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 } \{ {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 {P_t};\;F_t^n({P_t}) = {P_t} , and | det D F t n ( P t ) | > 1 |\det DF_t^n({P_t})| > 1 . We also require the stable and unstable manifolds of P t {P_t} to form homoclinic tangencies as the parameter t t varies through t 0 {t_0} . Our emphasis is the exploration of the behavior of families of diffeomorphisms for parameter values t t near t 0 {t_0} . We show that there are parameter values t t near t 0 {t_0} at which F t {F_t} has infinitely many co-existing periodic sinks.