ar X iv : 0 70 9 . 11 16 v 2 [ m at h . D S ] 9 S ep 2 00 7 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 + ez) where a, b, c, d, e are real with bd 6= 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.