Features of the Computational Implementation of the Algorithm for Estimating the Lyapunov Exponents of Systems with Delay

摘要

The computational implementation of the algorithm for estimating the spectrum of Lyapunov exponents for systems of differential equations with delayed arguments is considered. Taking into account that for such systems, as well as for boundary value problems, it is not possible to prove the well-known Oseledets theorem, which makes it possible to efficiently calculate the required values, we only have to talk about estimates of characteristic exponents, in a sense close to Lyapunov ones. In this paper, we propose two methods for processing solutions of systems linearized on an attractor, one of which is based on a set of impulse functions; the other one is based on a set of trigonometric functions. The flexibility of application of the indicated algorithms is demonstrated in the case of quasi-stable structures, when several Lyapunov exponents are close to zero. The developed methods are tested on the logistic equation with delay. The results illustrate the "closeness" of the estimated characteristics and Lyapunov exponents.