Dissipative homoclinic loops and rank one chaos

TL;DR

This paper demonstrates that certain dissipative systems with homoclinic loops can exhibit strange attractors with SRB measures under periodic forcing, using rank one map theory.

Contribution

It applies the recent rank one map theory to second order systems with homoclinic loops, establishing conditions for chaos under periodic forcing.

Findings

Existence of strange attractors with SRB measures for positive measure parameter sets

Application of rank one map theory to dissipative homoclinic systems

Extension of chaos results to second order differential equations

Abstract

We prove that when subjected to periodic forcing of the form $p_{\mu, \rh, \om} (t) = \mu (\rh h(x,y) + \sin (\om t))$, certain second order systems of differential equations with dissipative homoclinic loops admit strange attractors with SRB measures for a set of forcing parameters $(\mu, \rh, \om)$ of positive measure. Our proof applies the recent theory of rank one maps, developed by Wang and Young based on the analysis of strongly dissipative H\'enon maps by Benedicks and Carleson.