Monotonicity and non-monotonicity regions of topological entropy for Lorenz-like families with infinite derivatives

We study behavior of the topological entropy as the function of parameters for two-parameter family of symmetric Lorenz maps Tc,ɛ(x) = (−1 + c|x|1−ɛ) · sgn(x). This is the normal form for splitting the homoclinic loop in systems which have a saddle equilibrium with one-dimensional unstable manifold and zero saddle value. Due to L.P. Shilnikov results, such a bifurcation corresponds to the birth of Lorenz attractor (when the saddle value becomes positive). We indicate those regions in the bifurcation 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.

ON GEOMETRICAL PROPERTIES OF TWO-DIMENSIONAL DIFFEOMORPHISMS WITH HOMOCLINIC TANGENCIES

Two-dimensional diffeomorphisms with a quadratic tangency of invariant manifolds of a saddle fixed point are considered in the cases where the saddle value σ is either less than 1 or equal to it. A description of the structure of hyperbolic subsets is given. In the case σ=1, it is shown that almost all such diffeomorphisms admit the complete description in distinction with the case σ<1.