Discrete Lorenz attractors in homoclinic bifurcations

Abstract:

Bifurcations of homoclinic tangencies in three-dimensional maps give rise to various dynamical phenomena. However, in order to obtain three-dimensional chaotic behavior under the breakup of a homoclinic tangency, one needs a special setup, preventing from the existence of two- and one-dimensional center manifolds. First of all, the Jacobian at the saddle should be equal to one. The list of possible second required conditions include resonant saddles, saddle focuses and non-simple homoclinic tangencies. In the talk I will give an overview of all these cases and demonstrate the proof of the birth of Lorenz attractors in them.