Introduction to Dynamical Systems

Abstract

In this chapter we present without proofs notions and facts on the dynamical systems which are necessary to understand this book. We recall the notion of an invariant set and show the most important examples of such sets: fixed and periodic points, \(\omega -\) and \(\alpha -\)limit sets, wandering and nonwandering sets, chain recurrent sets, topologically transitive sets. We discuss the notion of stability of a dynamical system with respect to one of its characteristics, structural stability and \(\varOmega -\)stability in particular. We consider hyperbolic invariant sets, recall the theorem on existence of the stable and the unstable manifold for a point of such a set. We recall Hartman-Grobman theorem that a diffeomorphism in a neighborhood of a hyperbolic periodic point is topologically conjugate to its linearizion. We give the topological classification of hyperbolic fixed points. We present a brief explanation of the results of the “epoch of the hyperbolic revolution” begun in 1960s with the classical works by S. Smale and D. Anosov. We show the relations between the nonwandering set, the chain recurrent set and the limit set of a dynamical system (diffeomorphism). We present Smale’s spectral decomposition theorem which allows us to represent a hyperbolic nonwandering set of a diffeomorphism as the union of the disjoint closed transitive (basic) sets if the nonwandering set is the closure of the periodic points. We recall the criteria of structural and \(\varOmega -\)stability. The concepts of the symbolic dynamics, the reverse limit and the solenoid are presented as the means to describe the restriction of a dynamical system to its invariant set with complex dynamics. For more details see for example the books [17, 24, 28, 33, 43, 46], the surveys [3, 4, 5, 6, 48] and the papers [1, 7, 8, 9, 11, 13, 14, 18, 23, 26, 27, 29, 30, 34, 47, 51].