flow we discuss some of the families of knotted periodic orbits carried by this template in particular we obtain theorems of existence uniqueness and non existence for families of torus knots we relate these families to resonant hamiltonian bifurcations which occur as horseshoes are created in a one parameter family of area preserving maps and we also relate them to bifurcations of families of one dimensional quadratic like maps which can be studied by kneading theory thus using knot theory kneading theory and hamiltonian bifurcation theory we are able to connect a countable subsequence of one dimensional bifurcations with a subsequence of area preserving bifurcations in a two parameter family of suspensions in which horseshoes are created as the parameters vary one implication is that infinitely many bifurcation sequences are reversed as one passes from the one dimensional to the area preserving family there are no universal routes to chaos

unimodal polynomials associated with lie algebras and superalgebras it is well known that the theory of lie algebras can be used to prove the unimodality of certain polynomials recently it was shown that lie superalgebras also give rise to unimodal polynomials in this paper we present a unified approach to the question we do not discuss the technical details of lie algebras and superalgebras but give several old and new examples of unimodal polynomials associated with these structures

embeddings of low dimensional strange attractors topological invariants and degrees of freedom when a low dimensional chaotic attractor is embedded in a three dimensional space its topological properties are embedding dependent we show that there are just three topological properties that depend on the embedding parity global torsion and knot type we discuss how they can change with the embedding finally we show that the mechanism that is responsible for creating chaotic behavior is an invariant of all embeddings these results apply only to chaotic attractors of genus one which covers the majority of cases in which experimental data have been subjected to topological analysis this means that the conclusions drawn from previous analyses for example that the mechanism generating chaotic behavior is a smale horseshoe mechanism a reverse horseshoe a gateau roul e an s template branched manifold etc are not artifacts of the embedding chosen for the analysis

multiscale entropy analysis of complex physiologic time series there has been considerable interest in quantifying the complexity of physiologic time series such as heart rate however traditional algorithms indicate higher complexity for certain pathologic processes associated with random outputs than for healthy dynamics exhibiting long range correlations this paradox may be due to the fact that conventional algorithms fail to account for the multiple time scales inherent in healthy physiologic dynamics we introduce a method to calculate multiscale entropy mse for complex time series we find that mse robustly separates healthy and pathologic groups and consistently yields higher values for simulated long range correlated noise compared to uncorrelated noise

nonlinear similarity analysis for epileptic seizures prediction abstract the prediction of epileptic seizures can promise a new diagnostic application and a novel approach for seizure control this paper proposes an improved dynamical similarity measure to predict epileptic seizures in electroencephalographic eeg first mutual information and cao s method are employed to reconstruct a phase space of preprocessed eeg recordings by using the positive zero crossing method second a gaussian function replaces the heavyside function within correlation integral at calculating a similarity index the crisp boundary of the heavyside function is eliminated because of the gaussian function s smooth boundary third an adaptive detection method based on the similarity index is proposed to predict the epileptic seizures in light of test results of eeg recordings of rats it is found that the new dynamical similarity index is insensitive to the selection of the radius value of gaussian function and the size of segmented eeg recordings comparing with the dynamical similarity index proposed by le van quyen et al anticipating epileptic seizures in real time by a non linear analysis of similarity between eeg recordings neuroreport 10 1999 2149 2155 the tests of twelve rats show the new dynamical similarity index is better to predict the epileptic seizures

homoclinic tangencies generating partitions and curvature of invariant manifolds a method to compute the curvature of the unstable manifold is introduced and applied to henon map and duffing attractor showing that it allows the authors to locate the homoclinic tangencies and in turn to construct a generating partition the probability distribution of curvature values is investigated showing a power law decay finally the shape of the multifractal spectrum of effective liapunov exponents in nonhyperbolic systems is discussed

determining lyapunov exponents from a time series we present the first algorithms that allow the estimation of non negative lyapunov exponents from an experimental time series lyapunov exponents which provide a qualitative and quantitative characterization of dynamical behavior are related to the exponentially fast divergence or convergence of nearby orbits in phase space a system with one or more positive lyapunov exponents is defined to be chaotic our method is rooted conceptually in a previously developed technique that could only be applied to analytically defined model systems we monitor the long term growth rate of small volume elements in an attractor the method is tested on model systems with known lyapunov spectra and applied to data for the belousov zhabotinskii reaction and couette taylor flow

constrained equations of motion in multibody dynamics as odes on manifolds a convergence theorem for applying linear multistep numerical integration methods to constrained equations of motion in mechanical systems is presented using a differential geometric approach eul