Scaling Structures of Chaotic Attractors and q-Phase Transitions at Crises in the Henon and the Annulus Maps

Abstract

Singular local structures of the chaotic attractors at the bifurcation points for three types of crises in the Henon and the annulus maps are studied theoretically and numerically in terms of the spectrum heAl of the coarse· grained local expansion rates A of nearby orbits along the local unstable manifold, and are shown to produce linear parts in heAl with slopes qa=2, qp< 1/2 and qT= 1. These linear slopes bring about three types of discontinuous phase transitions of the q-weighted average A(q), (-oo<q<oo) of A at q=qa, qp, qT, respectively, as q is varied. The q-phase transition at q = qa is due to the homoclinic tangencies, whereas that at q = qp is caused by a collision of the chaotic attractor with an unstable periodic orbit, where the slope qp and the singularity exponent a of the natural invariant measure at the periodic orbit are given in terms of the eigenvalues of certain periodic orbits. Just after the merging of two chaotic attractors into one chaotic attractor, a q-phase transition occurs at q=qT due to the intermittent hopping motions between two phase-space regions formerly occupied by the old attractors.