rotational kinematics of pelvis and upper trunk at butterfly stroke can fins affect the dynamics of the system the purpose of the present study was to investigate the rotational kinematics pattern of the upper trunk and the pelvis and the complexity variability of their movement during the sprint butterfly stroke between male and female swimmers with long fins 18 26 cm and without fins two pairs of 3d accelerometers and gyroscopes were used to measure segments flexion extension angles there were no gender differences apart from the record the amplitude values at the pelvis were significantly larger when swimming without fins while at the c7 they were significantly larger when swimming with them autocorrelation coefficients of angles time histories were higher when swimming with fins for both segments the power spectrums revealed a dominant frequency representing the stroke period which grew significantly stronger for the fin condition correlation dimension verified a lower dimensionality for the fin condition for the c7 segment movement pattern overall fin use seems to offer strength to the swimmer s movement pattern this does not imply better movement coordination but rather a trend toward a more stable attractor

the delay vector variance method for detecting determinism and nonlinearity in time series a novel delay vector variance dvv method for detecting the presence of determinism and nonlinearity in a time series is introduced the method is based upon the examination of local predictability of a signal additionally it spans the complete range of local linear models due to the standardisation to the distribution of pairwise distances between delay vectors this provides consistent and easy to interpret diagrams which convey information about the nature of a time series in order to gain further insight into the technique a dvv scatter diagram is introduced which plots the dvv curve against that for a globally linear model surrogate data this way the deviation from the bisector line represents a qualitative measure of nonlinearity which can be used additionally for constructing a quantitative measure for statistical testing the proposed method is compared to existing methods on synthetic as well as standard real world signals and is shown to provide more consistent results overall compared to other established nonlinearity analysis methods

influence of observational noise on the recurrence quantification analysis in this paper we estimate the errors due to observational noise on the recurrence quantification analysis rqa based on this estimation we present ways to minimize these errors we give a criterion to choose the threshold e needed for the optimal computation of the recurrence plot rp one important point is to show the limits of interpretability of the results of the rqa if it is applied to measured time series we show that even though the rqa is very susceptible to observational noise it can yield reliable results for an optimal choice of e if the noise level is not too high we apply the results to typical models such as white noise the logistic map and the lorenz system and to experimental laser data

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

nonlinear dynamics delay times and embedding windows in order to construct an embedding of a nonlinear time series one must choose an appropriate delay time d often d is estimated using the autocorrelation function however this does not treat the nonlinearity appropriately and it may yield an incorrect value for d on the other hand the correct value of d can be found from the mutual information but this process is rather cumbersome computationally here we suggest a simpler method for estimating d using the correlation integral we call this the c c method and we test it on several nonlinear time series obtaining estimates of d in agreement with those obtained using the mutual information furthermore some researchers have suggested that one should not choose a fixed delay time d independent of the embedding dimension m but rather one should choose an appropriate value for the delay time window w d m 1 which is the total time spanned by the components of each embedded point unfortunately w cannot be estimated using the autocorrelation function or the mutual information and no standard procedure for estimating w has emerged however we show that the c c method can also be used to estimate w basically w is the optimal time for independence of the data while d is the first locally optimal time as tests we apply the c c method to the lorenz system a three dimensional irrational torus the rossler system and the rabinovich fabrikant system we also demonstrate the robustness of this method to the presence of noise c 1999 elsevier science b v all rights reserved

encoding tasks and renyi entropy a task is randomly drawn from a finite set of tasks and is described using a fixed number of bits all the tasks that share its description must be performed upper and lower bounds on the minimum pth moment of the number of performed tasks are derived the case where a sequence of tasks is produced by a source and n tasks are jointly described using nr bits is considered if r is larger than the renyi entropy rate of the source of order 1 1 provided it exists then the th moment of the ratio of performed tasks to n can be driven to one as n tends to infinity if r is smaller than the renyi entropy rate this moment tends to infinity the results are generalized to account for the presence of side information in this more general setting the key quantity is a conditional version of renyi entropy that was introduced by arimoto for iid sources two additional extensions are solved one of a rate distortion flavor and the other where different tasks may have different nonnegative costs finally a divergence that was identified by sundaresan as a mismatch penalty in the massey arikan guessing problem is shown to play a similar role here

