A study about complex motion caused by hierarchical structure and intermittency, in nonlinear dynamical system with large degree of freedom

非线性动力系统中隐藏着各种层次结构和运动，因此建立新的方法和总体思路是理解非线性动力系统的重要问题。本文以大自由度非线性动力系统中不变流形的层次结构为研究对象，从理论上和数值上理解其运动是本研究的目的。(1)为了理解高维混沌的复杂行为，我们已经开始研究它们周围的不变流形和开关间歇。我们开展了这项研究。然后，我们发现大自由度混沌（混沌流动）的复杂行为是在不变流形之间的不规则运动，并表明我们通过统计粗粒化得到了清晰的表征；（ii）我们用摄动理论论证了(i)基础的开关间歇性的性质。我们用非防腐剂的方法来处理它们，从而获得了新的知识。（iii）许多非混沌吸引子（不动点、周期轨道和准周期轨道）可以在大自由度下存在（吸引子的数量随着系统大小的增加而突然增加）。我们特别从噪声响应的角度研究了其特性，发现足够小的噪声会束缚吸引子并引起间歇性现象。此外，我们还发现了在系统宏观量中可以描述为反常扩散的运动。(4)研究了保守动力系统混沌的间断性与低维不变结构的关系。在对上述动力系统行为的研究之后，我们可以开发更复杂行为的方法。我们开始(a)用异常扩散表征广义位移映射中的复杂行为，(b)用时间序列熵分析脑电图模式，(c)樱岛火山爆发的间歇性分析，(d)研究具有不变流形中似乎有不变流形的分层结构的动力系统的复杂行为。少

Various hierarchical structures and motion hide in nonlinear dynamical systems, and so it is an important problem to build new methodology and general idea to understand it. We paid our attention to the hierarchical structure of invariant manifolds in nonlinear dynamical systems with the large degree of freedom as its approach, and understanding the motion theoretically and numerically is our aim of this research.( i ) To understand the complicated behavior of high dimensional chaos, we began studies invariant manifolds and On-off intermittency around them, already. We developed this study. Then we found that the complex behavior of the large degree of freedom chaos (chaotic itinerancy) is wondering motion between invariant manifolds and shown that we get clear characterization by statistical coarse-graining,( ii ) The properties of On-off intermittency that is the base of ( i ) had been argued with perturbation theory. We got new knowledge by treating them by non-preservative methods. … More ( iii ) Many non-chaotic attractors (fixed points, periodic orbits, and quasi-periodic orbits) can exist in a large degree of freedom (the number of attractor suddenly increases with increase of system size). We studied the properties from a viewpoint of response for noise in particular and was found that enough small noise tied attractor and caused intermittent phenomena. In addition, we found what we could describe the motion as anomalous diffusion in macro-quantities of system.( iv ) We studied also relation intermittency of chaos in conservative dynamical systems and low dimensional invariant structure.Next to a study about the behavior of dynamical systems mentioned above, we can develop approach for more complicated behavior. We started (a) a characterization of complicated behavior in generalized shift maps by anomalous diffusion, (b) an analysis of EEG pattern by time-series' entropy, (c) an analysis of intermittency of explosions of SAKURAJIMA volcano and (d) a study for the complicated behavior of dynamical systems with the hierarchical structure which seemed to have invariant manifold in an invariant manifold. Less