Existence of generic cubic homoclinic tangencies for Hénon maps

Abstract

In this paper, we show that the Hénon map ${\phi }_{a,b}$ has a generically unfolding cubic tangency for some $(a,b)$ arbitrarily close to $(-2,0)$ by applying results of Gonchenko-Shilnikov-Turaev [12]-[16]. Combining this fact with theorems in Kiriki-Soma [20], one can observe the new phenomena in the Hénon family, appearance of persistent antimonotonic tangencies and cubic polynomial-like strange attractors.

arXiv commentaccepted in Ergod. Th. & Dynam. Sys. 16 Feb 2012