Persistent antimonotonic bifurcations and strange attractors for cubic homoclinic tangencies

TL;DR

This paper investigates complex dynamical behaviors in a family of two-dimensional maps with cubic homoclinic tangencies, revealing persistent bifurcations and strange attractors with SRB measures, advancing understanding of non-monotonic bifurcations.

Contribution

It introduces conditions under which cubic homoclinic tangencies lead to persistent bifurcations and strange attractors in dissipative systems.

Findings

Existence of open sets with persistent contact bifurcations.

Verification of Wang-Young's conditions for strange attractors.

Identification of cubic polynomial-like strange attractors with SRB measures.

Abstract

In this paper, we study a two-parameter family of two-dimensional diffeomorphisms such that it has a cubic homoclinic tangency unfolding generically which is associated with a dissipative saddle point. Our first theorem presents an open set in the parameter-plane such that, for any parameter value in the open set, there exists a one-parameter subfamily through this value exhibiting cubically related persistent contact-making and contact-breaking quadratic tangencies. Moreover, the second theorem shows that any such two-parameter family satisfies Wang-Young's conditions which guarantee that it exhibits a cubic polynomial-like strange attractor with an SRB measure.