Dynamics near homoclinic bifurcations of three-dimensional dissipative diffeomorphisms

Abstract

We explore the complicated dynamics arising in a neighbourhood of a homoclinic point associated with a homoclinic bifurcation of a two-parameter family of three-dimensional dissipative diffeomorphisms. We address the case in which the unstable manifold of the periodic saddle involved in the homoclinic bifurcation has dimension two. Besides proving the existence of strange attractors with two positive Lyapounov exponents for the associated limit return map, we also select a curve in the space of parameters in order to numerically detect the presence of possible new families of one-dimensional and two-dimensional strange attractors. The end of this curve of parameters corresponds to a return map which is conjugate to a 'bidimensional tent map'.