The Periodic Points and Symbolic Dynamics

Abstract

At present the methods of symbolic dynamics have wide applications. One of the variants of the junctions of the symbolic dynamics and the numerical methods of differential equations is presented. Usually the constructions of periodic trajectories of dynamical systems are based on the theory of perturbations. A general method of their construction is the following. Assuming a periodic orbit of an unperturbed system to be known one constructs a sequence of approximate periodic orbits which tend to a periodic one of perturbed system. Thus knowledge of a periodic orbit of unperturbed system is a necessary condition for the construction of such a sequence. The aim of present paper is to give an algorithm for the determination of periodic trajectory without any preliminary information about a system. A proposed construction is based on the methods of symbolic dynamics. The common scheme of our method is the following. At first, we associate to a dynamical system an oriented graph which is called symbolic image of the system. It can be said that the symbolic image is a finite approximation of the dynamical system. The investigation of the symbolic image gives an opportunity to separate the points which periodic trajectories may pass through from those which periodic trajectories may not pass through. To be more precise the method gives a possibility to locate a chain recurrent set which includes returning trajectories of all types. During the process of localization of chain recurrent points we obtain the approximate periodic trajectories.