Birth of discrete Lorenz attractors at the bifurcations of 3D maps with homoclinic tangencies to saddle points

TL;DR

This paper proves the emergence of discrete Lorenz attractors in 3D maps with non-simple homoclinic tangencies to saddle points, extending previous results to a broader class of bifurcations.

Contribution

It establishes the birth of Lorenz-like attractors at bifurcations involving non-simple homoclinic tangencies in three-dimensional diffeomorphisms.

Findings

Lorenz-like strange attractors arise from non-simple homoclinic tangencies.

Extension of bifurcation theory to new classes of 3D maps.

Confirmation of Lorenz attractor formation in broader dynamical systems.

Abstract

It was established in 2006 that bifurcations of three-dimensional diffeomorphisms with a homoclinic tangency to a saddle-focus fixed point with the Jacobian equal to 1 can lead to Lorenz-like strange attractors. In the present paper we prove an analogous result for three-dimensional diffeomorphisms with a homoclinic tangency to a saddle fixed point with the Jacobian equal to 1, provided the quadratic homoclinic tangency under consideration is non-simple.