Structural Stability

Abstract

In this chapter, we discuss the concept of structural stability as a criterion for robustness of invariant sets of dynamical systems. In Sec. 4.1, we define and state important properties of this notion, along with its conditions as given in Sec. 4.1.1. An example of a proof of Anosov's theorem on structural stability of diffeomorphisms of a compact C ∞ manifold M without boundary due to Robinson and Verjovsky [Robinson & Verjovsky (1971)] is given and discussed in Sec. 4.1.2. At the end of this chapter, we present a weaker version called Ω-stability and then give a set of exercises and open problems concerning structural stability along with some suggested references. The concept of structural stabilityThe concept of structural stability was introduced by Andronov and Pontryagin in 1937, and it plays an important role in the development of the theory of dynamical systems. Conditions for structural stability of highdimensional systems were formulated by Smale in [Smale (1967)]. These conditions are the following: A system must satisfy both axiom A (recall Definition 6.3) and the strong transversality condition. Mathematically, let C r (R n , R n ) denote the space of C r vector fields of R n into R n . Let Dif f r (R n , R n ) be the subset of C r (R n , R n ) consisting of the C r diffeomorphisms.Definition 4.1. (a) Two elements of C r (R n , R n ) are C r ε-close (k ≤ r), or just C k close, if they, along with their first k derivatives, are within ε as measured in some norm. (b) A dynamical system (vector field or map) is structurally stable if nearby systems have the same qualitative dynamics. 61Robust Chaos and Its Applications Downloaded from www.worldscientific.com by DEAKIN UNIVERSITY on 03/15/15. For personal use only.