Heterodimensional tangencies on cycles leading to strange attractors

TL;DR

This paper investigates heterodimensional cycles in 3D diffeomorphisms, showing they can be approximated by systems with quadratic homoclinic tangencies, leading to the emergence of strange attractors.

Contribution

It demonstrates that heterodimensional cycles can be approximated by systems with quadratic homoclinic tangencies, revealing the presence of strange attractors near such cycles.

Findings

Heterodimensional cycles can be approximated by systems with quadratic homoclinic tangencies.

Strange attractors can be detected near systems with heterodimensional cycles.

The tangency unfolds generically with respect to the parameter family.

Abstract

In this paper, we study heterodimensional cycles of two-parameter families of 3-dimensional diffeomorphisms some element of which contains nondegenerate heterodimensional tangencies of the stable and unstable manifolds of two saddle points with different indexes, and prove that such diffeomorphisms can be well approximated by another element which has a quadratic homoclinic tangency associated to one of these saddle points. Moreover, it is shown that the tangency unfolds generically with respect to the family. This result together with some theorem in Viana, we detect strange attractors appeared arbitrarily close to the original element with the heterodimensional cycle.