on the condition numbers of large semidefinite toeplitz matrices abstract this paper is devoted to asymptotic estimates for the spectral or euclidean condition numbers t n a t n a t 1 n a of large n n toeplitz matrices t n a in the case where the symbol a is an l function and re a 0 almost everywhere we describe several classes of symbols a for which t n a increases like log n x n x or even e xn

lyapunov exponents for a duffing oscillator with the help of a parallel computer we perform a systematic computation of lyapunov exponents for a duffing oscillator driven externally by a force proportional to cos t in contrast to the familiar situation in discrete time systems where one finds windows of regularity embedded in intervals of chaos we find the continuous time duffing oscillator to contain a quite regular epetition of relatively self similar islands of chaos i e regions characterized by positive exponents embedded in large seas of regularity negative exponents we also investigate the effect of driving the oscillator with a jacobian elliptic function cn t m for m 0 one has cn t 0 cos t the usual trigonometric pumping for m 1 one has cn t 1 sech t a hyperbolic pumping when 0 m 1 the jacobian function is an intermediary double periodic function with periodicity depending on m and with fourier spectrum consisting of a regular train of narrow lines with varying envelope using m to tune the drive appropriately one may displace the islands of chaos in parameter space thus jacobian pumping provides a possible way of cleaning chaos in regions of the parameter space for periodically driven systems

correlation dimension of electroglottographic data from healthy and pathologic subjects this paper considers the effects of nonstationarity noise and finite data sets on the estimation of the correlation dimension of time series data characterizing the vibratory behavior of the vocal folds the electroglottographic signal of sustained a phonations from 10 healthy subjects and 20 subjects with vocal fold pathologies were used to reconstruct the state space in successively higher embeddings using the method of lags the dimension values were calculated from the scaling region the level area of the slope plots which did not increase for higher embeddings reasonably defined scaling regions were found in all of the data from the healthy subjects and from five of the pathologic subjects with values saturating between the first and second embeddings the egg data from those five pathologic subjects were nearly periodic from the remaining 15 subjects the scaling regions were highly constricted with nonconstant slopes so that dimension values could not be confidently estimated the results suggest that the correlation dimension is a highly subjective measure which is not usefully applied to abnormal egg data it is recommended that if used correlation dimension statistics need to be presented cautiously and graphical presentation of the data should be included

calculation of lyapunov exponents avoiding spurious elements abstract a new approach to evaluate lyapunov exponents from time series is discussed with the help of this approach the number of calculated lyapunov exponents can be reduced to the physically relevant ones calculations in high dimensional embedding spaces are performed on smaller data bases with improved speed more direct information is obtained from the dynamics of the underlying dynamical system

a matrix method for estimating the liapunov exponent of one dimensional systems let 0 1 0 1 be a piecewise monotonie expanding map then admits an absolutely continuous invariant measure a result of kosyakin and sandler shows that can be approximated by a sequence of absolutely continuous measures n invariant under piecewise linear markov maps itn each itn is constructed on the inverse images of the turning points of the easily computable measures n are used to estimate the liapunov exponent of the idea of using markov maps for estimating the liapunov exponent is applied to both expanding and nonexpanding maps

modelling and simulating higher dimensional chaotic data abstract we consider time series data whose estimated lyapunov exponents are positive this data is represented by a probability density function on a suitably fine partition of the phase space an algorithm is presented for constructing a transformation which has a unique absolutely continuous invariant measure whose density function is and which has the same lyapunov exponents as the original time series an example is given

dynamic instability and chaos of empty and fluid filled circular cylindrical shells under periodic axial loads abstract in the present paper the dynamic stability of circular cylindrical shells subjected to static and dynamic axial loads is investigated both donnell s nonlinear shallow shell and sanders koiter shell theories have been applied to model finite amplitude static and dynamic deformations results are compared in order to evaluate the accuracy of these theories in predicting instability onset and post critical nonlinear response the effect of a contained fluid on the stability and the post critical behaviour is analyzed in detail geometric imperfections are considered and their influence on the dynamic instability and post critical behaviour is investigated chaotic dynamics of pre compressed shells is investigated by means of nonlinear time series techniques extracting correlation dimension and lyapunov exponents

cauchy toeplitz matrices and some applications abstract some properties of cauchy toeplitz matrices of the form tn 1 i j 1 2 are investigated it is shown that minimal singular values of tn converge to zero with increasing n some applications of cauchy toeplitz matrices to the solution of singular integral equations by the method of discrete singularities are considered

a note on singular values of cauchy toeplitz matrices abstract we consider the asymptotic behaviour of the smallest singular values of the n n sections of a general infinite cauchy toeplitz matrix

strange attractors of uniform flows on considere des attracteurs orbitalement stables de ces flots sur le tore solide ouvert d 2 s 1 qui ont une vitesse dans la direction s 1 on trouve qu un tel attracteur est la frontiere d une suite strictement emboitee de tores solides ouverts positivement invariants

theoretical time series analysis from electric field oscillations generated by rate equations of generation recombination processes in n type semiconductors abstract we have analyzed the nonlinear electric field time series theoretically generated by a set of three coupled first order differential equations in order to interpret the chaotic behavior present in measurements of low frequency current oscillations in semiconductor samples in the present work we carry out a direct investigation of the theoretical time series applying techniques of experimental nonlinear time series analysis we have reconstructed the phase space of the system and calculated some parameters as the correlation dimension of the reconstructed attractors the lyapunov exponent s spectrum and the kaplan yorke dimension the conclusions are consistent with the theory of chaotic systems and with the available experimental results for semiconductors

matrix pencil methodologies for computing the greatest common divisor of polynomials hybrid algorithms and their performance the computation of the greatest common divisor gcd of several polynomials is a problem that emerges in many fields of applications the gcd computation has a non generic nature and thus its numerical computation is a hard problem in this paper we examine the family of matrix pencil methods for gcd computation and investigate their performance as far as their complexity error analysis and their effectiveness for evaluating approximate solutions the relative merits of the various variants of such methods are examined for the different cases of sets of polynomials with varying number of elements and degree the developed algorithms combine symbolical and numerical programming and this is what we define here as hybrid computations the combination of numerical operations with symbolical programming can improve the nature of the methods and guarantees the stability of the algorithm furthermore it emphasizes the significance of hybrid computations in complex problems such as the computation of gcd all m

