Infinitely many co-existing sinks from degenerate homoclinic tangencies

Abstract:

The evolution of a horseshoe is an interesting and important phenomenon in Dynamical Systems as it represents a change from a nonchaotic state to a state of chaos. As we are interested in determining how this transition takes place, we are studying certain families of diffeomorphisms. We restrict our attention to certain one-parameter families $\{ {F_t}\}$ of diffeomorphisms in two dimensions. It is assumed that each family has a curve of dissipative periodic saddle points, ${P_t};\;F_t^n({P_t}) = {P_t}$, and $|\det DF_t^n({P_t})| < 1$. We also require the stable and unstable manifolds of ${P_t}$ to form homoclinic tangencies as the parameter $t$ varies through ${t_0}$. Our emphasis is the exploration of the behavior of families of diffeomorphisms for parameter values $t$ near ${t_0}$. We show that there are parameter values $t$ near ${t_0}$ at which ${F_t}$ has infinitely many co-existing periodic sinks.