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. Contents 1.

We construct Sinai-Ruelle-Bowen (SRB) measures supported on partially hyperbolic sets of diffeomorphisms -- the tangent bundle splits into two invariant subbundles, one of which is uniformly contracting -- under the assumption that the complementary subbundle is non-uniformly expanding. If the rate of expansion (Lyapunov exponents) is bounded away from zero, then there are only finitely many SRB measures. Our techniques extend to other situations, including certain maps with singularities or critical points, as well as diffeomorphisms having only a dominated splitting (and no uniformly hyperbolic subbundle). 1 Introduction The following approach has been most effective in studying the dynamics of complicated systems: one tries to describe the average time spent by typical orbits in different regions of the phase space. According to the ergodic theorem of Birkhoff, such times are well defined for almost all point, with respect to any invariant probability measure. However, the...

After the stock market crash of October 19, 1987, interest in nonlinear dynamics, especially deterministic chaotic dynamics, has increased in both the financial press and the academic literature. This has come about because the frequency of large moves in stock markets is greater than would be expected under a normal distribution. There are a number of possible explanations. A popular one is that the stock market is governed by chaotic dynamics. What exactly is chaos and how is it related to nonlinear dynamics? How does one detect chaos? Is there chaos in financial markets? Are there other explanations of the movements of financial prices other than chaos? The purpose of this paper is to explore these issues. -1Chaos has captured the fancy of many macroeconomists and financial economists. The attractiveness of chaotic dynamics is its ability to generate large movements which appear to be random, with greater frequency than linear models. As a result, there has been an explosion of pa...

A Bayesian ensemble learning method is introduced for unsupervised extraction of dynamic processes from noisy data. The data are assumed to be generated by an unknown nonlinear mapping from unknown factors. The dynamics of the factors are modeled using a nonlinear statespace model. The nonlinear mappings in the model are represented using multilayer perceptron networks. The proposed method is computationally demanding, but it allows the use of higher dimensional nonlinear latent variable models than other existing approaches. Experiments with chaotic data show that the new method is able to blindly estimate the factors and the dynamic process which have generated the data. It clearly outperforms currently available nonlinear prediction techniques in this very di#cult test problem.

A novel method for regression has been recently proposed by V. Vapnik et al. [8, 9]. The technique, called Support Vector Machine (SVM), is very well founded from the mathematical point of view and seems to provide a new insight in function approximation. We implemented the SVM and tested it on the same data base of chaotic time series that was used in [1] to compare the performances of different approximation techniques, including polynomial and rational approximation, local polynomial techniques, Radial Basis Functions, and Neural Networks. The SVM performs better than the approaches presented in [1]. We also study, for a particular time series, the variability in performance with respect to the few free parameters of SVM.

The analysis of uni- or multivariate time series provides crucial information to describe, understand, and predict climatic variability. The discovery and implementation of a number of novel methods for extracting useful information from time series has recently revitalized this classical eld of study. Considerable progress has also been made in interpreting the information so obtained in terms of dynamical systems theory.

this paper the extended Kalman filter is used with a nonlinear multilayer quasi-geostrophic (QG) model. This provides us with both a realistic ocean model and a very sophisticated error statistics scheme. The extended Kalman filter is an extension of the common Kalman filter and may be used when the model dynamics or the measurement equation is nonlinear. It consists of an approximative equation for the propagation of error covariances, and also approximative filter equations if the measurement equation is nonlinear. When changing from a linear system to nonlinear dynamics the possible existence of a wide variety of phenomena which are nonexistent in the linear theory is introduced. Nonlinear systems may have solutions with multiple equilibria, where the solutions sometimes abruptly undergo transitions from one equilibrium to another as parameters change (bifurcations). Also chaotic behavior occurs in many deterministic systems, where solutions exhibit an apparently random behavior. The Lorenz [1963] model is probably the best known example of chaotic systems. It has solutions which undergo "unpredictable" transitions between two different equilibria (chaos). As discussed by Miller and Ghil

In this paper, we consider discrete and continuous QR algorithms for computing all of the Lyapunov exponents of a regular dynamical system. We begin by reviewing theoretical results for regular systems and present general perturbation results for Lyapunov exponents. We then present the algorithms, give an error analysis of them, and describe their implementation. Finally, we give several numerical examples and some conclusions.

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Secure communication using synchronization between chaotic systems (chaotic secure communication, for short) is a new concept of secure communication. The great potentials of this kind of ``hardware key'' secure communication systems had driven the progress of this field rapidly. Since 1992, chaotic secure communication had evolved four generations. In this paper, a detailed history of chaotic secure communication systems is given. The disadvantage of the first three generations of chaotic secure communication schemes is low efficiency of channel usage. To overcome this disadvantage, a chaotic communication scheme, which belongs to the fourth generation, using impulsive synchronization of chaotic systems is presented. In this paper, impulsive synchronization of two chaotic systems is reformulated as impulsive stabilization of a synchronization error system to the origin. Based on the theory of impulsive differential equations, we present theoretical results on the asymptotic synchronization of two chaotic systems by using synchronization impulses. An estimate of the upper bound of impulse interval is given for the purpose of asymptotic synchronization. The robustness of impulsive synchronization to additive channel noise and parameter mismatch is also studied. We conclude that impulsive synchronization is more robust than continuous synchronization. Combining both conventional cryptographic method and impulsive synchronization of chaotic systems, we propose a new chaotic secure communication scheme. We use this new chaotic secure communication scheme to transmit a speech signal. Computer simulation results based on Chua's oscillators are given. (c)2003 Yang's Scientific Research Institute, LLC.