Construction of attractors and filtrations

This paper is a study of the global structure of the attractors of a dynamical system. The dynamical system is associated with an oriented graph called a Symbolic Image of the system. The symbolic image can be considered as a finite discrete approximation of the dynamical system flow. Investigation of the symbolic image provides an opportunity to localize the attractors of the system and to estimate their domains of attraction. A special sequence of symbolic images is considered in order to obtain precise knowledge about the global structure of the attractors and to get filtrations of the system. Introduction. Our purpose is to study the structure of attractors without any preliminary information about the system. The investigation is based on the methods of symbolic dynamics and all needed estimations can be obtained by traditional numerical methods. The common scheme of the investigation is the following. By using a covering of phase space by cells the dynamical system is associated with an oriented graph called the Symbolic Image of the system. Valuable information about the global structure of the system may come from analysis of this symbolic image. By investigating the symbolic image, one can obtain neighborhoods of the attractors and estimate their domains of attraction. This allows us to construct a filtration of the dynamical system. By applying a subdivision of the covering, a fine sequence of filtrations is constructed. It must be emphasized that our investigation was stimulated by the basic ideas of Charles Conley [5] of the chain recurrent set, the Morse decomposition and the Lyapunov functions. We will consider a discrete dynamical system governed by a homeomorphism X defined on a compact C manifold M . To describe the continuous version, consider a shift operator along trajectories of the system of differential equations defined as follows. Let x = f(t, x) be a system of ordinary differential equations, where x ∈ M, f(t, x) is a C 1991 Mathematics Subject Classification: Primary 58F12; Secondary 54H20, 34C35. Supported by the Russian Foundation for Basic Research under Grant 97-07-90088 and in part by Grant No NWJOOO from the International Science Foundation. The paper is in final form and no version of it will be published elsewhere.