on the dimension of complex responses in nonlinear structural vibrations abstract the ability to accurately model engineering systems under extreme dynamic loads would prove a major breakthrough in many aspects of aerospace mechanical and civil engineering extreme loads frequently induce both nonlinearities and coupling which increase the complexity of the response and the computational cost of finite element models dimension reduction has recently gained traction and promises the ability to distill dynamic responses down to a minimal dimension without sacrificing accuracy in this context the dimensionality of a response is related to the number of modes needed in a reduced order model to accurately simulate the response thus an important step is characterizing the dimensionality of complex nonlinear responses of structures in this work the dimensionality of the nonlinear response of a post buckled beam is investigated significant detail is dedicated to carefully introducing the experiment the verification of a finite element model and the dimensionality estimation algorithm as it is hoped that this system may help serve as a benchmark test case it is shown that with minor modifications the method of false nearest neighbors can quantitatively distinguish between the response dimension of various snap through non snap through random and deterministic loads the state space dimension of the nonlinear system in question increased from 2 to 10 as the system response moved from simple low level harmonic to chaotic snap through beyond the problem studied herein the techniques developed will serve as a prescriptive guide in developing fast and accurate dimensionally reduced models of nonlinear systems and eventually as a tool for adaptive dimension reduction in numerical modeling the results are especially relevant in the aerospace industry for the design of thin structures such as beams panels and shells which are all capable of spatio temporally complex dynamic responses that are difficult and computationally expensive to model

kernel based approaches to local nonlinear non parametric variable selection in this paper we consider the variable selection problem for a nonlinear non parametric system two approaches are proposed one top down approach and one bottom up approach the top down algorithm selects a variable by detecting if the corresponding partial derivative is zero or not at the point of interest the algorithm is shown to have not only the parameter but also the set convergence this is critical because the variable selection problem is binary a variable is either selected or not selected the bottom up approach is based on the forward backward stepwise selection which is designed to work if the data length is limited both approaches determine the most important variables locally and allow the unknown non parametric nonlinear system to have different local dimensions at different points of interest further two potential applications along with numerical simulations are provided to illustrate the usefulness of the proposed algorithms

the awgn red alert problem consider the following unequal error protection scenario one special message dubbed the red alert message is required to have an extremely small probability of missed detection the remainder of the messages must keep their average probability of error and probability of false alarm below a certain threshold the goal then is to design a codebook that maximizes the error exponent of the red alert message while ensuring that the average probability of error and probability of false alarm go to zero as the blocklength goes to infinity this red alert exponent has previously been characterized for discrete memoryless channels this paper completely characterizes the optimal red alert exponent for additive white gaussian noise channels with block power constraints

quantitative analysis of financial system fragility based on manifold curvature abstract manifold curvature is an intrinsic attribute of a dynamical system by analyzing curvature characteristics of embedding attractor manifold resistance and resilience of financial dynamic system are investigated in the face of disturbances thus system fragility is discussed quantitatively from a systemic perspective first financial time series data are reconstructed into a high dimensional phase space topologically equivalent to its original dynamical system then manifold learning is employed to extract attractor manifold embedding in financial dynamic system last by numerical calculation the manifold curvature is obtained and further visualized with spatial mapping in empirical study shanghai composite index and dow jones industrial index are adopted as experimental data and then a more comprehensive understanding about financial systems fragility is obtained by exploring the dynamical properties related to manifold curvature

distributed source coding using syndromes discus design and construction we address the problem of compressing correlated distributed sources i e correlated sources which are not co located or which cannot cooperate to directly exploit their correlation we consider the related problem of compressing a source which is correlated with another source that is available only at the decoder this problem has been studied in the information theory literature under the name of the slepian wolf 1973 source coding problem for the lossless coding case and as rate distortion with side information for the lossy coding case we provide a constructive practical framework based on algebraic trellis codes dubbed as distributed source coding using syndromes discus that can be applicable in a variety of settings simulation results are presented for source coding of independent and identically distributed i i d gaussian sources with side information available at the decoder in the form of a noisy version of the source to be coded our results reveal the promise of this approach using trellis based quantization and coset construction the performance of the proposed approach is 2 5 db from the wyner ziv 1976 bound

the method of types information theory the method of types is one of the key technical tools in shannon theory and this tool is valuable also in other fields in this paper some key applications are presented in sufficient detail enabling an interested nonspecialist to gain a working knowledge of the method and a wide selection of further applications are surveyed these range from hypothesis testing and large deviations theory through error exponents for discrete memoryless channels and capacity of arbitrarily varying channels to multiuser problems while the method of types is suitable primarily for discrete memoryless models its extensions to certain models with memory are also discussed

analogy between a 10d model for nonlinear wave wave interaction in a plasma and the 3d lorenz dynamics this paper investigates nonlinear wave wave interactions in a system that describes a modified decay instability and consists of three langmuir and one ion sound waves as a means to establish that the underlying dynamics exists in a 3d space and that it is of the lorenz type both continuous and discrete time multivariable global models were obtained from data these data were obtained from a 10d dynamical system that describes the modified decay instability obtained from zakharov s equations which characterise langmuir turbulence this 10d model is equivariant under a continuous rotation symmetry and a discrete order 2 rotation symmetry when the continuous rotation symmetry is modded out that is when the dynamics are represented with the continuous rotation symmetry removed under a local diffeomorphism it is shown that a 3d system may describe the underlying dynamics for certain parameter values the models obtained using global modelling techniques from three time series from the 10d dynamics with the continuous rotation symmetry modded out generate attractors which are topologically equivalent these models can be simulated easily and due to their simplicity are amenable for analysis of the original dynamics after symmetries have been modded out moreover it is shown that all of these attractors are topologically equivalent to an attractor generated by the well known lorenz system 2003 elsevier science b v all rights reserved

