measuring the strangeness of strange attractors we study the correlation exponent v introduced recently as a characteristic measure of strange attractors which allows one to distinguish between deterministic chaos and random noise the exponent v is closely related to the fractal dimension and the information dimension but its computation is considerably easier its usefulness in characterizing experimental data which stem from very high dimensional systems is stressed algorithms for extracting v from the time series of a single variable are proposed the relations between the various measures of strange attractors and between them and the lyapunov exponents are discussed it is shown that the conjecture of kaplan and yorke for the dimension gives an upper bound for v various examples of finite and infinite dimensional systems are treated both numerically and analytically

exploratory analysis of spatiotemporal patterns of cellular automata by clustering compressibility in this paper we study the classification of spatiotemporal pattern of one dimensional cellular automata ca whereas the classification comprises ca rules including their initial conditions we propose an exploratory analysis method based on the normalized compression distance ncd of spatiotemporal patterns which is used as dissimilarity measure for a hierarchical clustering our approach is different with respect to the following points first the classification of spatiotemporal pattern is comparative because the ncd evaluates explicitly the difference of compressibility among two objects e g strings corresponding to spatiotemporal patterns this is in contrast to all other measures applied so far in a similar context because they are essentially univariate second kolmogorov complexity which underlies the ncd was used in the classification of ca with respect to their spatiotemporal pattern third our method is semiautomatic allowing us to investigate hundreds or thousands of ca rules or initial conditions simultaneously to gain insights into their organizational structure our numerical results are not only plausible confirming previous classification attempts but also shed light on the intricate influence of random initial conditions on the classification results

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

differentiability of palmer s linearization theorem and converse result for density functions we study differentiability properties in a particular case of the palmer s linearization theorem which states the existence of a homeomorphism h between the solutions of a linear ode system having exponential dichotomy and a quasilinear system indeed if the linear system is uniformly asymptotically stable sufficient conditions ensuring that h is a c2 preserving orientation diffeomorphism are given as an application we generalize a converse result of density functions for a nonlinear system in the nonautonomous case

momentary information transfer as a coupling measure of time series we propose a method to analyze couplings between two simultaneously measured time series our approach is based on conditional mutual sorting information it is related to other concepts for detecting coupling directions the old idea of marko for directed information and the more recent concept of schreiber s transfer entropy by setting suitable conditions we first of all consider momentary information in both time series this enables the detection not only of coupling directions but also delays sorting information refers to ordinal properties of time series which makes the analysis robust with respect to strictly monotonous distortions and thus very useful in the analysis of proxy data in climatology fortunately ordinal analysis is easy and fast to compute we consider also the problem of reliable estimation from finite time series

alternating adaptive projections in antenna synthesis the projection operator is a basic building block in the application of the alternating projections method to antenna synthesis in general it is a non linear operator that is repeatedly applied in the course of a single synthesis process thus having a considerable impact on the convergence properties of the resulting algorithm a novel approach to the computation of these projections is presented which exploits a simple definition of the relevant spaces particularly that of radiated fields characterization of field mask specification as scale invariant under this definition adds a further degree of freedom namely reference level which impacts on the sensitive to scaling projector in order to compute the optimum reference level an iterative procedure is proposed which is simple to implement easily integrable in standard alternating projection routines and adds negligible computational burden numerical tests confirm an improved performance with respect to the fixed scaling projection operator in terms of convergence rate and robustness against the initial guess supporting our approach as a valid aid in overcoming the drawbacks of the alternating projections based antenna synthesis

turbulent pattern bases for cellular automata abstract unpredictable patterns generated by cellular automata ca can be decomposed with respect to a turbulent positive entropy rate pattern basis the resulting filtered patterns uncover significant structural organization in a ca s dynamics and information processing capabilities we illustrate the decompostion technique by analyzing a binary range 2 cellular automaton having two invariant chaotic domains of different complexities and entropies once identified the domains are seen to organize the ca s state space and to dominate its evolution starting from the domains structures we show how to construct a finite state transducer that performs nonlinear spatial filtering such that the resulting space time patterns reveal the domains and the intervening walls and dislocations to show the statistical consequences of domain detection we compare the entropy and complexity densities of each domain with the globally averaged quantities a more graphical comparison uses difference patterns and difference plumes which trace the space time influence of a single site perturbation we also investigate the diversity of walls and particles emanating from the interface between two adjacent domains

