Type-I intermittency for the Hénon-map family

Abstract

Type-I intermittency was defined as the precursor quasiregular behavior to every tangent bifurcation for the one-dimensional (1D) quadratic-map family. We have extended the definition to the two-dimensional H\'enon-map family with the following results. The gradual increase of bistability and multistability that accompanies the reduction of dissipation from the 1D limit carries with it the eventual decrease and disappearance of the intermittency associated with the occurrence of each stable periodic orbit. Numerically we observe qualitative changes in the invariant orbit density and the Lyapunov exponent in the transition region, the latter showing a discontinuous, or first-order, rather than the usual continuous phase transition. A hysteretic response of the dynamics to slow parameter change, which usually accompanies a first-order transition, is also noted. The changes associated with various periodic orbits, which are tabulated, show both similarities and differences. A considerable understanding of these phenomena is achieved by an in-depth study of the topology of, and dynamics in, the phase plane of the H\'enon system. Heuristic pictures are developed for some surprising bifurcation structures.