maximum entropy and conditional probability it is well known that maximum entropy distributions subject to appropriate moment constraints arise in physics and mathematics in an attempt to find a physical reason for the appearance of maximum entropy distributions the following theorem is offered the conditional distribution of x l given the empirical observation 1 n sum n i l h x i alpha where x 1 x 2 cdots are independent identically distributed random variables with common density g converges to f lambda x e lambda t h x g x suitably normalized where lambda is chosen to satisfy int f lambda x h x dx alpha thus the conditional distribution of a given random variable x is the normalized product of the maximum entropy distribution and the initial distribution this distribution is the maximum entropy distribution when g is uniform the proof of this and related results relies heavily on the work of zabell and lanford

difference equations versus differential equations a possible equivalence for the rossler system abstract when a set of nonlinear differential equations is investigated most of time there is no analytical solution and only numerical integration techniques can provide accurate numerical solutions in a general way the process of numerical integration is the replacement of a set of differential equations with a continuous dependence on the time by a model for which the time variable is discrete in numerical investigations a fourth order runge kutta integration scheme is usually sufficient nevertheless sometimes a set of difference equations may be required and in this case standard schemes like the forward euler backward euler or central difference schemes are used the major problem encountered with these schemes is that they offer numerical solutions equivalent to those of the set of differential equations only for sufficiently small integration time steps in some cases it may be of interest to obtain difference equations with the same type of solutions as for the differential equations but with significantly large time steps nonstandard schemes as introduced by mickens nonstandard finite difference models of differential equations world scientific 1994 allow to obtain more robust difference equations in this paper using such nonstandard scheme we propose some difference equations as discrete analogues of the rossler system for which it is shown that the dynamics is less dependent on the time step size than when a nonstandard scheme is used in particular it has been observed that the solutions to the discrete models are topologically equivalent to the solutions to the rossler system as long as the time step is less than the threshold value associated with the nyquist criterion

stabilizing embedology geometry preserving delay coordinate maps delay coordinate mapping is an effective and widely used technique for reconstructing and analyzing the dynamics of a nonlinear system based on time series outputs the efficacy of delay coordinate mapping has long been supported by takens embedding theorem which guarantees that delay coordinate maps use the time series output to provide a reconstruction of the hidden state space that is a one to one embedding of the system s attractor while this topological guarantee ensures that distinct points in the reconstruction correspond to distinct points in the original state space it does not characterize the quality of this embedding or illuminate how the specific parameters affect the reconstruction in this paper we extend takens result by establishing conditions under which delay coordinate mapping is guaranteed to provide a stable embedding of a system s attractor beyond only preserving the attractor topology a stable embedding preserves the attractor geometry by ensuring that distances between points in the state space are approximately preserved in particular we find that delay coordinate mapping stably embeds an attractor of a dynamical system if the stable rank of the system is large enough to be proportional to the dimension of the attractor the stable rank reflects the relation between the sampling interval and the number of delays in delay coordinate mapping our theoretical findings give guidance to choosing system parameters echoing the tradeoff between irrelevancy and redundancy that has been heuristically investigated in the literature our initial result is stated for attractors that are smooth submanifolds of euclidean space with extensions provided for the case of strange attractors

quantitative theory of driven nonlinear brain dynamics abstract strong periodic stimuli such as bright flashing lights evoke nonlinear responses in the brain and interact nonlinearly with ongoing cortical activity but the underlying mechanisms for these phenomena are poorly understood at present the dominant features of these experimentally observed dynamics are reproduced by the dynamics of a quantitative neural field model subject to periodic drive model power spectra over a range of drive frequencies show agreement with multiple features of experimental measurements exhibiting nonlinear effects including entrainment over a range of frequencies around the natural alpha frequency f subharmonic entrainment near 2 f and harmonic generation further analysis of the driven dynamics as a function of the drive parameters reveals rich nonlinear dynamics that is predicted to be observable in future experiments at high drive amplitude including period doubling bistable phase locking hysteresis wave mixing and chaos indicated by positive lyapunov exponents moreover photosensitive seizures are predicted for physiologically realistic model parameters yielding bistability between healthy and seizure dynamics these results demonstrate the applicability of neural field models to the new regime of periodically driven nonlinear dynamics enabling interpretation of experimental data in terms of specific generating mechanisms and providing new tests of the theory

