Coexistence phenomena in the H\'enon family

TL;DR

This paper demonstrates that within the classical Hénon family, there exist parameter sets of positive measure where multiple attractive periodic orbits and strange attractors coexist, revealing complex dynamical phenomena.

Contribution

It proves the existence of parameter sets with positive measure where multiple attractors coexist in the Hénon family, including multiple periodic orbits and strange attractors.

Findings

Existence of parameters with multiple attractive periodic orbits.

Presence of parameters with coexisting strange attractors.

Positive Lebesgue measure sets of parameters exhibiting complex dynamics.

Abstract

We study the classical H\'enon family $f_{a,b}:(x,y)\mapsto(1-ax^2+y,bx)$, $0<a<2$, $0<b<1$, and prove that given an integer $k\geq 1$, there is a set of parameters $E_k$ of positive two-dimensional Lebesgue measure so that $f_{a,b}$, for $(a,b)\in E_k$, has at least $k$ attractive periodic orbits and one strange attractor. A corresponding statement also holds for the H\'enon-like families. The final main result of the paper is the existence, within the classical H\'enon family, of a positive Lebesgue measure set of parameters whose corresponding maps have two coexisting strange attractors.