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

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Singular local structures of the chaotic attractors at bifurcation points for three types of crises in the Henon and the annulus maps are studied theoretically and numerically in terms of the spectrum h(Λ) of the coarse-grained local expansion rates Λ of nearby orbits along the local unstable manifold, and are shown to produce linear parts in h(Λ) with slopes q_α =2, q_β< 1/2 and q_γ =1. These linear slopes bring about three types of discontinuous phase transitions of the q-weighted average Λ(q), (-∞ < q < ∞) of Λ at q=q_α, q_β, q_γ, respectively, as q is varied. The q-phase transition at q =q_α is due to the homoclinic tangencies, whereas that at q=q_β is caused by a collision of the chaotic attractor with an unstable periodic orbit, where the slope q_β and the singularity exponent α 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=q_γ due to the intermittent hopping motions between two phase-space regions formerly occupied by the old attractors.
理論物理学刊行会の論文
1988-12-25