On birth of discrete Lorenz attractors under bifurcations of 3D maps with nontransversal heteroclinic cycles

Ivan Ovsyannikov

Published 2017-05-12Version 1

In this paper new global bifurcations of three-dimensional diffeomorphisms leading to the birth of discrete Lorenz attractors are studied. We consider the case of a heteroclinic cycle having one non-transversal heteroclinic orbit (quadratic tangency). Additional conditions are imposed onto the system, namely, the Jacobians in the saddles are taken such that the phase volumes near one point are expanded and contracted near another point. Thus here we are dealing with the contracting-expanding, or mixed case. Also it is assumed that one of the heteroclinic orbits (either the transverse one, or the one corresponding to a quadratic tangency) is non-simple. These conditions prevent from existence of a two-dimensional global center manifold and thus keeps the dynamics effectively three-dimensional, thus giving a possibility for Lorenz attractors to exist. The analogous case of a non-simple homoclinic tangency was studied in \cite{GOT14}, but the birth of Lorenz attractor in the bifurcations of heteroclinic intersections with a non-simple geometry was not studied before.