Three-dimensional dissipative diffeomorphisms with homoclinic tangencies

Abstract

Given a two-parameter family of three-dimensional diffeomorphisms \{f_{a,b}\}_{a,b} , with a dissipative (but not sectionally dissipative) saddle fixed point, assume that a special type of quadratic homoclinic tangency of the invariant manifolds exists. Then there is a return map f^n_{a,b} , near the homoclinic orbit, for values of the parameter near such a tangency and for n large enough, such that, after a change of variables and reparametrization depending on n , this return map tends to a simple quadratic map. This implies the existence of strange attractors and infinitely many sinks as in other known cases. Moreover, there appear attracting invariant circles, implying the existence of quasi-periodic behaviour near the homoclinic tangency.