Bifurcation of the ACT map

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.