Bifurcation of the ACT map

TL;DR

This paper analyzes the bifurcation phenomena of the Arneodo-Coullet-Tresser map, revealing stability regions, chaotic dynamics, and complex invariant structures through theoretical and numerical methods.

Contribution

It provides new insights into the bifurcation and chaotic behavior of the ACT map, including stability analysis and the existence of hyperbolic invariant sets.

Findings

Identified stability regions for fixed points and period-2 points.

Established the presence of hyperbolic invariant sets conjugate to full shifts.

Numerical evidence of Hopf bifurcations, strange attractors, and nested invariant tori.

Abstract

In this paper, we study the Arneodo-Coullet-Tresser map $ F(x,y,z)=(ax-b(y-z), bx+a(y-z), cx-dx^k+e z)$ where $a,b,c,d,e$ are real with $bd\neq 0$ and $k>1$ is an integer. We obtain stability regions for fixed points of $F$ and symmetric period-2 points while $c$ and $e$ vary as parameters. Varying $a$ and $e$ as parameters, we show that there is a hyperbolic invariant set on which $F$ is conjugate to the full shift on two or three symbols. We also show that chaotic behaviors of $F$ while $c$ and $d$ vary as parameters and $F$ is near an anti-integrable limit. Some numerical results indicates $F$ has Hopf bifurcation, strange attractors, and nested structure of invariant tori.