Statistical-Physical Study for Chaos and Turbulence.

“OJI非线性非平衡统计力学研讨会”由H.Mori教授组织，1978年在京都举行(见Prog。西奥。太棒了。补编第64号(1978年))。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。目前，耗散动力系统中的混沌现象或情景被分为几种类型，即众所周知的倍周期路径、环面崩塌和间歇性。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，我们成功地完成了以下工作：1)I型和III型间歇混沌全局谱结构的统计物理理论和周期3窗口坍塌引起的间歇混沌；2)混沌奇异吸引子奇异性谱的动力学理论。3)奇异吸引子的时空标度性质及其不稳定周期轨道的表示；4)混沌吸引子的局部结构的粗粒度局部扩张率的表征；5)正弦圆映射危机的粗粒度扩张率的涨落及其加权平均的不连续跃迁；6)Henon映射和环形映射危机时混沌吸引子的局部结构；7)间歇性混沌局部扩张率的反常动态标度律和时间关联。

" Oji Seminar on Non-Linear Non-Equilibrium Statistical Mechanics" was organized by Professor H. Mori 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 trubulence or the scenario to chaos in dissipative 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) Dynamical theory of the spectra of singularities of strange attractors of chaos f(<alpha>).3) Spatial and temporal scaling properties of strange attractors and their representation by unstable periodic orbits.4) Characterization of local structures of chaotic attractors in terms of coarse-grained local expansion rates.5) Fluctuations of coarse-grained expansion rates and discontinuous transitions in their weighted averages for crises of sine-circle maps.6) Local structures of chaotic attractors at crises in the Henon and annulus maps.7) Anomalous dynamic scaling laws and time correlations of local expansion rates for intermittent chaos.