Structure change of stable and unstable manifolds in two-dimensional maps II. Island-merging and boundary crisis

Abstract

A family of dissipative orientation preserving two-dimensional maps is considered. The structure change of stable and unstable manifolds of saddles born in the sequence of the period-doubling bifurcation is investigated in relation to the island-merging and boundary crisis of the strange attractor. First, several relations among various critical parameter values are obtained. Second, a conjecture that the strange attractor is equal to the minimal set of unstable manifolds is proposed and theoretical and numerical evidence supporting this conjecture is exhibited. Third, a new interpretation of the mechanism of the boundary crisis is proposed.