Entropy charts and bifurcations for Lorenz maps with infinite derivatives.

Abstract

This paper deals with one-dimensional factor maps for the geometric model of Lorenz-type attractors in the form of two-parameter family of Lorenz maps on the interval I=[-1,1] given by Tc,ν(x)=(-1+c⋅|x|ν)⋅sign(x). This is the normal form for splitting the homoclinic loop with additional degeneracy in flows with symmetry that have a saddle equilibrium with a one-dimensional unstable manifold. Due to L. P. Shilnikov' results, such a bifurcation (under certain conditions) corresponds to the birth of the Lorenz attractor. We indicate those regions in the parameter plane where the topological entropy depends monotonically on the parameter c, as well as those for which the monotonicity does not take place. Also, we indicate the corresponding bifurcations for the Lorenz attractors.