J. Ezzine, AN ASYMPTOTICALLY EXACT STOPPING RULE FOR THE NUMERICAL COMPUTATION OF THE LYAPUNOV SPECTRUM, Chaos, solitons and fractals, 7(8), 1996, pp. 1213-1225
It is in general not possible to analytically compute the Lyapunov spe
ctrum of a given dynamical system. This has been achieved for a few sp
ecial cases only. Therefore, numerical algorithms have been devised fo
r this task. However, one major drawback of these numerical algorithms
is their lack of stopping rules. In this paper, an asymptotically exa
ct stopping rule is proposed to alleviate this shortcoming while compu
ting the Lyapunov spectrum of linear discrete-time random dynamical sy
stems (i.e., linear systems with random parameters). The proposed stop
ping rule provides an estimate of the least number of iterations, for
which the probability of incurring a prescribed error, in the numerica
l computation of the Lyapunov spectrum, is minimized. It exploits simp
le upper bounds on the Lyapunov exponents, along with some results fro
m finite state Markov chains. The accuracy of the stopping rule, and t
he computational load, is proportional to the tightness of the bound.
In fact, a series of increasingly tighter bounds are proposed, yieldin
g an asymptotically exact stopping rule for the tightest one. It is de
monstrated via an example, that the proposed stopping rule is applicab
le to nonlinear dynamics as well. Copyright (C) 1996 Elsevier Science
Ltd