Bifurcations of Morse–Smale Dynamical Systems

Abstract

This chapter describes a phenomenon that occurs in the bifurcation theory of one-parameter families of diffeomorphisms. This is the appearance of infinitely many different topological conjugacy classes of structurally stable diffeomorphisms, each class containing a diffeomorphism with an infinite nonwandering set, in every neighborhood of certain diffeomorphisms in the boundary of the Morse–Smale diffeomorphisms. This phenomenon occurs quite frequently in the following sense: any Morse–Smale diffeomorphism, on a manifold of dimension greater than one, can be moved through a one-parameter family which first exhibits some simple phase portrait changes to a new Morse–Smale diffeomorphism. The natural place where the phenomenon appears is in the construction of a cycle, and it occurs in that situation generically. On the other hand, under a reasonably general condition if one approaches the boundary of the Morse–Smale diffeomorphisms without creating a cycle, then the only new structurally stable diffeomorphisms one can find nearby will also be Morse–Smale. In this case as well, one can encounter an infinite number of topologically different Morse–Smale diffeomorphisms. It is shown that all of the conditions assumed for these results are true for open subsets of the space of one-parameter families of diffeomorphisms.