estimating entropy rates with bayesian confidence intervals the entropy rate quantifies the amount of uncertainty or disorder produced by any dynamical system in a spiking neuron this uncertainty translates into the amount of information potentially encoded and thus the subject of intense theoretical and experimental investigation estimating this quantity in observed experimental data is difficult and requires a judicious selection of probabilistic models balancing between two opposing biases we use a model weighting principle originally developed for lossless data compression following the minimum description length principle this weighting yields a direct estimator of the entropy rate which compared to existing methods exhibits significantly less bias and converges faster in simulation with monte carlo techinques we estimate a bayesian confidence interval for the entropy rate in related work we apply these ideas to estimate the information rates between sensory stimuli and neural responses in experimental data shlens kennel abarbanel chichilnisky in preparation

information and entropy in strange attractors a technique for analyzing time series data from experiments is presented that provides estimates of four basic characteristics of a system 1 the measure theoretic entropy 2 the accuracy of the measurements 3 the number of measurements necessary to specify a system state and 4 the best delay time t to use in order to construct phase space portraits by the method of delays these characteristics are obtained by separating the entropy of measurements into a part due to noise and parts due to deterministic effects for the technique to work the noise associated with each measurement must be independent of the noise associated with all other measurements an algorithm for implementing the analysis is presented with three examples

ordinary dichotomy and global linearization abstract suppose that f t x t t d t f t x 1 f t x 2 r t x 1 x 2 r t d t c c is some constant then if system x a t x has an ordinary dichotomy then x a t x f t x is topologically equivalent to x a t x if system x a t x has an ordinary dichotomy with asymptotically stable manifolds then x a t x f t x is strongly topologically equivalent to x a t x x f t x can be considered as a perturbation of the linear system x 0 which has an ordinary dichotomy the structure of the solution set of x f t x is clear since x f t x is strongly topologically equivalent to x 0

the equality of kolmogorov sinai entropy and metric permutation entropy generalized abstract in a 2005 paper the author and collaborators proposed an approach to permutation entropy based on symbolic dynamics this approach allowed us to prove the equality of metric permutation entropy to the conventional metric entropy for symbolic dynamics and as a consequence also for n dimensional interval maps under the assumption of ergodicity in this paper we generalize our approach and extend that equality both to general i e not necessarily ergodic symbolic dynamics and to just measurable maps on not necessarily ordered finite measure spaces arguably the most general setting possible

symbolic diffusion entropy rate of chaotic time series as a surrogate measure for the largest lyapunov exponent existing methods for estimating the largest lyapunov exponent from a time series rely on the rate of separation of initially nearby trajectories reconstructed from the time series in phase space according to ueda chaotic dynamical behavior is viewed as a manifestation of random transitions between unstable periodic orbits in a chaotic attractor which are triggered by perturbations due to experimental observation or the roundoff error characteristic of the computing machine and consequently consists of a sequence of piecewise deterministic processes instead of an entirely deterministic process chaotic trajectories might have no physical reality here we propose a mathematical method for estimating a surrogate measure for the largest lyapunov exponent on the basis of the random diffusion of the symbols generated from a time series in a chaotic attractor without resorting to initially nearby trajectories we apply the proposed method to numerical time series generated by chaotic flow models and verify its validity

permutation complexity via duality between values and orderings abstract we study the permutation complexity of finite state stationary stochastic processes based on a duality between values and orderings between values first we establish a duality between the set of all words of a fixed length and the set of all permutations of the same length second on this basis we give an elementary alternative proof of the equality between the permutation entropy rate and the entropy rate for a finite state stationary stochastic processes first proved in j m amigo m b kennel l kocarev the permutation entropy rate equals the metric entropy rate for ergodic information sources and ergodic dynamical systems physica d 210 2005 77 95 third we show that further information on the relationship between the structure of values and the structure of orderings for finite state stationary stochastic processes beyond the entropy rate can be obtained from the established duality in particular we prove that the permutation excess entropy is equal to the excess entropy which is a measure of global correlation present in a stationary stochastic process for finite state stationary ergodic markov processes

ordinal analysis of time series abstract in order to develop fast and robust methods for extracting qualitative information from non linear time series bandt and pompe have proposed to consider time series from the pure ordinal viewpoint on the basis of counting ordinal patterns which describe the up and down in a time series they have introduced the concept of permutation entropy for quantifying the complexity of a system behind a time series the permutation entropy only provides one detail of the ordinal structure of a time series here we present a method for extracting the whole ordinal information