refinements of the gibbs conditioning principle refinements of sanov s large deviations theorem lead via csiszar s information theoretic identity to refinements of the gibbs conditioning principle which are valid for blocks whose length increase with the length of the conditioning sequence sharp bounds on the growth of the block length with the length of the conditioning sequence are derived

postprocessing methods for finding the embedding dimension of chaotic time series one problem when using the global false nearest neighbors gfnn method and cao s method to estimate embedding dimension is that their effectiveness is affected by the ratio of signal power to noise power snr simple models are proposed to explain the curves commonly obtained when using the gfnn method and cao s method methods are proposed for systematically estimating the embedding dimension prior information is incorporated to improve the estimates

estimating correlation dimension from a chaotic time series when does plateau onset occur abstract suppose that a dynamical system has a chaotic attractor a with a correlation dimension d 2 a common technique to probe the system is by measuring a single scalar function of the system state and reconstructing the dynamics in an m dimensional space using the delay coordinate technique the estimated correlation dimension of the reconstructed attractor typically increases with m and reaches a plateau on which the dimension estimate is relatively constant for a range of large enough m values the plateaued dimension value is then assumed to be an estimate of d 2 for the attractor in the original full phase space in this paper we first present rigorous results which state that for a long enough data string with low enough noise the plateau onset occurs at m ceil d 2 where ceil d 2 standing for ceiling of d 2 is the smallest integer greater than or equal to d 2 we then numerical examples illustrating the theoretical prediction in addition we discuss new findings showing how practical factors such as a lack of data and observational noise can produce results that may seem to be inconsistent with the theoretically predicted plateau onset at m ceil d 2

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

entropic nonextensivity as a measure of time series complexity abstract information entropy is applied to the analysis of time series generated by dynamical systems complexity of a temporal or spatio temporal signal is defined as the difference between the sum of entropies of the local linear regions of the trajectory manifold and the entropy of the globally linearized manifold when the entropies are tsallis entropies the complexity is characterized by the value of q

universal prediction this paper consists of an overview on universal prediction from an information theoretic perspective special attention is given to the notion of probability assignment under the self information loss function which is directly related to the theory of universal data compression both the probabilistic setting and the deterministic setting of the universal prediction problem are described with emphasis on the analogy and the differences between results in the two settings

practical method for determining the minimum embedding dimension of a scalar time series abstract a practical method is proposed to determine the minimum embedding dimension from a scalar time series it has the following advantages 1 does not contain any subjective parameters except for the time delay for the embedding 2 does not strongly depend on how many data points are available 3 can clearly distinguish deterministic signals from stochastic signals 4 works well for time series from high dimensional attractors 5 is computationally efficient several time series are tested to show the above advantages of the method

subset source coding this paper studies the fundamental limits of storage for structured data where statistics and structure are both critical to the application accordingly a framework is proposed for optimal lossless and lossy compression of subsets of the possible realizations of a discrete memoryless source dms for the lossless subset compression problem it turns out that the optimal source code may not index the conventional source typical sequences but rather index certain subset typical sequences consistent with the subset of interest building upon an achievability and a strong converse an analytic expression is given based on the shannon entropy relative entropy and subset entropy which identifies such subset typical sequences for a broad class of subsets of a dms intuitively subset typical sequences belong to those typical sets which highly intersect the subset of interest but are still closest to the source distribution in the sense of relative entropy for the lossy subset compression problem an upper bound is derived on the subset rate distortion function in terms of the subset mutual information optimized over the set of conditional distributions that satisfy the expected distortion constraint with respect to the subset typical distribution and over a set of certain auxiliary subsets by proving a strong converse result this upper bound is shown to be tight for a class of symmetric subsets as shown in our numerical examples more often than not one achieves a gain in the fundamental limits in that the optimal compression rate for the subset in both the lossless and lossy settings can be strictly smaller than the source entropy and the source rate distortion function respectively although exceptions are also possible

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

on the non equivalence of observables in phase space reconstructions from recorded time series in practical problems of phase space reconstruction it is usually the case that the reconstruction is much easier using a particular recorded scalar variable this seems to contradict the general belief that all variables of a dynamical system are equivalent in phase space reconstruction problems this paper will argue that in many cases the choice of a particular scalar time series from which to reconstruct the original dynamics could be critical it is argued that different dynamical variables do not provide the same level of information observability of the underlying dynamics and as a consequence the quality of a global reconstruction critically depends on the recorded variable examples in which the choice of observables is critical are discussed and the level of information contained in a given variable is quantified in the case where the original system is known a clear example of such a situation arises in the r ossler system for which the performance of a global vector field reconstruction technique is investigated using time series of variables x y or z taken one at a time

optimal topology design using a global self organisational approach abstract a method based on a self organisational approach has been developed where a local state operator defines the state of each finite element at each iteration the algorithm is based on the principle of local adaptation with global feedback in the form of a derivative equation based on von mises stresses that allows the local remodelling function to vary at each time step the state operator has the form of a nonlinear differential equation solved iteratively based on the material density and strain energy density within each element a constraint equation is formulated based on a maximum deviatoric strain energy criteria with the objective to minimise the mass of the design domain subject to the above constraint a number of examples are presented to demonstrate the use of this approach

