On a particle stopping rule for the numerical computation of the Lyapunov spectrum

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. One major drawback of these numerical algorithms is the lack of an adequate stopping rule. In this paper, a 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 is based on upper bounds on the Lyapunov exponents, along with some results from finite state Markov chains and ergodic stochastic processes. However, only the largest Lyapunov exponent is address in this paper, for the computation of the remaining exponents follows a similar procedure.