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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



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