the influence of noise on the logistic model the behavior of the logistic system which is generated by the functionf x ax 1 x changes in an interesting way if it is perturbed by external noise it turns out that the chaotic behavior which was predicted by li and yorke for orbits of period 3 becomes visible and that a sequence of mergence transitions occurs at the critical parameter the change of the invariant probability density and the lyapunov exponents are examined numerically the power spectrum for the period 3 orbit for different fluctuations is calculated and a recursion formula for the time evolution of the probability density is presented as a discrete time analog of a chapman kolmogorov equation

a nonequilibrium entropy for dynamical systems it is proposed to define entropy for nonequilibrium ensembles using a method of coarse graining which partitions phase space into sets which typically have zero measure these are chosen by considering the totality of future possibilities for observation on the system it is shown that this entropy is necessarily a nondecreasing function of the timet there is no contradiction with the reversibility of the laws of motion because this method of coarse graining is asymmetric under time reversal under suitable conditions which are stated explicitly this entropy approaches the equilibrium entropy ast and the fine grained entropy ast in particular the conditions can always be satisfied if the system is ak system as in the sinai billiard models some theorems are given which give information about whether it is possible to generate the partition used here for coarse graining from time translates of a finite partition and at the same time elucidate the connection between our concept of entropy and the entropy invariant of kolmogorov and sinai

forbidden patterns permutation entropy and stock market inefficiency abstract in this paper we introduce two new quantifiers for the stock market inefficiency the number of forbidden patterns and the normalized permutation entropy they are model independent measures thus they have more general applicability we find robust evidence that degree of market inefficiency is positively correlated with the number of forbidden patterns and negatively correlated with the permutation entropy our empirical results suggest that these two physical tools are useful to discriminate the stage of stock market development and can be easily implemented

observations of order and chaos in nonlinear systems experiments on nonlinear electrical oscillators the belousov zhabotinskii reaction rayleigh benard convection and couette taylor flow have revealed several common routes to chaos that have also been found in numerical studies of models with a few degrees of freedom experimental results are presented illustrating the following transition sequences period doubling and the u sequence intermittency the periodic quasiperiodic chaotic sequence frequency locking and an alternating periodic chaotic sequence

entropy and local uncertainty of data from sensory neurons we present an empirical comparison between neural interspike interval sequences obtained from two different kinds of sensory receptors both differ in their internal structure as well as in the strength of correlations and the degree of predictability found in the respective spike trains as a further tool in this context we suggest the local uncertainty assigning a well defined predictability to individual spikes the local uncertainty is demonstrated to reveal significant patterns within the interspike interval sequences even when its overall structure is almost random our approach is based on the concept of symbolic dynamics and information theory

statistically relaxing to generating partitions for observed time series data we introduce a relaxation algorithm to estimate approximations to generating partitions for observed dynamical time series generating partitions preserve dynamical information of a deterministic map in the symbolic representation our method optimizes an essential property of a generating partition avoiding topological degeneracies we construct an energylike functional and use a nonequilibrium stochastic minimization algorithm to search through configuration space for the best assignment of symbols to observed data as each observed point may be assigned a symbol the partitions are not constrained to an arbitrary parametrization we further show how to select particular generating partition solutions which also code low order unstable periodic orbits in a given way hence being able to enumerate through a number of potential generating partition solutions

computing the topological entropy of maps we give an algorithm for determining the topological entropy of a unimodal map of the interval given its kneading sequence we also show that this algorithm converges exponentially in the number of letters of the kneading sequence

is every approximate trajectory of some process near an exact trajectory of a nearby process this paper deals with the problem can a noisy orbit be tracked by a real orbit in particular we will study the one parameter family of tent maps and the one parameter family of quadratic maps we writeg for eitherf orf withf x x forx 1 2 andf x 1 x forx 1 2 andf x x 1 x for a given we will say g permits increased parameter shadowing if for each x 0 there exists some 0 and some f 0 such that every f pseudog orbit starting in some invariant interval can be x shadowed by a realg orbit with we show thatg typically permits increased parameter shadowing

dynamical entropy for systems with stochastic perturbation dynamics of deterministic systems perturbed by random additive noise is characterized quantitatively since for such systems the kolmogorov sinai ks entropy diverges if the diameter of the partition tends to zero we analyze the difference between the total entropy of a noisy system and the entropy of the noise itself we show that this quantity is finite and non negative and we call it the dynamical entropy of the noisy system in the weak noise limit this quantity is conjectured to tend to the ks entropy of the deterministic system in particular we consider one dimensional systems with noise described by a finite dimensional kernel for which the frobenius perron operator can be represented by a finite matrix