smoothing data with local instabilities for the identification of chaotic systems chaotic systems are characterized by the local divergence of nearby orbits in state space this provokes a sensitive dependence on initial conditions and in turn drastically limits the accuracy of long term predictions this has important implications in the filtering of data generated by such systems in the present paper the use of global smoothers for chaotic data is investigated the ultimate objective is to be able to identify dynamically valid models from smoothed data when the identification from the original noisy data has completely failed the objective is to produce identified models that faithfully reproduce the dynamical invariants of the original system such as the geometry of the attractors in state space the largest lyapunov exponent fractal dimensions and poincare sections numerical examples are included which illustrate the main points of the paper

generalized one parameter bifurcation diagram reconstruction using time series abstract in this paper we look into the use of time series in reconstructing bifurcation diagrams of systems that cannot be modeled directly from first principles in this reconstruction problem time series at different unknown parameter values are available these time series are then used to obtain a suitable family of nonlinear predictor functions which exhibits qualitatively similar bifurcations as the original system to solve the reconstruction problem we propose a generalized one parameter reconstruction algorithm based on principal curves we illustrate its feasibility numerically by reconstructing the bifurcation diagrams of the fitzhugh nagumo equations and the lorenz equations with parameter values mostly in the fixed point regime using the proposed algorithm we obtain a reconstruction which preserved the important features of the bifurcation diagrams of the original systems such as the hopf bifurcations of the two systems the limit cycle of the fitzhugh nagumo equations and the symmetry of the fixed points of the lorenz equations among others moreover the reconstructed systems also exhibit the same sequence of behavior as the original systems with respect to the changes in the bifurcation parameters

robust estimation of tangent maps and liapunov spectra abstract in this paper we describe three algorithms for estimating the spectrum of liapunov exponents which we have found to be robust and reasonably accurate when used in practical circumstances attention is focussed on least squares and total least squares methods for estimating tangent maps from time series data the properties of the algorithms are demonstrated by application to a range of data sets we consider numerical and experimental data and discuss the effect of additive noise and system noise on the proposed methods

stochastic spatial behaviour in deterministic pattern formation the authors introduce a new approach to the analysis of growing unstable interfaces in order to characterise the degree of their randomness highly ramified viscous fingering patterns are digitised and the related surface curvature data are evaluated using techniques common in the investigations of dynamical systems their results for the associated correlation dimension indicate that the experimentally observed deterministic patterns correspond to a spatial behaviour which is more complex than a low dimensional chaotic geometry

detecting switch dynamics in chaotic time waveform using a parametrized family of nonlinear predictors abstract an algorithm is presented for detecting switch dynamics in chaotic time waveform by the switch dynamics we mean that the chaotic time waveform is measured from a dynamical system whose bifurcation parameters are occasionally switched among a set of slightly different parameter values first the switched chaotic time waveform is divided into windows of short term time waveforms from the set of windowed time waveforms qualitatively similar parametrized family of nonlinear predictors is constructed qualitatively similar parametrized family means that the family of nonlinear predictors exhibits qualitatively similar bifurcation phenomena as the original by characterizing the windows of short term chaotic time waveforms in terms of the qualitative parameters of nonlinear predictors switch dynamics of their associated bifurcation parameters are detected for the lorenz equations the rossler equations and the mackey glass equations efficiency of the algorithm is demonstrated in the experiment chaotic time waveforms contaminated with observational noise is considered

on stability analysis via lyapunov exponents calculated based on radial basis function networks the concept of lyapunov exponents is a powerful tool for analysing the stability of nonlinear dynamic systems especially when the mathematical models of the systems are available for real world systems such models are often unknown estimating lyapunov exponents using a time series has the advantage in that no mathematical model is required however the time series based methods are believed to be reliable only for estimating positive exponents furthermore when nonlinear mapping is applied for deriving the neighbourhood to neighbourhood matrices the loads of mathematical deduction and programming increase significantly which makes it unfeasible to nonlinear systems with high dimensions in contrary the model based methods are constructive and reliable for calculating both positive and non positive exponents the use of the system jacobians is the key to the advantage of the model based methods in this article a novel approach is proposed where the system jacobians are derived based on system ap

assessing nonlinear structures in real exchange rates using recurrence plot strategies purchasing power parity ppp is an important theory at the basis of a large number of economic models however the implication derived from the theory that real exchange rates must follow stationary processes is not conclusively supported by empirical studies in a recent paper serletis and gogas appl finance econ 10 2000 615 show evidence of deterministic chaos in several oecd exchange rates as a consequence ppp rejections could be spurious in this work we follow a two stage testing procedure to test for nonlinearities and chaos in real exchange rates using a new set of techniques designed by webber and zbilut j appl physiol 76 1994 965 called recurrence quantification analysis rqa our conclusions differ slightly from serletis and gogas appl finance econ 10 2000 615 but they are also supportive of chaos for some exchange rates

recognizing chaotic time waveforms in terms of a parametrized family of nonlinear predictors abstract consider a chaotic dynamical system which exhibits a variety of chaotic time waveforms with a change in the bifurcation parameters this paper presents an algorithm for estimating the underlying bifurcation parameters of the chaotic time waveforms in experimental situation in which no a priori analytical knowledge of the dynamical system is available first we construct qualitatively similar parametrized family of nonlinear predictors only from several sets of chaotic time waveforms qualitatively similar parametrized family means that the family of nonlinear predictors exhibits qualitatively similar bifurcation phenomena as the original chaotic time waveforms are then characterized in terms of the qualitatively similar bifurcation parameters of the nonlinear predictors we call the characterization of chaotic time waveforms in terms of the underlying bifurcation parameters chaotic time waveform recognition several numerical experiments using the rossler equations show the efficiency of the algorithm the effect of observational noise included in chaotic time waveforms is also considered

