Analysis of Phase Space Structure of a 1-D Discrete System Using Global and Local Symbolic Dynamics

摘要

Symbolic dynamics, in which the system trajectory is represented as a string of symbols, appears as a convenient method for the analysis of properties of chaotic attractors. In this paper, we show that, using a non-canonical coding scheme based on a moving partition point, we are able to access such properties of the phase space of a dynamical system as the localisation of unstable periodic orbits and of their stable invariant manifolds. Applying different coding schemes enables us to extract different information about the phase space structure from the chaotic trajectory. A judicial choice of the method of symbolic coding allows to obtain information which may be missing in the symbolic dynamics from the generating partition. We present results for the 1-D case taking the logistic map as a numerical example. The extension to higher dimension is also discussed. The theoretical background of the methods used is also given.