Scaling Structures and Statistical Mechanics of Type I Intermittent Chaos : General and Mathematical Physics

概要

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Intermittent chaos exhibits regular laminar motions and irregular turbulent bursts alternately, indicating that its chaotic attractors has two different types of local structures. For type I intermittency just before the saddle--node bifurcation, it is shown that the two types of local structures can be captured by the fluctuation spectrum h(Λ) of the coarse-grained local expansion rates Λ of nearby orbits and their q-weighted average Λ(q), (-∞<q<∞). The spectrum h(Λ) and the average Λ(q) are obtained analytically for a piecewise linear Markov map which exhibits type I intermittency, and numerically for the logistic map just before the period-three window. Thus it turns out that h(Λ) has a linear part between Λ=Λ_1>0 and Λ=Λ_2=0 with a slope q_δ, leading to a discontinuous transition of Λ(q) from Λ_1 to Λ_2 at q=q_δ as q is increased across q_δ. This represents a phase transition between the laminar motions and the turbulent bursts, and give a new type of q-phase transition with 1.0>q>0.5 in contrast to other three types of q-phase transitions with transition points q_α=2.0, q_β<0.5 and q_γ=1.0. Thus statistical mechanics of chaotic attractors at the bifurcation points is fully developed for the nontrivial case.
理論物理学刊行会の論文
1989-01-25