constrained realization monte carlo method for hypothesis testing abstract we compare two theoretically distinct approaches to generate artificial or surrogate data for testing hypotheses about a given data set the first and more straightforward approach is to fit a single best model to the original data and then to generate surrogate data sets that are typical realizations of that model the second approach concentrates not on the model but directly on the original data it attempts to constrain the surrogate data sets so that they exactly agree with the original data for a specified set of sample statistics examples of these two approaches are provided for two simple cases a test for deviations from a gaussian distribution and a test for serial dependence in a time series additionally we consider test for nonlinearity in time series based on a fourier transform ft method and on more conventional autoregressive moving average arma fits to the data the comparative performance of hypothesis testing schemes based on these two approaches is found to depend on whether or not the discriminating statistic is pivotal a statistic is pivotal if its distribution is the same for all processes consistent with the null hypothesis the typical realization method requires that the discriminating statistic satisfy this property the constrained realization approach on the other hand does not share this requirement and can provide an accurate and powerful test without having to sacrifice flexibility in the choice of discriminating statistic

analysis of positive lyapunov exponents from random time series abstract the conventional method for estimating lyapunov spectra can give spurious positive lyapunov exponents when applied to random time series we analyze this phenomenon by considering a situation in which the method is applied to completely random time series produced by a simple stochastic model we show that the possible estimation of spurious positive lyapunov exponents is due to the statistical nature and the finiteness of data we also derive an upper bound of the largest lyapunov exponent for the model which is useful in testing positive lyapunov exponents with random shuffled surrogate data the results suggest that the method should be applied very carefully to experimentally obtained chaotic time series with possible random contamination so as to avoid spurious estimation of positive lyapunov exponents as evidences of deterministic chaos

testing for chaos in deterministic systems with noise abstract recently we introduced a new test for distinguishing regular from chaotic dynamics in deterministic dynamical systems and argued that the test had certain advantages over the traditional test for chaos using the maximal lyapunov exponent in this paper we investigate the capability of the test to cope with moderate amounts of noisy data comparisons are made between an improved version of our test and both the tangent space method and direct method for computing the maximal lyapunov exponent the evidence of numerical experiments ranging from the logistic map to an eight dimensional lorenz system of differential equations the lorenz 96 system suggests that our method is superior to tangent space methods and that it compares very favourably with direct methods

the topological invariance of lyapunov exponents in embedded dynamics abstract there are two contributions of this paper one is to provide a theoretical ground for the jacobian methods by showing that n of the lyapunov exponents of the estimated jacobian are the lyapunov exponents of an n dimensional unknown dynamical system the second is to show that it is necessary that the lyapunov exponents be calculated in the tangent space of the attractor in the embedding the method of reconstruction is to form vectors of m consecutive observations which for m 2 n is generically an embedding this is takens result a function of m variables is then fit to the data and the jacobian matrix is constructed at each point in the orbit of the data when embedding occurs at dimension m n then the lyapunov exponents of the reconstructed dynamics are the lyapunov exponents of the original dynamics this is the case for the henon the lorenz and the mackey glass systems with a first component observer however if embedding only occurs for an m n then the jacobian method yields m lyapunov exponents only n of which are the lyapunov exponents of the original system the problem is that as currently used the jacobian method is applied to the full m dimensional space of the reconstruction and not just to the n dimensional manifold that is the image of the embedding map our examples show that it is possible to get spurious lyapunov exponents that are even larger than the largest lyapunov exponent of the original system

on kicked systems modulated along the thue morse sequence we give necessary conditions for the fourier transform of the time evolution operator of kicked systems modulated along the thue morse sequence to present dirac s delta functions the case of two level systems is analysed in detail and it is found based on the trace map and numerical investigations that generically its time evolution has no quasi periodic component

a nonlinear dynamical systems analysis of fricative consonants acoustic waveforms of the strident fricatives s z sh and yog spoken by two native american english speakers are analyzed using modern chaotic analysis techniques fricative data are extracted from both intervocalic and sustained utterances for comparison acoustic waveforms of the vowels a i and u are also analyzed for 44 of the unvoiced fricative tokens in vcv contexts and 59 of the sustained voiced fricatives indications of low dimensional dynamics could be found with the given limitations of stationarity the low dimensional chaotic behavior is exhibited by a correlation dimension d2 ranging between 3 and 7 2 and by positive maximum lyapunov exponents les for the remaining fricatives results suggest that the dimensional complexity therein is greater than the maximum d2 value that could be reliably estimated from the available data about 7 8 for the intervocalic cases and 9 for the sustained cases intervocalic voiced fricatives are excluded from the analysis due to stat

measurement of the lyapunov spectrum from a chaotic time series the exponential divergence or convergence of nearby trajectories lyapunov exponents is conceptually the most basic indicator of deterministic chaos we propose a new method to determine the spectrum of several lyapunov exponents including positive zero and even negative ones from the observed time series of a single variable we have applied the method to various known model systems and also to the rayleigh b enard experiment and have elucidated the dependence of the lyapunov exponents on the rayleigh number

absolutely continuous invariant measures for one parameter families of one dimensional maps given a one parameter familyf x of maps of the interval 0 1 we consider the set of parameter values for whichf has an invariant measure absolutely continuous with respect to lebesgue measure we show that this set has positive measure for two classes of maps i f x f x where 0 4 andf x is a functionc3 near the quadratic mapx 1 x and ii f x f x mod 1 wheref isc3 f 0 f 1 0 andf has a unique nondegenerate critical point in 0 1

generalised resultants dynamic polynomial combinants and the minimal design problem the theory of dynamic polynomial combinants is linked to the linear part of the dynamic determinantal assignment problems dap which provides the unifying description of the dynamic as well as static pole and zero dynamic assignment problems in linear systems the assignability of spectrum of polynomial combinants provides necessary conditions for solution of the original dap this paper demonstrates the origin of dynamic polynomial combinants from linear systems examines issues of their representation and the parameterisation of dynamic polynomial combinants according to the notions of order and degree and examines their spectral assignment central to this study is the link of dynamic combinants to the theory of generalised resultants which provide the matrix representation of the dynamic combinants the paper considers the case of coprime set of polynomials for which spectral assignability is always feasible and provides a complete characterisation of all assignable combinants with order above an