unequal error protection an information theoretic perspective an information theoretic framework for unequal error protection is developed in terms of the exponential error bounds the fundamental difference between the bit wise and message wise unequal error protection uep is demonstrated for fixed length block codes on discrete memoryless channels dmcs without feedback effect of feedback is investigated via variable length block codes it is shown that feedback results in a significant improvement in both bit wise and message wise ueps except the single message case for missed detection the distinction between false alarm and missed detection formalizations for message wise uep is also considered all results presented are at rates close to capacity

maximum approximate entropy and r threshold a new approach for regularity changes detection abstract approximate entropy apen has been widely used as an estimator of regularity in many scientific fields it has proved to be a useful tool because of its ability to distinguish different system s dynamics when there is only available short length noisy data incorrect parameter selection embedding dimension m threshold r and data length n and the presence of noise in the signal can undermine the apen discrimination capacity in this work we show that r m a x a p e n m r m a x n a p e n m a x can also be used as a feature to discern between dynamics moreover the combined use of a p e n m a x and r m a x allows a better discrimination capacity to be accomplished even in the presence of noise we conducted our studies using real physiological time series and simulated signals corresponding to both low and high dimensional systems when a p e n m a x is incapable of discerning between different dynamics because of the noise presence our results suggest that r m a x provides additional information that can be useful for classification purposes based on cross validation tests we conclude that for short length noisy signals the joint use of a p e n m a x and r m a x can significantly decrease the misclassification rate of a linear classifier in comparison with their isolated use

difficulties in detecting chaos in a complex system abstract the sequences given by a 7d map are analysed by means of the methods widely used to detect chaos in the real world in order to test their sensitivity to chaotic features of a non linear system determined by comparatively high number of parameters the same diagnostic approaches are applied to the 3d lorenz map for comparison the results show that for some of the sequences yielded from the 7d map the adopted methods are not able to give a straightforward answer to the question if the system is chaotic as in the 3d case since the sequences subject to the analysis are not contaminated by noise and are sufficiently long it could be assumed that such difficulties arise likely due to specific internal features of the more complex system it is found also that an increase from 0 01 to 0 5 of the sampling step determining the sequences obtained from the 7d map masks the chaos in some of them

simulation of random processes and rate distortion theory we study the randomness necessary for the simulation of a random process with given distributions on terms of the finite precision resolvability of the process finite precision resolvability is defined as the minimal random bit rate required by the simulator as a function of the accuracy with which the distributions are replicated the accuracy is quantified by means of various measures variational distance divergence orstein 1973 prohorov 1956 and related measures of distance between the distributions of random process in the case of ornstein prohorov and other distances of the kantorovich vasershtein type we show that the finite precision resolvability is equal to the rate distortion function with a fidelity criterion derived from the accuracy measure this connection leads to new results on nonstationary rate distortion theory in the case of variational distance the resolvability of stationary ergodic processes is shown to equal entropy rate regardless of the allowed accuracy in the case of normalized divergence explicit expressions for finite precision resolvability are obtained in many cases of interest and connections with data compression with minimum probability of block error are shown

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

topological properties and fractal analysis of a recurrence network constructed from fractional brownian motions many studies have shown that we can gain additional information on time series by investigating their accompanying complex networks in this work we investigate the fundamental topological and fractal properties of recurrence networks constructed from fractional brownian motions fbms first our results indicate that the constructed recurrence networks have exponential degree distributions the average degree exponent increases first and then decreases with the increase of hurst index h of the associated fbms the relationship between h and can be represented by a cubic polynomial function we next focus on the motif rank distribution of recurrence networks so that we can better understand networks at the local structure level we find the interesting superfamily phenomenon i e the recurrence networks with the same motif rank pattern being grouped into two superfamilies last we numerically analyze the fractal and multifractal properties of recurrence networks we find that the average fractal dimension db of recurrence networks decreases with the hurst index h of the associated fbms and their dependence approximately satisfies the linear formula db 2 h which means that the fractal dimension of the associated recurrence network is close to that of the graph of the fbm moreover our numerical results of multifractal analysis show that the multifractality exists in these recurrence networks and the multifractality of these networks becomes stronger at first and then weaker when the hurst index of the associated time series becomes larger from 0 4 to 0 95 in particular the recurrence network with the hurst index h 0 5 possesses the strongest multifractality in addition the dependence relationships of the average information dimension d 1 and the average correlation dimension d 2 on the hurst index h can also be fitted well with linear functions our results strongly suggest that the recurrence network inherits the basic characteristic and the fractal nature of the associated fbm series