generalized exponential dichotomy and global linearization abstract hartman s linearization theorem says that if all eigenvalues of matrix a have no zero real part and f x is small lipschitzian then nonlinear system x a x f x and its linear system x a x are topologically equivalent in 1970s palmer extended the theorem to nonautonomous systems in this paper we extend hartman s theorem to the systems with generalized exponential dichotomy

symbol statistics and spatio temporal systems abstract we consider the problem of estimating parameters from time series observations of spatio temporal systems two types of models are considered a a one dimensional coupled map lattice with nearest neighbor diffusive coupling and b the complex ginzburg landau equation in one and two spatial dimensions model parameters are to be estimated using time series observations from only a few sites a symbolic partition of the time series is introduced and the probabilities of observing various symbol sequences in the data are measured the parameter fitting is accomplished by adjusting parameters of the model until it produces time series whose symbol sequences have the same probabilities as the data we show that it is possible to reliably estimate the parameters from a single time series when the spatio temporal dynamics is turbulent i e it displays a wide range of space and time scales with no discernible patterns

a model for prejudiced learning in noisy environments based on the heuristics that maintaining presumptions can be beneficial in uncertain environments we propose a set of basic requirements for learning systems to incorporate the concept of prejudice the simplest memoryless model of a deterministic learning rule obeying the axioms is constructed and shown to be equivalent to the logistic map the system s performance is analysed in an environment in which it is subject to external randomness weighing learning defectiveness against stability gained the corresponding random dynamical system with inhomogeneous additive noise is studied and shown to exhibit the phenomena of noise induced stability and stochastic bifurcations the overall results allow for the interpretation that prejudice in uncertain environments can entail a considerable portion of stubbornness as a secondary phenomenon

the calculi of emergence computation dynamics and induction abstract defining structure and detecting the emergence of complexity in nature are inherently subjective though essential scientific activities despite the difficulties these problems can be analyzed in terms of how model building observers infer from measurements the computational capabilities embedded in nonlinear processes an observer s notion of what is ordered what is random and what is complex in its environment depends directly on its computational resources the amount of raw measurement data of memory and of time available for estimation and inference the discovery of structure in an environment depends more critically and subtlely though on how those resources are organized the descriptive power of the observer s chosen or implicit computational model class for example can be an overwhelming determinant in finding regularity in data this paper presents an overview of an inductive framework hierarchical machine reconstruction in which the emergence of complexity is associated with the innovation of new computational model classes complexity metrics for detecting structure and quantifying emergence along with an analysis of the constraints on the dynamics of innovation are outlined illustrative examples are drawn from the onset of unpredictability in nonlinear systems finitary nondeterministic processes and cellular automata pattern recognition they demonstrate how finite inference resources drive the innovation of new structures and so lead to the emergence of complexity

monotonicity of the lozi family near the tent maps we show the monotonicity of the lozi family when the jacobian determinant is close to zero the main ingredients of our proof are the pruning pair method and a detailed analysis of the parameter dependence of the kneading invariant of the tent map family

fast iterative method of power synthesis for antenna arrays a fast and simple method of power synthesis for antenna arrays of arbitrary geometry is presented the method adopts an auxiliary function having the desired amplitude pattern and a generic phase pattern and determines in closed form the array pattern approximating the auxiliary function in both amplitude and phase a cost function involving the auxiliary function and the approximating pattern is minimized by modifying the phase pattern of the auxiliary function using a simple and fast iterative technique once the optimal phase pattern has been found the array pattern approximating the corresponding auxiliary function is the synthesized pattern the proposed method allows to satisfy stringent requirements on the amplitude pattern including multibeam synthesis and radiation suppression within large angular regions numerical results show the effectiveness of the algorithm and are compared to those obtained using other existing methods in order to achieve a more significant comparison such methods have been properly modified to improve their performances such comparison shows that the presented algorithm allows a significant reduction of cpu time with respect to the other considered methods as well as a weaker dependence on the starting point finally the presented method is extended in such a way as to reduce the dynamic range ratio drr of the array excitations