dispersive graded entropy on computing dynamical complexity abstract we propose a phase space based statistical disorder to investigate the dynamical complexity of chaotic models the statistical disorder is defined by introducing a grade function inversely maps the mean dispersion of the trajectories in the phase space we denote the associate entropy by the dispersive graded entropy dge numerical investigation shows that dge can quantify the dynamical complexity of discrete as well as continuous chaotic systems a comparative study is also made with the other phase space based entropy measures finally the proposed measure has been applied on three types of heart rate variability hrv signals the results support the clinical observations related to the dynamics of healthy and congestive hearts

stability criteria of matrix polynomials abstractin this paper the stability of matrix polynomials is investigated first upper and lower bounds are derived for the eigenvalues of a matrix polynomial the bounds are based on the spectra

strange attractors and asymptotic measures of discrete time dissipative systems we investigate asymptotic properties of certain discrete time dynamical systems in two and three dimensions with solenoidal attractor it is proved that the asymptotic measures relevant for the generalized version of the ergodic theorem all derive from one haar measure

computation of the gcd of polynomials using gaussian transformations and shifting abstract a new numerical method for the computation of the greatest common divisor gcd of an m set of polynomials of r s pm d of maximal degree d is presented this method is based on a recently developed theoretical algorithm karcanias 1987 that uses elementary transformations and shifting operations the present algorithm takes into account the non generic nature of gcd and thus uses steps which minimize the introduction of additional errors and defines the gcd in an approximate sense for a given set pm d with a basis matrix pm the method defines first the most orthogonal uncorrupted base pr from the rows of pm where r rank pm m by applying successively gaussian transformations and shifting on the basis matrix pr e rr d 1 we produce each time a new basis matrix pz with z rank pz r the method terminates when the rank of pz is approximately equal to 1 the coefficient vector of the gcd is then defined as a row of the unit rank matrix pz the method defines the exact degree of t

index calculus for approximation methods and singular value decomposition abstract we introduce two classes of sequences a n of approximation operators moore penrose and fredholm sequences which correspond in a sense to normally solvable and fredholm operators respectively for moore penrose sequences we derive a certain qualitative behavior of the singular values of the approximation operators a n and we describe this behavior quantitatively for fredholm sequences by defining the number and the index of an approximation method

a statistical framework for testing chaotic dynamics via lyapunov exponents abstract one common criticism of the existing lyapunov exponents algorithms is the absence of a distributional theory which provides a framework for the statistical hypothesis testing for the calculated lyapunov exponents this paper presents a methodology to calculate the empirical distributions of lyapunov exponents by using a bootstrapping technique the methodology of the paper provides a formal test of chaos under the null hypothesis the numerical examples show that the method works well with small data sets

relationship of calculating the jacobian matrices of nonlinear systems and population coding algorithms in neurobiology abstract in this paper we examine how the calculation of jacobian matrices in nonlinear systems is related to population coding algorithms in neurobiology population coding algorithms have been designed to retrodict sensory stimuli or predict motor behavior from neuronal responses we simulate the visuomotor hand movement task of reaching in a plane adaptive feedback control updating of joint angles of a three joint arm was used in the model the control signal is the dot product between the visual error signal and the jacobian matrix of the direct kinematic equation of hand movement the x and y hand trajectories can follow the x and y time series of lorenz and ro ssler systems of coupled nonlinear equations in this particular example the elements of the jacobian matrix can be estimated from observed changes in joint angles and cartesian coordinate values of the hand we also point out how the jacobian matrices in nonlinear biological systems can be estimated in general from observed time series e g joint angles cartesian coordinate values of hand movement neural or muscle activity estimating jacobian matrices of nonlinear functions from experimental observations can provide not only data analysis but also signal synthesis and can lead to a more realistic modeling of sensorimotor transformations

new resampling method to assess the accuracy of the maximal lyapunov exponent estimation abstract in this paper we propose a novel method to assess standard errors and confidence intervals for the maximal lyapunov exponent estimated on time continuous chaotic systems the method is based on resampling the original series by means of spline interpolation in the time domain in such a way new time series of increased size are obtained and the sample distribution of the estimators is constructed the method is explained and tested on the basis of computer simulations both for clean and noisy series we give evidence that the distribution of the maximal lyapunov exponent calculated by this method fairly agrees with the one obtained by true series with different initial conditions an empirical criterion for the choice of the parameters of the resampling is also suggested

algebraic decomposition of fat and water in mri the decomposition of magnetic resonance imaging mri data to generate water and fat images has several applications in medical imaging including fat suppression and quantification of visceral fat we introduce a novel algorithm to overcome some of the problems associated with current analytical and iterative decomposition schemes in contrast to traditional analytical schemes our approach is general enough to accommodate any uniform echo shift pattern any number of metabolites and signal samples in contrast to region growing method that use a smooth field map initialization to resolve the ambiguities with the ideal algorithm we propose to use an explicit smoothness constraint on the final fieldmap estimate towards this end we estimate the number of feasible solutions at all the voxels prior to the evaluation of the roots this approach enables the algorithm to evaluate all the feasible roots thus avoiding the convergence to the wrong solution the estimation procedure is based on a modification of the harmonic retrieval hr framework to account for the chemical shift dependence in the frequencies in contrast to the standard linear hr framework we obtain the frequency shift as the common root of a set of quadratic equations on most of the pixels with multiple feasible solutions the correct solution can be identified by a simple sorting of the solutions we use a region merging algorithm to resolve the remaining ambiguity and phase wrapping experimental results indicate that the proposed algebraic scheme eliminates most of the difficulties with the current schemes without compromising the noise performance moreover the proposed algorithm is also computationally more efficient

the bootstrap and lyapunov exponents in deterministic chaos inasmuch as lyapunov exponents provide a necessary condition for chaos in a dynamical system confidence bounds on estimated lyapunov exponents are of great interest estimates derived either from observations or from numerical integrations are limited to trajectories of finite length and it is the uncertainties in the distribution of these finite time lyapunov exponents which are of interest within this context a bootstrap algorithm for quantifying sampling uncertainties is shown to be inappropriate for multiplicative ergodic statistics of deterministic chaos this result remains unchanged in the presence of observational noise as originally proposed the algorithm is also inappropriate for general nonlinear stochastic processes a modified version is presented which may prove of value in the case of stochastic dynamics a new approach towards quantifying the minimum duration of observations required to estimate global lyapunov exponents is suggested and is explored in a companion paper

