Characterization of nonstationary chaotic systemsRuth Serquina,1Ying-Cheng Lai,2,3and Qingfei Chen21Department of Mathematics, MSU-Iligan Institute of Technology, the Philippines2Department of Electrical Engineering, Arizona State University, Tempe, Arizona 85287, USA3Department of Physics and Astronomy, Arizona State University, Tempe, Arizona 85287, USA共Received 5 July 2007; revised manuscript received 2 October 2007; published 12 February 2008兲Nonstationary dynamical systems arise in applications, but little has been done in terms of the characteriza-tion of such systems, as most standard notions in nonlinear dynamics such as the Lyapunov exponents andfractal dimensions are developed for stationary dynamical systems. We propose a framework to characterizenonstationary dynamical systems. A natural way is to generate and examine ensemble snapshots using a largenumber of trajectories, which are capable of revealing the underlying fractal properties of the system. Bydefining the Lyapunov exponents and the fractal dimension based on a proper probability measure from theensemble 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共s兲: 05.45.⫺aI. INTRODUCTIONIn many previous studies of nonlinear dynamical systems,stationarity is assumed. That is, the underlying system equa-tions and parameters are assumed to be fixed in time. Onecan then define asymptotic invariant sets such as unstableperiodic orbits, attractors, chaotic saddles 共nonattracting in-variant sets兲, study their properties such as the spectra ofLyapunov exponents and of fractal dimensions, and searchfor various bifurcations that concern how the time-asymptotic behaviors of the system vary with parameters 关1兴.There are, however, practical situations where the assump-tion of stationarity does not hold. For a nonstationary dy-namical system, many notions that are fundamental to thedevelopment of nonlinear dynamics such as periodic orbitsand attractors, are no longer meaningful. The purpose of thispaper is to develop a systematic and physically meaningfulway to characterize nonstationary dynamical systems.We shall be concerned with typical nonlinear systemswhich, when being stationary, can have both chaos and peri-odic motions depending on the parameters. Without loss ofgenerality we will focus on the relatively simple situationwhere a single parameter of the system varies with time. Indiscrete time, our model system can be represented byxt+1= f共xt,pt兲, 共1兲where x is a d-dimensional dynamical variable, f is a non-linear mapping function, and ptis a time-dependent param-eter. In a time interval of interest, the parameter can vary ina range, say 关pa, pb兴共pa⬍ pb兲 where for any p 苸 关pa, pb兴, thecorresponding stationary dynamical system xt+1=f共xt, p兲 canpossess a chaotic attractor, or a periodic attractor, or evenmultiple coexisting attractors. Because of the time variationof the parameter, a long trajectory originated from a singleinitial condition typically appears random and exhibits nofractal structure. To reveal the intrinsic fractal structure as-sociated with the deterministic but nonstationary chaotic sys-tem, a viable approach is to examine simultaneously the evo-lution of a large number of trajectories from an ensemble ofinitial conditions. At a given instant of time, the trajectoriestend to form a pattern that can be apparently fractal. Suchpatterns are called snapshot attractors in the context of ran-dom dynamical systems that have proven effective to revealthe underlying fractal structure 关2–7兴. For a nonstationarysystem, the notion of “attractor” is no longer meaningful as atrajectory will in general not have sufficient time to settledown to any asymptotic state of the system. We shall call thephase-space images of an ensemble of trajectories at a giventime ensemble snapshots. The question to be addressed inthis paper concerns the dynamical properties of such en-semble snapshots. In particular, we will focus on theirLyapunov exponents and the fractal dimensions.Due to nonstationarity, we are restricted to examining thedynamical evolution of an ensemble of trajectories in shorttime intervals, during which the system can be regarded as“stationary.” The lengths of these time intervals depend onthe rate of change of the system parameter: a slower ratewould give a relatively longer interval and vice versa. Forconvenience, we call them adiabatic time intervals. Since therate of parameter change is in general time-dependent andcan even be random, in a long experimental time the adia-batic time intervals are not necessarily uniform. Nonetheless,assuming adiabatic time intervals allows the Lyapunov expo-nents of an ensemble snapshot to be defined as the ensemble-averaged values of the corresponding short-time Lyapunovexponents from all trajectories comprising the snapshot. Dueto nonstationarity, the exponents exhibit random fluctuationswith time. If for any given time all trajectories in the en-semble snapshot are contained in a single basin of attractionfor the “frozen” dynamical system at that time, the varianceof the fluctuations of the exponents is independent of time.However, if the trajectories can be in different basins of at-traction for the frozen system, the magnitude of the fluctua-tions of the exponents will depend on the value of the instan-taneous Lyapunov exponents 关8兴 and therefore can vary withtime.For a stationary dynamical system, the Kaplan-Yorke for-mula holds, which relates the information dimension of anattractor to its Lyapunov spectrum 关9兴. Thus, for our nonsta-tionary system, after the Lyapunov exponents of an ensemblesnapshot have been calculated, one may wonder whether thePHYSICAL REVIEW E 77, 026208 共2008兲1539-3755/2008/77共2兲/026208共5兲 ©2008 The American Physical Society026208-1information dimension of the snapshot can be defined andrelated to the exponents. We shall argue that it is possible todefine a dimension spectrum for an ensemble snapshot. Themain result of this paper is that, if all trajectories constitutingthe ensemble snapshot are contained in a single basin of theunderlying temporarily stationary dynamical system, theKaplan-Yorke formula still holds in the sense that the infor-mation dimension obtained by a straightforward box-counting procedure