existence and robustness of nonuniform h k μ ν dichotomies for nonautonomous impulsive differential equations abstract for the linear nonautonomous impulsive differential equation we propose a new notion called the nonuniform h k dichotomy which extends the notion of dichotomy of uniform and nonuniform types and is related to the theory of nonuniform hyperbolicity we establish a sufficient criterion for the existence of the nonuniform h k dichotomy in terms of appropriate lyapunov exponents for the linear perturbation we develop the robustness or roughness of nonuniform h k dichotomies in the sense that a nonuniform h k dichotomy persists under a sufficiently small linear perturbation

the permutation entropy rate equals the metric entropy rate for ergodic information sources and ergodic dynamical systems abstract permutation entropy quantifies the diversity of possible orderings of the values a random or deterministic system can take as shannon entropy quantifies the diversity of values we show that the metric and permutation entropy rates measures of new disorder per new observed value are equal for ergodic finite alphabet information sources discrete time stationary stochastic processes with this result we then prove that the same holds for deterministic dynamical systems defined by ergodic maps on n dimensional intervals this result generalizes a previous one for piecewise monotone interval maps on the real line c bandt g keller b pompe entropy of interval maps via permutations nonlinearity 15 2002 1595 1602 at the expense of requiring ergodicity and using a definition of permutation entropy rate differing modestly in the order of two limits the case of non ergodic finite alphabet sources is also studied and an inequality developed finally the equality of permutation and metric entropy rates is extended to ergodic non discrete information sources when entropy is replaced by differential entropy in the usual way

on the relation of ks entropy and permutation entropy abstract since bandt et al have shown that the permutation entropy and the kolmogorov sinai entropy coincide for piecewise monotone interval maps the relationship of both entropies for time discrete dynamical systems is of a certain interest the aim of the present paper is a discussion of this relationship on the basis of an ordinal characterization of the kolmogorov sinai entropy recently given

a method for the optimal pattern synthesis of linear arrays with prescribed nulls a new l sub spl infin optimal method for the synthesis of equispaced linear array functions with asymmetrical far field pattern functions is proposed this iterative method provides for the exact specification of the beam width while at the same time allowing for the specification of the relative levels of individual sidelobes by index or as a function of bearing e g angularly extended nulls as well as the realization of specified narrowband nulls the resulting array factors are optimal in the weighted l sub spl infin sense and in general have complex coefficients this new remez type method employs multiple objective functions to provide the degrees of freedom that are required for exact null placement examples which demonstrate the design flexibility offered by the method are included for various sum and difference patterns including superdirective and shaped beam arrays

estimating the entropy rate of spike trains via lempel ziv complexity normalized lempel ziv complexity which measures the generation rate of new patterns along a digital sequence is closely related to such important source properties as entropy and compression ratio but in contrast to these it is a property of individual sequences in this article we propose to exploit this concept to estimate or at least to bound from below the entropy of neural discharges spike trains the main advantages of this method include fast convergence of the estimator as supported by numerical simulation and the fact that there is no need to know the probability law of the process generating the signal furthermore we present numerical and experimental comparisons of the new method against the standard method based on word frequencies providing evidence that this new approach is an alternative entropy estimator for binned spike trains

forbidden ordinal patterns in higher dimensional dynamics abstract forbidden ordinal patterns are ordinal patterns or rank blocks that cannot appear in the orbits generated by a map taking values on a linearly ordered space in which case we say that the map has forbidden patterns once a map has a forbidden pattern of a given length l 0 it has forbidden patterns of any length l l 0 and their number grows superexponentially with l using recent results on topological permutation entropy in this paper we study the existence and some basic properties of forbidden ordinal patterns for self maps on n dimensional intervals our most applicable conclusion is that expansive interval maps with finite topological entropy have necessarily forbidden patterns although we conjecture that this is also the case under more general conditions the theoretical results are nicely illustrated for n 2 both using the naive counting estimator for forbidden patterns and chao s estimator for the number of classes in a population the robustness of forbidden ordinal patterns against observational white noise is also illustrated

conditional entropy of ordinal patterns abstract in this paper we investigate a quantity called conditional entropy of ordinal patterns akin to the permutation entropy the conditional entropy of ordinal patterns describes the average diversity of the ordinal patterns succeeding a given ordinal pattern we observe that this quantity provides a good estimation of the kolmogorov sinai entropy in many cases in particular the conditional entropy of ordinal patterns of a finite order coincides with the kolmogorov sinai entropy for periodic dynamics and for markov shifts over a binary alphabet finally the conditional entropy of ordinal patterns is computationally simple and thus can be well applied to real world data

