On a method of applied symbolic dynamics for investigation of dynamical systems

Abstract

We consider a method of applied symbolic dynamics which may be used to obtain wide spectrum of characteristics of complex dynamical systems: approximation of invariant sets, estimation of topological entropy and approximation of invariant measures. The method has received wide acceptance in studying complex dynamical systems. The main idea of the method is to describe the system behaviour approximately by means of an oriented graph (called symbolic image). Such a graph is a representation of symbolic dynamical system which is more appropriately known as topological Markov chain. There exists a correspondence between trajectories of a given system and paths on the graph, which allows us to design algorithms for estimation of topological entropy, approximate invariant measures of a given system by using stationary flows on the graph and calculate corresponding metric entropies. These values characterize complex behaviour of dynamical systems such as the existence of trajectories with large periods and chaotic regimes. The results of experiments are given for systems with chaotic dynamics.