Local Structures of Chaotic Attractors and q-Phase Transitions at Attractor-Merging Crises in the Sine-Circle Maps: General and Mathematical Physics

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A merging of two coexisting phase-locked chaotic attractors into one phase-unlocked chaotic attractor occurs in the region where two Arnold tongues overlap each other. This merging is studied in terms of a weighted average Λ (q),(-∞< q< ∞) of the coarse-grained local expansion rates Λ of nearby orbits with a q-dependent weight. Just before the merging, each of the two coexisting attractors has local structures similar to that of the attractor of the logistic map F(x) =1-2x^2, and, as q is varied, Λ(q) for each exhibits two discontinuous transitions at q=q_α=2.0 and q=q_β≃0.0. The phase for q_α > q > q_β is dominated by the normal orbits with Λ equal to a positive Liapunov exponent Λ_k, (k=1,2). A theory of the q-phase transitions is proposed. Just after the merging, the phase-unlocked attractor has a singular local structure, and Λ(q) exhibits a new discontinuous transition from Λ_1 to Λ_2 at q=q_γ=1.0 due to the intermittent hopping motions between two repellers with Λ≃Λ_1 and Λ≃ Λ_2. These features of the attractors just before and after the merging are quite universal and can be summarized in terms of linear parts of the expansion-rate spectrum h(Λ).
理論物理学刊行会の論文
1988-11-25