noise induced convergence of the low flow rate chaos in the belousov zhabotinsky reaction abstract the effect of noise on the low flow rate chaos in the belousov zhabotinsky bz reaction was studied the chaos was simulated using the three variable model of gyorgyi and field gaussian white noise was imposed on the flow rate of the reactant solutions fed into cstr to simulate the so called type p noise the range of average noise amplitudes was chosen between 0 01 and 1 related to the inverse residence time the calculated time series were analyzed on the basis of their fourier spectra maximum lyapunov exponent kolmogorov entropies return maps and invariant density we found that the noise induces partial order of the period 3 like oscillations in the low flowrate chaos

c algebras and numerical linear algebra given a self adjoint operator a on a hilbert space suppose that one wishes to compute the spectrum of a numerically in practice these problems often arise in such a way that the matrix of a relative to a natural basis is sparse for example discretized second order differential operators can be represented by doubly infinite tridiagonal matrices in these cases it is easy and natural to compute the eigenvalues of large n n submatrices of the infinite operator matrix and to hope that if n is large enough then the resulting distribution of eigenvalues will give a good approximation to the spectrum of a numerical analysts call this the galerkin method while this hope is often realized in practice it often fails as well and it can fail in spectacular ways the sequence of eigenvalue distributions may not converge as n or they may converge to something that has little to do with the original operator a at another level even the meaning of convergence has not been made precise in general in this paper we determine the proper general setting in which one can expect convergence and we describe the asymptotic behavior of the n n eigenvalue distributions in all but the most pathological cases under appropriate hypotheses we establish a precise limit theorem which shows how the spectrum of a is recovered from the sequence of eigenvalues of the n n compressions in broader terms our results have led us to the conclusion that numerical problems involving infinite dimensional operators require a reformulation in terms ofc algebras indeed it is only when the single operator a is viewed as an element of an appropriate c algebra a that one can see the precise nature of the limit of the n n eigenvalue distributions the limit is associated with a tracial state on a normally a is highly noncommutative and in our main applications it is a simple c algebra having a unique tracial state we obtain precise asymptotic results for operators which represent discretized hamiltonians of one dimensional quantum systems with arbitrary continuous potentials

tsallis non extensive statistics intermittent turbulence soc and chaos in the solar plasma part one sunspot dynamics abstract in this study the non linear analysis of the sunspot index is embedded in the non extensive statistical theory of tsallis 1988 2004 2009 7 9 10 the q triplet of tsallis as well as the correlation dimension and the lyapunov exponent spectrum were estimated for the svd components of the sunspot index timeseries also the multifractal scaling exponent spectrum f a the generalized renyi dimension spectrum d q and the spectrum j p of the structure function exponents were estimated experimentally and theoretically by using the q entropy principle included in tsallis non extensive statistical theory following arimitsu and arimitsu 2001 2000 76 77 our analysis showed clearly the following a a phase transition process in the solar dynamics from high dimensional non gaussian soc state to a low dimensional non gaussian chaotic state b strong intermittent solar turbulence and anomalous multifractal diffusion solar process which is strengthened as the solar dynamics makes a phase transition to low dimensional chaos in accordance to ruzmaikin zeleny and milovanov s studies zelenyi and milovanov 1991 21 milovanov and zelenyi 1993 22 ruzmakin et al 1996 26 c faithful agreement of tsallis non equilibrium statistical theory with the experimental estimations of i non gaussian probability distribution function p x ii multifractal scaling exponent spectrum f a and generalized renyi dimension spectrum d q iii exponent spectrum j p of the structure functions estimated for the sunspot index and its underlying non equilibrium solar dynamics

controlling chaos with weak periodic signals optimized by a genetic algorithm in the present study we develop a relatively novel and effective chaos control approach with a multimode periodic disturbance applied as a control signal and perform an in depth analysis on this nonfeedback chaos control strategy different from previous chaos control schemes the present method is of two characteristic features 1 the parameters of the controlling signal are optimized by a genetic algorithm ga with the largest lyapunov exponent used as an index of the stability and 2 the optimization is justified by a fitness function defined with the target lyapunov exponent and the controlling power this novel method is then tested on the noted rossler and lorenz systems with and without the presence of noise the results disclosed that compared to the existing chaos control methods the present ga based control needs only significantly reduced signal power and a shorter transient stage to achieve the preset control goal the switching control ability and the robustness of the proposed method for cases with sudden change in a system parameter and or with the presence of noise environment are also demonstrated

invariance properties and characterization of the greatest common divisor of a set of polynomials on m sets of polynomials from r s of degree d x1d4c5 m d the notion of extended r equivalence emd under x1d4c5 m d is characterized by a complete invariant and a canonical form it is shown that the greatest common divisor g c d of x1d4c5 m d is an invariant of x1d4c5 m d for proper sets x1d4c5 m d it is also shown that the g c d is invariant under the combined action of e r e transformations and shifting operations finally a systematic procedure for the computation of the g c d of x1d4c5 mdis given this procedure is based on the use of elementary row operations on real matrices and shifting operations

characterizing the clarinet tone measurements of lyapunov exponents correlation dimension and unsteadiness the clarinet tone is produced by a self sustained oscillation involving nonlinearity between the flow through the reed and the mechanical response of the reed and acoustic coupling via the air column response regimes of oscillation include periodic biperiodic and other quasi periodic signals yet even a nominally periodic tone has small but perceptually and musically important deviations that are obscured in a conventional power spectrum such deviations may be due to the nonlinear dynamics underlying sound production or to perturbations in the performer s control of the instrument via changes in lip embouchure blowing pressure and vocal tract configuration techniques based upon experimental nonlinear dynamics and short time signal processing are applied to the acoustic signal measured within the clarinet mouthpiece to assess the role of these additional deviations on sound production these include the lyapunov exponent correlation dimension and a normalized period synchronous energy variance

noise reduction finding the simplest dynamical system consistent with the data abstract a novel method is described for noise reduction in chaotic experimental data whose dynamics are low dimensional in addition we show how the approach allows experimentalists to use many of the same techniques that have been essential for the analysis of nonlinear systems of ordinary differential equations and difference equations

