Rank-one strange attractors versus Heteroclinic tangles

Abstract

We present a mechanism for the emergence of strange attractors (observable chaos) in a two-parameter periodically-perturbed family of differential equations on the plane. The two parameters are independent and act on different ways in the invariant manifolds of consecutive saddles in the cycle. When both parameters are zero, the flow exhibits an attracting heteroclinic cycle associated to two equilibria. The first parameter makes the two-dimensional invariant manifolds of consecutive saddles in the cycle to pull apart; the second forces transverse intersection. These relative positions may be determined using the Melnikov method. Extending the previous theory on the field, we prove the existence of many complicated dynamical objects in the two-parameter family, ranging from "large" strange attractors supporting SRB (Sinai-Ruelle-Bowen) measures to superstable sinks and H\'enon-type attractors. We draw a plausible bifurcation diagram associated to the problem under consideration and we show that the occurrence of heteroclinic tangles is a \emph{prevalent} phenomenon.