Chaos, Solitons & Fractals Vol. 7, No. 8, pp. 1213-1225, 1996 Coowieht CT, 19% Elsevier Science Ltd Printed I; &a;-Britain. All rights reserved o!+5om79/96 $15.00 + 0.00 09fio-0779(95)00107-7 An Asymptotically Exact Stopping Rule for the Numerical Computation of the Lyapunov Spectrum JELEL EZZINE Department of Systems Engineering, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia (Accepted 9 November 1995) Abstract-It is in general not possible to analytically compute the Lyapunov spectrum of a given dynamical system. This has been achieved for a few special cases only. Therefore, numerical algorithms have been devised for this task. However, one rnajor drawback of these numerical algorithms is their lack of stopping rules. In this paper, an asymptotically exact stopping rule is proposed to alleviate this shortcoming while computing the Lyapunov spectrum of linear discrete- time random dynamical systems (i.e., linear systems with random parameters). The proposed stopping rule provides an estimate of the least number of iterations, for which the probability of incurring a prescribed error, in the numerical computation of the Lyapunov spectrum, is minimized. It exploits simple upper bounds on the Lyapunov exponents, along with some results from finite state Markov chains. The accuracy of the stopping rule, and the computational load, is proportional to the tightness of the bound. In fact, a series of increasingly tighter bounds are proposed, yielding an asymptotically exact stopping rule for the tightest one. It is demonstrated via an example, that the proposed stopping rule is applicable to nonlinear dynamics as well. Copyright @ 1996 Elsevier Science Ltd 1. INTRODUCTION One century ago, Lyapunov [l] attempted to define an eigenvalue-like concept for linear time-varying dynamical systems, but his findings applied only to a very special class of systems he called regular. Regularity is a property hard to check. However, recently, Oseledest [l] showed that it is an almost sure (a.s.) property for random dynamical systems, and developed a useful eigenstructure-like theory for this class of systems. Though Lyapunov exponents are a very powerful tool for the analysis of dynamical systems, they suffer from an important drawback as regards their analytical as well as numerical computations. The Lyapunov exponents were analytically computed for a few simple dynamical systems only. The only available means to compute these numbers is via numerical algorithms. These numerical algorithms use the actual time evolution of the states of the system, thus requiring extended simulation time of the dynamics in question. Theoretically, an infinite time horizon is required to compute these numbers, thus, practically, there is an acute need for adequate stopping rules to alleviate this major limitation. Recently, this absence of a stopping rule was clearly acknowledged in Lichten- berg, and Lieberman’s book [2], and Talay’s recent work, such as [3], where it was stated in the former: “This points out a major difficulty in numerically evaluating the Lyapunov exponents: there is no a priori condition for determining the number of iterations it that must be used.” In addition, Eckmann and Ruelle [4], questioned the validity of a number of recently published estimates of dimensions of attractors, and Lyapunov exponents, based on rather short time series. They concluded that long time series, of high quality, are 1213