Dynamic Description of the Critical 2^∞ Attractor and 2^m-Band Chaos : General and Mathematical Physics

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Motion on the critical 2^∞ attractor at the accumulation point of period doubling is studied by the sum of the local expansion rates of nearby orbits over time n; S_n(X_0)≡Σ^<n-1>_<t=0>λ_1(X_t). The sum S_n(X_0) fluctuates with n, depending upon the initial value X_0. The Liapunov number Λ^∞≡lim_<n⇾∞>S_n(X_0)/n is zero, indicating that the critical attractor has no mixing. Then it is found that, as n increases, the variance <{S_n(X)}^2> exhibits a remarkable self-similar temporal structure which can be described by a universal memory function. Further, the algebraic exponent β_n(X_0)≡S_n(X_0)/log n is introduced for large n, and its fluctuation spectrum ψ_β(β) is shown to exhibit a linear part with a universal slope S_C≃-0.756. These two phenomena characterize the critical attractor clearly. The critical attractor is also studied as the limit of the cascade of the band-splitting bifurcations of band chaos. It is found for the 2^m-band attractor that, as M≡2^m⇾∞, the spectrumψ_β(β) for n<M reduces to that of the critical 2^∞ attractor, whereas the fluctuation spectrum ψ(Λ) of Λ_n(X_0)≡S_n(X_0)/n for n≫M tends to satisfy a critical scaling law ψ(Λ)=M^<-1>φ(M(Λ-Λ^∞)) with a universal scaling function φ(z) which has two linear parts with slope s_α= -1 and s_β≃1.37. Thus we obtain a unifying description of the critical 2^∞ attractor and the 2^m-band attractor in terms of the fluctuations of the sum S_n(X_0).
理論物理学刊行会の論文
1989-11-25

Dynamic Description of the Critical 2^∞ Attractor and 2^m-Band Chaos : General and Mathematical Physics

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