chaotic attractors of an infinite dimensional dynamical system abstract we study the chaotic attractors of a delay differential equation the dimension of several attractors computed directly from the definition agrees to experimental resolution with the dimension computed from the spectrum of lyapunov exponents according to a conjecture of kaplan and yorke assuming this conjecture to be valid as the delay parameter is varied from computations of the spectrum of lyapunov exponents we observe a roughly linear increase from two to twenty in the dimension while the metric entropy remains roughly constant these results are compared to a linear analysis and the asymptotic behavior of the lyapunov exponents is derived

on runlength codes several results on binary d k codes are given first a novel derivation for the capacity of these codes based on information theoretic principles is given based on this result the spectrum of a d k code is computed finally the problem of computing the capacity of the binary symmetric channel under the condition that the input sequences satisfy the d k constraint is considered lower bounds on the capacity of such a channel are derived

the adaptation of bone apparent density to applied load a phenomenological theory of bone remodeling was developed with improved spatial stability compared to some of the more standard formulations the improved stability was created by changing the nature of the remodeling differential equation to have an exponential character as a result the theoretical predictions are consistent with the experimental observation that changes in bone density during disuse after hip surgery during growth and during aging are all consistent with an exponential dependence of density on time the new theory and the standard theory were both used to model the time course of bone changes in two animal models of bone loss during disuse the new theory was better able to model the results of the experiments than the standard theory the basic continuum theory underlying the remodeling theory was presented in some detail this presentation was used to motivate the development of the new theory as the standard theories can predict non smooth distributions of bone density rather than the expected smooth distributions it was shown that these non smooth distributions are a violation of the continuum assumption one of the bases for the theory of finite element stress analysis the new model s stability was investigated using example problems and shown to be improved compared to the standard model

markov types and minimax redundancy for markov sources redundancy of universal codes for a class of sources determines by how much the actual code length exceeds the optimal code length in the minimax scenario one designs the best code for the worst source within the class such minimax redundancy comes in two flavors average minimax or worst case minimax we study the worst case minimax redundancy of universal block codes for markovian sources of any order we prove that the maximal minimax redundancy for markov sources of order r is asymptotically equal to 1 2m sup r m 1 log sub 2 n log sub 2 a sub m sup r lnlnm sup 1 m 1 lnm o 1 where n is the length of a source sequence m is the size of the alphabet and a sub m sup r is an explicit constant e g we find that for a binary alphabet m 2 and markov of order r 1 the constant a sub 2 sup 1 16 spl middot g spl ap 14 655449504 where g is the catalan number unlike previous attempts we view the redundancy problem as an asymptotic evaluation of certain sums over a set of matrices representing markov types the enumeration of markov types is accomplished by reducing it to counting eulerian paths in a multigraph in particular we propose exact and asymptotic formulas for the number of strings of a given markov type all of these findings are obtained by analytic and combinatorial tools of analysis of algorithms

extracting qualitative dynamics from experimental data abstract we consider the notion of qualitative information and the practicalities of extracting it from experimental data our approach based on a theorem of takens draws on ideas from the generalized theory of information known as singular system analysis due to bertero pike and co workers we illustrate our technique with numerical data from the chaotic regime of the lorenz model

chaos in the discrete time algorithm for bone density remodeling rate equations we compare the predictions of the differential equation form of a class of bone density stress adaptation models with their associated discrete time computational algorithms although our considerations apply to the class of adaptation models based on bulk or apparent bone density remodeling we focus attention on a particular model in this class a model employed by weinans et al trans orthop res soc 14 310 1989 trans first world congress of biomechanics vol ii p 75 1990 we show that the discrete time computational algorithm of that stress adaptation model has a well known chaos mechanism for stress values of practical interest further we obtain a condition on the discrete time step that prevents the transition to chaos and conditions that insure monotonic convergence this chaos mechanism is only present in the discrete time computational algorithm we show that the corresponding differential equation form of the bone density stress adaptation model is smooth monotonic and nonchaotic

use of recurrence plot and recurrence quantification analysis in taiwan unemployment rate time series abstract the aim of the article is to answer the question if the taiwan unemployment rate dynamics is generated by a non linear deterministic dynamic process this paper applies a recurrence plot and recurrence quantification approach based on the analysis of non stationary hidden transition patterns of the unemployment rate of taiwan the case study uses the time series data of the taiwan s unemployment rate during the period from 1978 01 to 2010 06 the results show that recurrence techniques are able to identify various phases in the evolution of unemployment transition in taiwan

