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(本项目的调查人员之一)组织，1978年在京都举行(见Prog。西奥。太棒了。补编(1978))。从那时起，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，甚至社会主义者，开始并鼓励了基于耗散动力系统的非线性动力学的研究。现在人们知道，离散动力系统中的湍流或混沌场景可以分为几种类型，即众所周知的倍周期路径、环面崩溃和间歇性。混沌轨道的显著特征表现为空间和时间尺度上的几个标度律，以及由于混沌中的有序而产生的高度相干的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统计动力学范式，该范式给出了非线性非平衡系统中的普遍行为。基于上述指导原则，我们成功地实现了以下主题：1)I型和III型间歇性混沌的全局谱结构的统计-物理理论，以及由于周期-3窗口坍塌引起的间歇性混沌。2)用粗粒度局部扩张率及其谱psi(Lambda)和Q相变描述混沌吸引子的局部结构。3)临界轨道记忆效应的长时间关联及其局部扩张率谱的临界标度律。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.