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

“王子非线性非平衡统计力学研讨会”由H。森（本项目的调查员之一），并于1978年在京都举行（见Prog. Theor. Phys. Suppl. No. 64（1978））。从那时起，基于耗散动力系统的非线性动力学的研究开始了，并受到了日本科学家，特别是物理学家、数学家、生物学家、化学家、工程师、地质学家，甚至社会学家的鼓励。耗散动力系统中的湍流开始或走向混沌的情况现在已知分为几种类型，即著名的倍周期途径，环面的坍缩和不对称性。混沌轨道的突出性质表现为空间和时间尺度的标度律以及混沌中的有序性所导致的高度相干行为或强时间相关性，我们的目标是构建一个混沌或湍流运动的动力学范式，给出非线性-非平衡系统的普遍行为。基于上述指导原则，我们成功地执行了以下主题：1）I型和III型阵发混沌以及周期3窗口坍塌引起的阵发混沌的整体谱结构的统计物理理论。2）混沌吸引子的局部结构的粗粒度局部膨胀率表征，3）临界轨道的记忆效应引起的长时间关联及其局域膨胀率谱的临界标度律; 4）Hamilton动力系统中粒子的对流扩散和混合; 5）粒子间的相互作用混沌动力系统中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.