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

singular values of cauchy toeplitz matrices abstract the behavior of singular values of matrices a n 1 i j g n i j 1 with n is investigated for any real g which is not integer it is proved that the singular values are clustered at sin g which is their upper boundary the only o n singular values are those which lie outside a given e neighborhood of the clustering point o n n 0 as n o n o ln 2 n holds if g 1 2 also proved is that the minimum singular values of a n g tend to zero provided that g 1 2

nearest common root of a set of polynomials a structured singular value approach abstract the paper considers the problem of calculating the nearest common root of a polynomial set under perturbations in their coefficients in particular we seek the minimum magnitude perturbation in the coefficients of the polynomial set such that the perturbed polynomials have a common root it is shown that the problem is equivalent to the solution of a structured singular value problem arising in robust control for which numerous techniques are available it is also shown that the method can be extended to the calculation of an approximate gcd of fixed degree by introducing the notion of the generalized structured singular value of a matrix the work generalizes previous results by the authors involving the calculation of the approximate gcd of two polynomials although the general case considered here is considerably harder and relies on a matrix dilation approach and several preliminary transformations

approximate greatest common divisor of many polynomials generalised resultants and strength of approximation the computation of the greatest common divisor gcd of many polynomials is a nongeneric problem techniques defining approximate gcd solutions have been defined but the proper definition of the approximate gcd and the way we can measure the strength of the approximation has remained open this paper uses recent results on the representation of the gcd of many polynomials in terms of factorisation of generalised resultants to define the notion of approximate gcd and define the strength of any given approximation by solving an optimisation problem the newly established framework is used to evaluate the performance of alternative procedures which have been used for defining approximate gcds

how to extract lyapunov exponents from short and noisy time series in this correspondence we discuss an algorithm for the extraction of lyapunov spectra from short and noisy time series initially we briefly describe an existing algorithm and include a method for improving the robustness to noise of this algorithm the problem of short time series is then addressed and a novel concatenation technique is presented and combined with the extraction algorithm previously utilized results of applying this extended algorithm to chaotic time series are presented and the effects of noise and data length are discussed

bootstrap nonlinear prediction estimating the jacobian matrix of a nonlinear dynamical system through observed time series data is one of the important steps in predicting future states of the time series the jacobian matrix is estimated using local information about divergences of nearby trajectories although the basic algorithm for estimating the jacobian matrix generally works well it often fails for short or noisy data series in this paper we proposed a scheme to effectively use near neighbor information for more accurate estimation of the jacobian matrix using the bootstrap resampling method then to confirm the validity of the proposed method we applied it to a mathematical model and several real time series as a result we confirmed that the proposed method greatly improves nonlinear predictability not only for noise corrupted mathematical models but also for real time series

a practical method for calculating largest lyapunov exponents from small data sets detecting the presence of chaos in a dynamical system is an important problem that is solved by measuring the largest lyapunov exponent lyapunov exponents quantify the exponential divergence of initially close state space trajectories and estimate the amount of chaos in a system we present a new method for calculating the largest lyapunov exponent from an experimental time series the method follows directly from the definition of the largest lyapunov exponent and is accurate because it takes advantage of all the available data we show that the algorithm is fast easy to implement and robust to changes in the following quantities embedding dimension size of data set reconstruction delay and noise level furthermore one may use the algorithm to calculate simultaneously the correlation dimension thus one sequence of computations will yield an estimate of both the level of chaos and the system complexity

indications of nonlinearity intraindividual specificity and stability of human eeg the unfolding dimension abstract with the help of a parameter which we call the unfolding dimension ww provide a bivariate representation of dynamical eeg analysis applied to human eeg our approach successfully discriminates surrogate data from raw eeg and similarly shows human eeg to be both intraindividually specific and stable over time the heart of our approach is the grassberger procaccia algorithm for the determination of the correlation dimension d 2 within the context of the well known method of time delays to guarantee the reproducibility of results this algorithm as well as an estimate for the k 2 entropy and the determination of f spectra have been integrated into a computer program which encompasses an operator user independent automatic and reproducible specification of both an optimal time delay for calculating the correlation integral as well as an optimal scalar invariant plateau region for the extraction of d 2 and k 2 our embedding protocol applied to mathematical systems is shown to be consistent with findings based on other algorithms in particular with false nearest neighbor considerations and sauer s minimal embedding criteria particular intelligent features of the automation and optimization logic plus the inclusion of estimates of the inherent systematic error make the proposed algorithm especially appropriate for applications to biological psychological data

problems in estimating dynamics from data abstract this paper discusses some difficulties in estimating dynamics from time delay embeddings of experimental data that can be characterized as low dimensional a new procedure is described to reduce noise by exploiting the properties of saddle periodic orbits on the reconstructed attractor

shape distributions of nonlinear dynamical systems for video based inference this paper presents a shape theoretic framework for dynamical analysis of nonlinear dynamical systems which appear frequently in several video based inference tasks traditional approaches to dynamical modeling have included linear and nonlinear methods with their respective drawbacks a novel approach we propose is the use of descriptors of the shape of the dynamical attractor as a feature representation of nature of dynamics the proposed framework has two main advantages over traditional approaches a representation of the dynamical system is derived directly from the observational data without any inherent assumptions and b the proposed features show stability under different time series lengths where traditional dynamical invariants fail we illustrate our idea using nonlinear dynamical models such as lorenz and rossler systems where our feature representations shape distribution support our hypothesis that the local shape of the reconstructed phase space can be used as a discriminative feature our experimental analyses on these models also indicate that the proposed framework show stability for different time series lengths which is useful when the available number of samples are small variable the specific applications of interest in this paper are 1 activity recognition using motion capture and rgbd sensors 2 activity quality assessment for applications in stroke rehabilitation and 3 dynamical scene classification we provide experimental validation through action and gesture recognition experiments on motion capture and kinect datasets in all these scenarios we show experimental evidence of the favorable properties of the proposed representation

