Statistical-Physical Study for Global Spectral Structures of Intermittent Chaos and Its Spatio-Temporal Scaling Properties.

“Oji非线性非平衡统计力学研讨会”由H. Mori教授（本项目的研究人员之一）于1978年在京都举行(见Prog。定理。理论物理。增刊。第64（1978）号)。从那时起，以耗散动力系统为基础的非线性动力学研究开始了，并受到日本科学家的鼓励，特别是物理学家、数学家、生物学家、化学家、工程师、地质学家，甚至社会主义者。在耗散动力系统中，湍流的发生或混沌的情景现在已知可分为几种类型，即众所周知的周期加倍路线，环面坍缩和间歇性。混沌轨道的突出性质表现为空间和时间尺度上的若干标度定律以及混沌中由于有序而产生的高度相干行为或强时间相关性。我们的目标是为混沌或湍流运动建立一个统计动力学范式，给出非线性非平衡系统的普遍行为。基于上述指导原则，我们成功地完成了以下主题：1)I型和III型间歇混沌和3期窗口崩塌导致的间歇混沌的全局谱结构的统计物理理论。2)用粗粒度局部膨胀率表征混沌吸引子的局部结构及其谱psi （LAMBDA）和q相变。3)局部膨胀率谱的临界轨道记忆效应的长期相关性及其临界标度规律。4)哈密顿动力系统中粒子的平流扩散和混合。5)混沌动力系统的能量耗散及其涨落。

"Oji Seminar on Non-Linear Non-Equilibrium Statistical Mechanics" was organized by Professor H. Mori (one of the investigators of the present project) and held at Kyoto in 1978 (see Prog. Theor. Phys. Suppl. No. 64 (1978)). Since then the studies of nonlinear dynamics on the basis of dissipative dynamical systems were started and encouraged for Japanese scientists, particularly, physicists, mathematicians, biologists, chemists, engineers, geologists and even for socialists.Onset of turbulence or the scenario to chaos in dissiparatieve dynamical systems is now known to be classified into a few types, which are the well-known period-doubling route, the collapse of a torus and the intermittency. Prominent properties of chaotic orbits are represented by several scaling laws for spatial and temporal scales and highly coherent behaviors or strong time correlations due to order in chaos.Our goal is to construct a statistical-dynamical paradigm for chaotic or turbulent motions which gives universal behaviors in nonlinear-nonequilibrium systems. Due to the above guiding priciples, we have performed successfully the following themes :1) Statistical-physical theory of global spectral structures of type I and III intermittent chaos and the intermittent chaos due to the collapse of period-3 windows.2) Characterization of local structures of chaotic attractors in terms of coarse-grained local expansion rates, and its spectrum psi (LAMBDA) and q-phase transition.3) Long-time correlations due to memory effect of critical orbits and its critical scaling laws of local expansion rate spectrum.4) Advective diffusion and mixing of particles in Hamiltonian dynamical systems.5) Energy dissipation and its fluctuations in chaotic dynamical systems.