detection of synchronization between chaotic signals an adaptive similarity based approach we present an adaptive similarity based approach to detect generalized synchronization gs with n m phase synchronization ps where n and m are integers and one of them is 1 this approach is based on the similarity index si and gaussian mixture model with the minimum description length criterion the clustering method which is shown to be superior to the closeness and connectivity of a continuous function is employed in this study to detect the existence of gs with n m ps we conducted a computer simulation and a finger lifting experiment to illustrate the effectiveness of the proposed method in the simulation of a rossler lorenz system our method outperformed the conventional si and gs with 2 1 ps within the coupled system was found in the experiment of self paced finger lifting movement cortico muscular gs with 1 2 and 1 3 ps was found between the surface electromyogram signals on the first dorsal interossei muscle and the magnetoencephalographic data in the motor area the gs with n m ps n or m 1 has been simultaneously resolved from both simulation and experiment the proposed approach thereby provides a promising means for advancing research into both nonlinear dynamics and brain science

interpreting supply chain dynamics a quasi chaos perspective abstract chaotic phenomena chaos amplification and other interesting nonlinear behaviors have been observed in supply chain systems chaos can be defined theoretically if the dynamics under study are produced only by deterministic factors however deterministic settings rarely present themselves in reality in fact real data are typically unknown how can the chaos theory and its related methodology be applied in the real world when the demand is stochastic the interpretation and distribution of the lyapunov exponents derived from the effective inventory at different supply chain levels are not similar to those under deterministic demand settings are the observed dynamics of the effective inventory random chaotic or simply quasi chaos in this study we investigate a situation whereby the chaos analysis is applied to a time series as if its underlying structure deterministic or stochastic is unknown the result shows clear distinction in chaos characterization between the two categories of demand process deterministic vs stochastic it also highlights the complexity of the interplay between stochastic demand processes and nonlinear dynamics therefore caution should be exercised in interpreting system dynamics when applying chaos analysis to a system of unknown underlying structure by understanding this delicate interplay decision makers have the better chance to tackle the problem correctly or more effectively at the demand end or the supply end

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

optimal reconstruction of dynamical systems a noise amplification approach in this work we propose an objective function to guide the search for a state space reconstruction of a dynamical system from a time series of measurements these statistics can be evaluated on any reconstructed attractor thereby allowing a direct comparison among different approaches uniform or nonuniform delay vectors pca legendre coordinates etc it can also be used to select the most appropriate parameters of a reconstruction strategy in the case of delay coordinates this translates into finding the optimal delay time and embedding dimension from the absolute minimum of the advocated cost function its definition is based on theoretical arguments on noise amplification the complexity of the reconstructed attractor and a direct measure of local stretch which constitutes an irrelevance measure the proposed method is demonstrated on synthetic and experimental time series

detecting determinism in short time series using a quantified averaged false nearest neighbors approach we propose a criterion to detect determinism in short time series this criterion is based on the estimation of the parameter e 2 defined by the averaged false neighbors method for analyzing time series cao physica d 110 43 1997 using surrogate data testing with several chaotic and stochastic simulated time series we show that the variation coefficient of e 2 over a few values of the embedding dimension d defines a suitable statistic to detect determinism in short data sequences this result holds for a time series generated by a high dimensional chaotic system such as the mackey glass one different decreasing lengths of the time series are included in the numerical experiments for both synthetic and real world data we also investigate the robustness of the criterion in the case of deterministic time series corrupted by additive noise

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

influence of noise on the averaged false neighbors method for analyzing time series this paper deals with the influence of noise on the averaged false neighbors method afn proposed by l cao for analyzing time series and describing the dynamical properties of their underlying process first we give a theoretical justification of the afn method results for a pure random time series white gaussian noise then we present some numerical experiments corresponding to different known chaotic processes corrupted by noise simulations on real measured data are also presented eventually after discussing these simulations we are led to state some practical results on the critical noise level not to be exceeded for getting usable results of the afn algorithm

the error exponent for the noiseless encoding of finite ergodic markov sources a new approach to the classical fixed length noiseless source coding problem is proposed for the case of finite ergodic markov sources this approach is based on simple counting arguments the central notion of markov type a set containing all the source sequences having the same transition counts from letter to letter is introduced and the cardinality of such a set is evaluated via graph theoretical tools the error exponent is shown to be a weighted average of informational divergences and the universal character of the result is stressed as a corollary the classical source coding theorem determining the achievable rates is rederived

detecting intrinsic dynamics of traffic flow with recurrence analysis and empirical mode decomposition abstract in this paper we apply the empirical mode decomposition emd method to the recurrence plot rp and recurrence quantification analysis rqa to evaluate the frequency and time evolving dynamics of the traffic flow based on the cumulative intrinsic mode functions extracted by the emd the frequency evolving rp regarding different oscillation of modes suggests that apparent dynamics of the data considered are mainly dominated by its components of medium and low frequencies while severely affected by fast oscillated noises contained in the signal noises are then eliminated to analyze the intrinsic dynamics and consequently the denoised time evolving rqa diversely characterizes the properties of the signal and marks crucial points more accurately where white bands in the rp occur whereas a strongly qualitative agreement exists between all the non denoised rqa measures generally the emd combining with the recurrence analysis sheds more reliable abundant and inherent lights into the traffic flow which is meaningful to the empirical analysis of complex systems