on computing the entropy of the henon attractor in a recent article d ruelle inlecture notes in physics no 80 springer berlin 1978 has conjectured that for the henon attractor its measure theoretic entropy should be equal to its characteristic exponent this result is known to be true for systems which satisfy smale s axiom a in this article we report the results of our computations which suggest that ruelle s conjecture may be true for the henon attractor further in our study we are confronted with fundamental questions which suggest that certain existence theorems from ergodic theory are not sufficient from a computational point of view

small random perturbations of dynamical systems and the definition of attractors the strange attractors plotted by computers and seen in physical experiments do not necessarily have an open basin of attraction in view of this we study a new definition of attractors based on ideas of conley we argue that the attractors observed in the presence of small random perturbations correspond to this new definition

on the periodically perturbed logistic equation we study the logistic equation modified by a periodic time dependence the perturbation introduces bifurcation delays which can be calculated explicitly and we qualitatively explain how the bifurcation diagram deforms as the perturbation is increased

the noisy shift map starting from the topological conjugacy of a chaotic discrete map with the shift map on symbol space ideas from the baxter eight vertex model and a stochastic difference equation due to falk are used to define a notion of a noisy shift map on symbol space its long time behavior is determined along with a discussion of the ramifications of this result on the effects of noise on chaotic discrete dynamical systems

context tree modeling of observed symbolic dynamics modern techniques invented for data compression provide efficient automated algorithms for the modeling of the observed symbolic dynamics we demonstrate the relationship between coding and modeling motivating the well known minimum description length mdl principle and give concrete demonstrations of the context tree weighting and context tree maximizing algorithms the predictive modeling technique obviates many of the technical difficulties traditionally associated with the correct mdl analyses these symbolic models representing the symbol generating process as a finite state automaton with probabilistic emission probabilities provide excellent and reliable entropy estimations the resimulations of estimated tree models satisfying the mdl model selection criterion are faithful to the original in a number of measures the modeling suggests that the automated context tree model construction could replace fixed order word lengths in many traditional forms of empirical symbolic analysis of the data we provide an explicit pseudocode for implementation of the context tree weighting and maximizing algorithms as well as for the conversion to an equivalent markov chain

consistency of constraints on nulls and on dynamic range ratio in pattern synthesis for antenna arrays given an antenna array of n elements we deal with the problem of finding an array pattern that approximates a desired pattern while a null constraints are imposed in m les n 2 assigned directions b an upper bound d is imposed on the dynamic range ratio drr of the excitations at first we show that the system of constraint equations has solutions if and only if d ges d0 where d0 is a parameter depending on the null directions and describe a method to evaluate d0 then we propose two simple and fast algorithms to solve the synthesis problem and show their effectiveness by numerical examples statistical results show that usually d0 1 when m is small compared to n thus exact nulls can be formed for any d ges 1 this result provides an insight into the numerical results

strongly topological linearization with generalized exponential dichotomy abstract it has been proved that if x a t x has a generalized exponential dichotomy and f t x satisfies certain conditions then the nonlinear system x a t x f t x is topologically equivalent to its linear system x a t x in this paper we prove that if the condition a t m a t is added then x a t x f t x is strongly topologically equivalent to x a t x where m is some positive number and a t is the eigenfunction of the generalized exponential dichotomy and therefore the corresponding solutions of x a t x f t x and x a t x have the same stability

structural information in two dimensional patterns entropy convergence and excess entropy we develop information theoretic measures of spatial structure and pattern in more than one dimension as is well known the entropy density of a two dimensional configuration can be efficiently and accurately estimated via a converging sequence of conditional entropies we show that the manner in which these conditional entropies converge to their asymptotic value serves as a measure of global correlation and structure for spatial systems in any dimension we compare and contrast entropy convergence with mutual information and structure factor techniques for quantifying and detecting spatial structure

nonparametric entropy estimation for stationary processes and random fields with applications to english text we discuss a family of estimators for the entropy rate of a stationary ergodic process and prove their pointwise and mean consistency under a doeblin type mixing condition the estimators are cesaro averages of longest match lengths and their consistency follows from a generalized ergodic theorem due to maker 1940 we provide examples of their performance on english text and we generalize our results to countable alphabet processes and to random fields

topological permutation entropy permutation entropy quantifies the diversity of possible ordering of the successively observed values a random or deterministic system can take just as shannon entropy quantifies the diversity of the values themselves when the observable or state variable has a natural order relation making permutation entropy possible to compute then the asymptotic rate of growth in permutation entropy with word length forms an alternative means of describing the intrinsic entropy rate of a source herein extending a previous result on metric entropy rate we show that the topological permutation entropy rate for expansive maps equals the conventional topological entropy rate familiar from symbolic dynamics this result is not limited to one dimensional maps

