Three-dimensional dissipative diffeomorphisms with homoclinic tangencies. (English) Zbl 0972.37013

Tatjer, Joan Carles

Ergodic Theory Dyn. Syst. 21, No. 1, 249-302 (2001).

The author considers diffeomorphisms on three dimensional manifolds with a homoclinic tangency of stable and unstable manifolds of a saddle. It is assumed that, apart from the homoclinic tangency, there is an additional geometric degeneracy. In one of the studied cases, the unstable manifold is one dimensional and is assumed to also have a tangency with the strong stable foliation of the stable manifold. In two parameter unfoldings the existence is shown of Bogdanov-Takens bifurcations and invariant circles. Return maps on small domains near the tangency renormalize to small perturbations of simple quadratic maps such as \((x,y,z) \mapsto (z , bz , a + y + z^2)\). Numerical experiments suggest the occurrence of two dimensional strange attractors.

Reviewer: Ale Jan Homburg (Amsterdam)

MSC:

37C05

Dynamical systems involving smooth mappings and diffeomorphisms