the problem of spurious lyapunov exponents in time series analysis and its solution by covariant lyapunov vectors we briefly recall the methods for the determination of lyapunov exponents from time series data and their limitations one particular problem is given by the fact that reconstructed phase spaces have usually extra dimensions compared to the true phase space of a dynamical system leading to extra so called spurious lyapunov exponents several methods to identify the true ones have been proposed which do not give satisfactory results we show that the geometric information contained in covariant lyapunov vectors can be used to identify the true exponents we illustrate its use and its limitations by applying it to experimental nmr laser data this article is part of a special issue of journal of physics a mathematical and theoretical devoted to lyapunov analysis from dynamical systems theory to applications

stability and lyapunov stability of dynamical systems a differential approach and a numerical method abstract a set of differential equations for the eigenvalues and eigenvectors of the stability matrix of a dynamical system as well as for the lyapunov exponents and the corresponding eigenvectors is derived the rate of convergence of the lyapunov eigenvectors is shown to be exponential the eigenvectors of the stability matrix can be grouped into sets each spanning a subspace which converges at an exponential rate it is demonstrated that generically the real parts of the eigenvalues of the stability matrix equal the corresponding lyapunov exponents this statement has been tested numerically the values of the lyapunov exponents i are shown to be related to the corresponding finite time values of the lyapunov exponents e g those deduced from a finite time numerical simulation i t by i t i b i i t t the b i s are constants and i t are noise terms of zero mean this observation leads to a method of extrapolation which has been used to predict lyapunov exponents from a finite amount of data it is shown that the use of the standard numerical methods to compute lyapunov exponents introduces an error a i t in the value of i t where the a i s are constants thus the standard method has a rate of convergence which is the same as that of the exact i t s finally we have shown how one can compute the eigenvectors associated with each of the eigenvalues of the stability matrix as well as the lyapunov eigenvectors

a new test for chaos in deterministic systems we describe a new test for determining whether a given deterministic dynamical system is chaotic or nonchaotic in contrast to the usual method of computing the maximal lyapunov exponent our metho

a modified chua chaotic oscillator and its application to secure communications a new modification to the chua oscillator is proposed bifurcation diagrams poincare map and the lyapunov exponents are presented an application to secure communications is presented a synchronization scheme based on the modified oscillator is proposed system simulations are performed for analysis in this paper a new modified chua oscillator is introduced the original chua oscillator is well known for its simple implementation and mathematical modeling a modification of the oscillator is proposed in order to facilitate the synchronization and the encryption and decryption scheme the modification consists in changing the nonlinear term of the original oscillator to a smooth and bounded nonlinear function a bifurcation diagram a poincare map and the lyapunov exponents are presented as proofs of chaoticity of the newly modified oscillator an application to secure communications is proposed in which two channels are used numerical simulations are performed in order to analyze the communication system

distinguishing between low dimensional dynamics and randomness in measured time series the success of current attempts to distinguish between low dimensional chaos and random behavior in a time series of observations is considered first we discuss stationary stochastic processes which produce finite numerical estimates of the correlation dimension and k2 entropy under naive application of correlation integral methods we then consider several straightforward tests to evaluate whether correlation integral methods reflect the global geometry or the local fractal structure of the trajectory this determines whether the methods are applicable to a given series if they are we evaluate the significance of a particular result for example by considering the results of the analysis of stochastic signals with statistical properties similar to those of observed series from the examples considered it is clear that the correlation integral should not be used in isolation but as one of a collection of tools to distinguish chaos from stochasticity

chaos in segments from korean traditional singing and western singing the time series of the segments from a korean traditional song gwansanyungma and a western song la mamma morta were investigated using chaotic analysis techniques it is found that the phase portrait in the reconstructed state space of the time series of the segment from the korean traditional song has a more complex structure in comparison with the segment from the western songs the segment from the korean traditional song has the correlation dimension 4 4 and two positive lyapunov exponents which show that the dynamic related to the korean traditional song is a high dimensional hyperchaotic process on the other hand the segment from the western song with only one positive lyapunov exponent and the correlation dimension 2 5 exhibits low dimensional chaotic behavior

a statistical approach to estimate the lyapunov spectrum in disc brake squeal abstract the estimation of squeal propensity of a brake system from the prediction of unstable vibration modes using the linear complex eigenvalue analysis cea in the frequency domain has its fair share of successes and failures while the cea is almost standard practice for the automotive industry time domain methods and the estimation of lyapunov spectra have not received much attention in brake squeal analyses one reason is the challenge in estimating the true lyapunov exponents and their discrimination against spurious ones in experimental data a novel method based on the application of the eckmann ruelle matrices is proposed here to estimate lyapunov exponents by using noise in a statistical procedure it is validated with respect to parameter variations and dimension estimates by counting the number of non overlapping confidence intervals for lyapunov exponent distributions obtained by moving a window of increasing size over bootstrapped same length estimates of an observation function a dispersion measure s width is calculated and fed into a bayesian beta binomial model results obtained using this method for benchmark models of white and pink noise as well as the classical henon map indicate that true lyapunov exponents can be isolated from spurious ones with high confidence the method is then applied to accelerometer and microphone data obtained from brake squeal tests estimated lyapunov exponents indicate that the pad s out of plane vibration behaves quasi periodically on the brink to chaos while the microphone s squeal signal remains periodic

computation of lyapunov spectra effect of interactive noise and application to a chemical oscillator abstract a refined version of the sano sawada algorithm to compute lyapunov spectra has been implemented and tested with several models which include periodic chaotic and hyperchaotic attractors under conditions designed to mimic experimental situations it was found that reliable lyapunov exponents can be obtained for time series of moderate length and point density also in the presence of interactive noise as long as its amplitude does not exceed 1 of the oscillation amplitude the algorithm was then applied to experimental data from the co oxidation on pt 110 clearly showing the occurrence of chaos as well as hyperchaos

theoretical and experimental time series analysis of an inductorless chua s circuit abstract in this paper we have analyzed the time series obtained from an experimental realization of an inductorless version of chua s circuit we have also compared the experimental results with the results emerging by analysis of the time series obtained by numerical simulations of the respective model equations lyapunov exponent spectra the kaplan yorke dimension and phase space diagrams were used to show the similarities between theoretical and experimental results we were able to identify the effects of the operational amplifier simulated inductor internal resistance on the chua s circuit dynamics decreasing internal resistance for decreasing current through the circuit was responsible for the observed bifurcation route