testing nonlinear markovian hypotheses in dynamical systems abstract we present a statistical approach for detecting the markovian character of dynamical systems by analyzing their flow of information especially in the presence of noise which is mostly the case for real world time series the calculation of the information flow of the underlying system via the concept of symbolic dynamics is rather problematic since one has to use infinitesimal partitions we circumvent this difficulty by measuring the information flow indirectly more precisely we calculate a measure based on higher order cumulants which quantifies the statistical dependencies between the past values of the time series and the point r steps ahead as an extension of theiler s method of surrogate data theiler et al 1992 this cumulant based information flow a function of the look ahead r is used as the discriminating statistic in testing the observed dynamics against a hierarchy of null hypotheses corresponding to nonlinear markov processes of increasing order this procedure is iterative in the sense that whenever a null hypothesis is rejected new data sets can be generated corresponding to better approximations of the original process in terms of information flow since we use higher order cumulants for calculating the discriminating statistic our method is also applicable to small data sets numerical results on artificial and real world examples including non chaotic nonlinear processes autoregressive models and noisy chaos show the effectiveness of our approach

computational mechanics pattern and prediction structure and simplicity computational mechanics an approach to structural complexity defines a process s causal states and gives a procedure for finding them we show that the causal state representation an machine is the minimal one consistent with accurate prediction we establish several results on machine optimality and uniqueness and on how machines compare to alternative representations further results relate measures of randomness and structural complexity obtained from machines to those from ergodic and information theories

composition law of cardinal ordering permutations abstract in this paper the theorems that determine composition laws both cardinal ordering permutations and their inverses are proven so the relative position of points in an h s periodic orbit is completely known as well as which order those points are visited no matter how the h s periodic orbit emerges be it through a period doubling cascade s 2 n of as the h periodic orbit or a primary window like saddle node bifurcation cascade with h 2 n or a secondary window the birth of an s periodic window inside the h periodic one certainly period doubling cascade orbits are particular cases with h 2 and s 2 n both composition laws are also shown in algorithmic way for easy use

symbolic dynamics of noisy chaos abstract one model of randomness observed in physical systems is that low dimensional deterministic chaotic attractors underly the observations a phenomenological theory of chaotic dynamics requires an accounting of the information flow from the observed system to the observer the amount of information available in observations and just how this information affects predictions of the system s future behavior in an effort to develop such a description we discuss the information theory of highly discretized observations of random behavior metric entropy and topological entropy are well defined invariant measures of such an attractor s level of chaos and are computable using symbolic dynamics real physical systems that display low dimensional dynamics are however inevitably coupled to high dimensional randomless e g thermal noise we investigate the effects of such fluctuations coupled to deterministic chaotic systems in particular the metric entropy s response to the fluctuations we find that the entropy increases with a power law in the noise level and that the convergence of the entropy and the effect of fluctuations can be cast as a scaling theory we also argue that in addition to the metric entropy there is a second scaling invariant quantity that characterizes a deterministic system with added fluctuations i0 the maximum average information obtainable about the initial condition that produces a particular sequence of measurements or symbols

estimating the number of species a review how many kinds are there suppose that a population is partitioned into c classes in many situations interest focuses not on estimation of the relative sizes of the classes but on estimation of c itself for example biologists and ecologists may be interested in estimating the number of species in a population of plants or animals numismatists may be concemed with estimating the number of dies used to produce an ancient coin issue and linguists may be interested in estimating the size of an author s vocabulary in this article we review the problem of statistical estimation of c many approaches have been proposed some purely data analytic and others based in sampling theory in the latter case numerous variations have been considered the population may be finite or infinite if finite samples may be taken with replacement multinomial sampling or without replacement hypergeometric sampling or by bernoulli sampling if infinite sampling may be multinomial or bernoulli or the sample may be th

deterministic dynamics of neural activity during absence seizures in rats the study of brain electrical activities in terms of deterministic nonlinear dynamics has recently received much attention forbidden ordinal patterns fop is a recently proposed method to investigate the determinism of a dynamical system through the analysis of intrinsic ordinal properties of a nonstationary time series the advantages of this method in comparison to others include simplicity and low complexity in computation without further model assumptions in this paper the fop of the eeg series of genetic absence epilepsy rats from strasbourg was examined to demonstrate evidence of deterministic dynamics during epileptic states experiments showed that the number of fop of the eeg series grew significantly from an interictal to an ictal state via a preictal state these findings indicated that the deterministic dynamics of neural networks increased significantly in the transition from the interictal to the ictal states and also suggested that the fop measures of the eeg series could be considered as a predictor of absence seizures

