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Characterization of nonstationary chaotic systems



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PHYSICAL REVIEW E 77 026208 2008 Characterization of nonstationary chaotic systems 1 Ruth Serquina 1 Ying Cheng Lai 2 3 and Qingfei Chen2 Department of Mathematics MSU Iligan Institute of Technology the Philippines Department of Electrical Engineering Arizona State University Tempe Arizona 85287 USA 3 Department of Physics and Astronomy Arizona State University Tempe Arizona 85287 USA Received 5 July 2007 revised manuscript received 2 October 2007 published 12 February 2008 2 Nonstationary dynamical systems arise in applications but little has been done in terms of the characterization of such systems as most standard notions in nonlinear dynamics such as the Lyapunov exponents and fractal dimensions are developed for stationary dynamical systems We propose a framework to characterize nonstationary dynamical systems A natural way is to generate and examine ensemble snapshots using a large number of trajectories which are capable of revealing the underlying fractal properties of the system By defining the Lyapunov exponents and the fractal dimension based on a proper probability measure from the ensemble snapshots we show that the Kaplan Yorke formula which is fundamental in nonlinear dynamics remains valid most of the time even for nonstationary dynamical systems DOI 10 1103 PhysRevE 77 026208 PACS number s 05 45 a I INTRODUCTION In many previous studies of nonlinear dynamical systems stationarity is assumed That is the underlying system equations and parameters are assumed to be fixed in time One can then define asymptotic invariant sets such as unstable periodic orbits attractors chaotic saddles nonattracting invariant sets study their properties such as the spectra of Lyapunov exponents and of fractal dimensions and search for various bifurcations that concern how the timeasymptotic behaviors of the system vary with parameters 1 There are however practical situations where the assumption of stationarity does not hold For a nonstationary dynamical system many notions that are fundamental to the development of nonlinear dynamics such as periodic orbits and attractors are no longer meaningful The purpose of this paper is to develop a systematic and physically meaningful way to characterize nonstationary dynamical systems We shall be concerned with typical nonlinear systems which when being stationary can have both chaos and periodic motions depending on the parameters Without loss of generality we will focus on the relatively simple situation where a single parameter of the system varies with time In discrete time our model system can be represented by xt 1 f xt pt 1 where x is a d dimensional dynamical variable f is a nonlinear mapping function and pt is a time dependent parameter In a time interval of interest the parameter can vary in a range say pa pb pa pb where for any p pa pb the corresponding stationary dynamical system xt 1 f xt p can possess a chaotic attractor or a periodic attractor or even multiple coexisting attractors Because of the time variation of the parameter a long trajectory originated from a single initial condition typically appears random and exhibits no fractal structure To reveal the intrinsic fractal structure associated with the deterministic but nonstationary chaotic system a viable approach is to examine simultaneously the evolution of a large number of trajectories from an ensemble of initial conditions At a given instant of time the trajectories 1539 3755 2008 77 2 026208 5 tend to form a pattern that can be apparently fractal Such patterns are called snapshot attractors in the context of random dynamical systems that have proven effective to reveal the underlying fractal structure 2 7 For a nonstationary system the notion of attractor is no longer meaningful as a trajectory will in general not have sufficient time to settle down to any asymptotic state of the system We shall call the phase space images of an ensemble of trajectories at a given time ensemble snapshots The question to be addressed in this paper concerns the dynamical properties of such ensemble snapshots In particular we will focus on their Lyapunov exponents and the fractal dimensions Due to nonstationarity we are restricted to examining the dynamical evolution of an ensemble of trajectories in short time intervals during which the system can be regarded as stationary The lengths of these time intervals depend on the rate of change of the system parameter a slower rate would give a relatively longer interval and vice versa For convenience we call them adiabatic time intervals Since the rate of parameter change is in general time dependent and can even be random in a long experimental time the adiabatic time intervals are not necessarily uniform Nonetheless assuming adiabatic time intervals allows the Lyapunov exponents of an ensemble snapshot to be defined as the ensembleaveraged values of the corresponding short time Lyapunov exponents from all trajectories comprising the snapshot Due to nonstationarity the exponents exhibit random fluctuations with time If for any given time all trajectories in the ensemble snapshot are contained in a single basin of attraction for the frozen dynamical system at that time the variance of the fluctuations of the exponents is independent of time However if the trajectories can be in different basins of attraction for the frozen system the magnitude of the fluctuations of the exponents will depend on the value of the instantaneous Lyapunov exponents 8 and therefore can vary with time For a stationary dynamical system the Kaplan Yorke formula holds which relates the information dimension of an attractor to its Lyapunov spectrum 9 Thus for our nonstationary system after the Lyapunov exponents of an ensemble snapshot have been calculated one may wonder whether the 026208 1 2008 The American Physical Society PHYSICAL REVIEW E 77 026208 2008 SERQUINA LAI AND CHEN information dimension of the snapshot can be defined and related to the exponents We shall argue that it is possible to define a dimension spectrum for an ensemble snapshot The main result of this paper is that if all trajectories constituting the ensemble snapshot are contained in a single basin of the underlying temporarily stationary dynamical system the Kaplan Yorke formula still holds in the sense that the information dimension obtained by a straightforward boxcounting procedure can be approximated by the value determined by the Lyapunov exponents In Sec II we propose a proper natural


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