a matrix pencil based numerical method for the computation of the gcd of polynomials the paper presents a new numerical method for the computation of the greatest common divisor gcd of an m set of polynomials of r s p sub m d of maximal degree d it is based on a previously proposed theoretical procedure karcanias 1989 that characterizes the gcd of p sub m d as the output decoupling zero polynomial of a linear system s a spl circ c spl circ that may be associated with p sub m d the computation of the gcd is thus reduced to finding the finite zeros of the pencil sw aw where w is the unobservable subspace of s a spl circ c spl circ if k dim w the gcd is determined as any nonzero entry of the kth compound c sub k sw a spl circ w the method defines the exact degree of gcd works satisfactorily with any number of polynomials and evaluates successfully approximate solutions

testing for nonlinearity in time series the method of surrogate data we describe a statistical approach for identifying nonlinearity in time series the method first specifies some linear process as a null hypothesis then generates surrogate data sets which are consistent with this null hypothesis and finally computes a discriminating statistic for the original and for each of the surrogate data sets if the value computed for the original data is significantly different than the ensemble of values computed for the surrogate data then the null hypothesis is rejected and nonlinearity is detected we discuss various null hypotheses and discriminating statistics the method is demonstrated for numerical data generated by known chaotic systems and applied to a number of experimental time series which arise in the measurement of superfluids brain waves and sunspots we evaluate the statistical significance of the evidence for nonlinear structure in each case and illustrate aspects of the data which this approach identifies

embedding dimension estimation of chaotic time series using self generating radial basis function network abstract in this paper we apply the self generating radial basis function network sgrbf to the dimension analysis of the nonlinear dynamical systems including chaotic time series firstly we formulate a nonlinear time series identification problem with a nonlinear autoregressive moving average narmax model secondly we propose an identification algorithm using sgrbf which is regarded as both a three layer network or a fuzzy model of class c with gaussian membership function we apply this method to the estimation of embedding dimension for chaotic time series since the embedding dimension plays an essential role for the identification and the prediction of nonlinear dynamical systems including chaos in this estimation method identification problems with gradually increasing embedding dimension are solved and the identified result is used for computing correlation coefficients between the predicted time series and the observed one we apply this method to the embedding dimension estimation of a henon map and a chaotic pulsation time series in a finger s capillary vessels

resultant properties of gcd of many polynomials and a factorization representation of gcd the properties of the greatest common divisor gcd of a set of polynomials of may be investigated using the sylvester resultant new properties of the sylvester resultant linked to gcd are established and these lead to canonical factorizations of resultants expressing the extraction of common divisors from the elements of the original set these results lead to a new representation of the gcd introduced in terms of a canonical factorization of the sylvester resultant into a reduced sylvester resultant and a toeplitz matrix representing the gcd the use of the sylvester resultant allows a simplification of the eres and matrix pencils computational procedures for gcd computation and provides the means for formulating simpler and robust computational procedures

a p ge semiconductor experiment showing chaos and hyperchaos abstract a p ge semiconductor experiment is investigated by the help of both probabilistic and dynamical characterization methods dimensions lyapunov exponents and the corresponding scaling functions are calculated two exemplary files of data from the p ge semiconductor experiment exhibiting spontaneous i e undriven resistance oscillations in the low temperature avalanche breakdown are shown to be chaotic and hyperchaotic respectively for the first file we obtained a fractal dimension between two and three and one positive lyapunov exponent whereas for the second file we found a fractal dimension between three and four and two positive lyapunov exponents adopting the terminology introduced by rossler the behavior corresponding to the latter file is called hyperchaotic furthermore using the language of the thermodynamical formalism the probabilistic scaling function evaluated for the hyperchaotic state indicates a phase transition like behavior

on the concept of attractor this note proposes a definition for the concept of attractor based on the probable asymptotic behavior of orbits the definition is sufficiently broad so that every smooth compact dynamical system has at least one attractor

numerical verification of a non residual orthogonal term by term stabilized finite element formulation for incompressible convective flow problems abstract the modeling of turbulent flows is relevant in many engineering applications and is an active field of research on numerical methods convergence and stability of proposed formulations are crucial to predict transitional flows from laminar to turbulent flows in this work a recently developed stabilized finite element formulation is used as a powerful tool to describe such kind of problems the essential point of the formulation is the time dependent nature of the subscales and contrary to residual based formulations the introduction of two velocity subscale components the theoretical rate of convergence of the method is verified numerically using linear and quadratic equal order finite element discretizations to this end a standard convergence test of l 2 norm is presented where the computed solutions are compared when manufactured solutions are imposed at gauss point level moreover the hopf bifurcation is studied for two well known benchmark problems flow past a cylinder and the three dimensional lid driven cavity flow for the flow past a cylinder case the hopf bifurcation is verified using dynamic subscales and is assessed so that they do not disturb the solution in particular a dominant convective problem re 4000 is solved using both the quasistatic and dynamic versions of the method evaluating the performance of each one in the quality of the solution and in the cpu time needed to obtain a converged solution for the 3d lid driven cavity flow problem the hopf bifurcation is determined using two different boundary conditions analyzing their effect on the dynamics of the problem and on the thickness and shape of the boundary layer the final test case is the turbulent 3d lid driven cavity problem re 12000 where velocity profiles are compared with experimental les and dns reference solutions additionally pressure and velocity spectra are shown at certain representative points of the domain as well as phase diagrams correlation function graphs the poincare map and the lyapunov exponent typical mathematical tools used in dynamical system analysis from the results the method is robust and accurate for all the numerical tests both in viscous dominant problems as in dominant convective ones

attractors and characteristic exponents a natural definition of an attractor as an invariant measure is given based on the ergodic theory of axiom a diffeomorphisms and some results are proved which support this definition it is also proved that if an attractor has every characteristic exponent less than zero in a set of nonzero measure then the support set of the attractor is an asymptotic stable periodic orbit

lyapunov exponents as a measure of the size of chaotic regions chaotic orbits in conservative dynamical systems are partitioned into a number of distinct regions by sets of cantori this causes long time scale oscillations in the corresponding lyapunov spectrums a direct correlation between the magnitude of the maximal lyapunov exponent and the size of the corresponding chaotic region in the surface of section appears to exist for the two dimensional conservative systems examined