Attractors for return maps near homoclinic tangencies of three-dimensional dissipative diffeomorphisms

Antonio Pumariño,Joan Carles Tatjer

doi:10.3934/dcdsb.2007.8.971

Abstract

We numerically analyse different kinds of one-dimensional and two-dimensional attractors for the limit return map associated to the unfolding of homoclinic tangencies for a large class of three-dimensional dissipative diffeomorphisms. Besides describing the topological properties of these attractors, we often numerically compute their Lyapunov exponents in order to clarify where two-dimensional strange attractors can show up in the parameter space. Hence, we are specially interested in the case in which the unstable manifold of the periodic saddle taking part in the homoclinic tangency has dimension two.

Highlights

From the time of Poincare [18] to the present days, the homoclinic phenomena have been extensively studied for parameter families of diffeomorphisms defined in two-dimensional manifolds, see [16] for a complete overview on the subject
As was announced in the Introduction, we have put special emphasis along the paper in order to state the resemblances between our family of limit return maps, Ta,b, and the well-known quadratic family Qa (x) = 1 − ax2, which is the family of limit return maps prevailing for homoclinic phenomena for two dimensional diffeomorphisms (m = 2)
Even Hopf bifurcations in dimension two have similarities with flip bifurcations in dimension one. If we consider both bifurcations for fixed points, in the Hopf case we have an invariant set after the bifurcation which coincides with the border of a topological disk, while in the flip case, the invariant set is the border of an interval

Summary

Introduction

From the time of Poincare [18] to the present days, the homoclinic phenomena have been extensively studied for parameter families of diffeomorphisms defined in two-dimensional manifolds, see [16] for a complete overview on the subject. We think that the dynamical richness exhibited for those parameters belonging to (4) will be enough to realize the great complexity arising in families of dissipative three-dimensional diffeomorphisms unfolding a homoclinic tangency associated to a saddle point whose unstable invariant manifold has dimension two. If the invariant manifolds of p0 have a generalized homoclinic tangency for (a, b) = (a0, b0) which unfolds generically, there exists a positive measure set E of parameter values near (a0, b0), such that for (a, b) ∈ E the diffeomorphism fa,b exhibits a strange attractor with two positive Lyapunov exponents.

One-dimensional attractors

Destruction of invariant curves

Joining strange attractors

Conclusions