quantifying the closeness of fractal measures numerically fractal measures are represented by finite sets of discrete points e g those generated by trajectories we introduce a cross correlation integral to evaluate the distinctness and overlap of two such sets and present a criterion to determine up to what precision the two sets are realizations of the same measure

comparison of three nonlinear seizure prediction methods by means of the seizure prediction characteristic epilepsy is characterized by the spontaneous and unforeseeable occurrence of seizures during which the perception or behavior of patients is disturbed the predictability of these seizures would render novel therapeutic approaches possible several prediction methods have claimed to be able to predict seizures based on eeg recordings minutes in advance however the term seizure prediction is not unequivocally defined different criteria to assess prediction methods exist and only little attention has been paid to issues of sensitivity and false prediction rate we introduce an assessment criterion called the seizure prediction characteristicthat incorporates the assessment of sensitivity and false prediction rate within this framework three nonlinear seizure prediction methods were evaluated on a large eeg data pool of 21 patients altogether 582 h intracranial eeg data and 88 seizures were examined with a rate of 1 3 6 false predictions per day the dynamical similarity index achieves a sensitivity between 21 and 42 which was the best result of the three methods sensitivity was between 18 and 31 for the extended prospective version of the accumulated energy and between 13 and 30 for the effective correlation dimension these results still are not sufficient for clinical applications

chaotic bayesian optimal prediction method and its application in hydrological time series the embedding dimension and the number of nearest neighbors are very important parameters in the prediction of chaotic time series to reduce the prediction errors and the uncertainties in the determination of the above parameters a new chaos bayesian optimal prediction method cbopm is proposed by choosing optimal parameters in the local linear prediction method llpm and improving the prediction accuracy with bayesian theory in the new method the embedding dimension and the number of nearest neighbors are combined as a parameter set the optimal parameters are selected by mean relative error mre and correlation coefficient cc indices according to optimization criteria real hydrological time series are taken to examine the new method the prediction results indicate that cbopm can choose the optimal parameters adaptively in the prediction process compared with several llpm models the cbopm has higher prediction accuracy in predicting hydrological time series

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

adaptive metrics in the nearest neighbours method we present a modification of the nearest neighbours method in order to improve accuracy of forecast of a time series this modification is characterized by an adaptive metrics which is able to adjust its parameters to each segment of a time series in order to assess the accuracy of the presented method synthetic and real time series were predicted we show that this modified metrics is advantageous in prediction in comparison with standard euclidean metrics or weighted metrics

structure selection techniques applied to continous time nonlinear models this paper addresses the problem of choosing the multinomials that should compose a polynomial mathematical model starting from data the mathematical representation used is a nonlinear differential equation of the polynomial type some approaches that have been used in the context of discrete time models are adapted and applied to continuous time models two examples are included to illustrate the main ideas models obtained with and without structure selection are compared using topological analysis the main differences between structure selected models and complete structure models are i the former are more parsimonious than the latter ii a predefined fixed point configuration can be guaranteed for the former and iii the former set of models produce attractors that are topologically closer to the original attractor than those produced by the complete structure models 2001 elsevier science b v all rights reserved

testing predictability of symbolic sequences using time delay embedding a criterion is proposed to determine the embedding dimension from symbolic sequences several artificial and dna sequences are tested by using this criterion and the likelihood of determinism in those sequences are investigated based on the results of the time delay embedding some interesting numerical results have been obtained

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

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

optimal embedding parameters a modelling paradigm the reconstruction of a dynamical system from a time series requires the selection of two parameters the embedding dimension de and the embedding lag many competing criteria to select these parameters exist and all are heuristic within the context of modelling the evolution operator of the underlying dynamical system we show that one only need be concerned with the product de we introduce an information theoretic criterion for the optimal selection of the embedding window dw de for infinitely long time series this method is equivalent to selecting the embedding lag that minimises the nonlinear model prediction error for short and noisy time series we find that the results of this new algorithm are data dependent and are superior to estimation of embedding parameters with the standard techniques

global asymptotic stability of bone remodeling theories a new approach based on non linear dynamical systems analysis abstract mathematical tools for the analysis of nonlinear dynamical systems are applied to the study of stability of bone remodeling theories as a practical application the same problem studied by harrigan and hamilton 1992 and cowin et al 1994b is analysed using these tools and their findings on the necessary and sufficient conditions to ensure local asymptotic stability are easily confirmed using a general approach based on lyapunov s method the same condition has been found to be necessary and sufficient also for the global asymptotic stability thus confirming a result obtained by harrigan and hamilton 1994 by variational methods applied to finite element models the proof is based on the discretization of the spatial domain but the results for the continuum can be easily extrapolated