inferring markov chains bayesian estimation model comparison entropy rate and out of class modeling markov chains are a natural and well understood tool for describing one dimensional patterns in time or space we show how to infer kth order markov chains for arbitrary k from finite data by applying bayesian methods to both parameter estimation and model order selection extending existing results for multinomial models of discrete data we connect inference to statistical mechanics through information theoretic type theory techniques we establish a direct relationship between bayesian evidence and the partition function which allows for straightforward calculation of the expectation and variance of the conditional relative entropy and the source entropy rate finally we introduce a method that uses finite data size scaling with model order comparison to infer the structure of out of class processes

estimating and improving the signal to noise ratio of time series by symbolic dynamics we investigate the effect of symbolic encoding applied to times series consisting of some deterministic signal and additive noise as well as time series given by a deterministic signal with randomly distributed initial conditions as a model of event related brain potentials we introduce an estimator of the signal to noise ratio snr of the system by means of time averages of running complexity measures such as shannon and r enyi entropies and prove its asymptotical equivalence with the linear snr in the case of shannon entropies of symbol distributions a snr improvement factor is defined exhibiting a maximum for intermediate values of noise amplitude in analogy to stochastic resonance phenomena we demonstrate that the maximum of the snr improvement factor can be shifted toward smaller noise amplitudes by using higher order r enyi entropies instead of the shannon entropy for a further improvement of the snr a half wave encoding of noisy time series is introduced finally we discuss the effect of noisy phases on the linear snr as well as on the snr defined by symbolic dynamics it is shown that longer symbol sequences yield an improvement of the latter

kolmogorov sinai entropy from the ordinal viewpoint abstract in the case of ergodicity much of the structure of a one dimensional time discrete dynamical system is already determined by its ordinal structure we generally discuss this phenomenon by considering the distribution of ordinal patterns which describe the ups and downs in the orbits of a borel measurable map on a borel subset of the real line on this base we give a natural ordinal description of the kolmogorov sinai entropy of one dimensional dynamical systems and relate the kolmogorov sinai entropy to the permutation entropy recently introduced by bandt and pompe

verification of chaotic behavior in an experimental loudspeaker the dynamics of an experimental electrodynamic loudspeaker is studied by using the tools of chaos theory and time series analysis delay time embedding dimension fractal dimension and other empirical quantities are determined from experimental data particular attention is paid to issues of stationarity in a system in order to identify sources of uncertainty lyapunov exponents and fractal dimension are measured using several independent techniques results are compared in order to establish independent confirmation of low dimensional dynamics and a positive dominant lyapunov exponent we thus show that the loudspeaker may function as a chaotic system suitable for low dimensional modeling and the application of chaos control techniques

finite statistical complexity for sofic systems we propose a measure of complexity for symbolic sequences which is based on conditional probabilities and captures computational aspects of complexity without the explicit construction of minimal deterministic finite automata dfa moreover if the sequence is obtained from a dynamical system through a suitable encoding and its equations of motion are known we show how to estimate the regions of phase space that correspond to computational states with statistically equivalent futures causal states

symbolic dynamics of event related brain potentials we apply symbolic dynamics techniques such as word statistics and measures of complexity to nonstationary and noisy multivariate time series of electroencephalograms eeg in order to estimate event related brain potentials erp their significance against surrogate data as well as between different experimental conditions is tested these methods are validated by simulations using stochastic dynamical systems with time dependent control parameters and compared with traditional erp analysis techniques continuous eeg data are cut into epochs according to stimuli events presented to the subjects these ensembles of time series can be considered as ensembles of trajectories given by some dynamical systems we employ a statistical mechanics approach motivated by the frobenius perron equation and apply it to coarse grained symbolic descriptions of the dynamics we develop time dependent measures of complexity founded on running cylinder sets and show that these quantities are able to distinguish simulated data obtained by different control parameters as well as experimental data between different experimental conditions as a first finding our approach restores the well known erp components and it reveals additionally qualitative changes in the eeg that cannot be detected by means of the traditional techniques we criticize the prerequisites of the traditional approach to erp analysis and propose to consider erp instead in terms of dynamical system theory and information theory