Study on the emergent mechanism of macroscopic dynamical structure in large dimensional dynamical systems

我们研究了大维动力系统相空间中的结构，这些结构在不完全稳定的情况下吸引并保持轨迹相当长的时间。这些结构被认为在时间相关中具有长尾的慢动力学的出现中起着重要作用。我们采用全局耦合混沌映射系统作为主要工作模型，通过研究，我们发现了大维混沌系统中一类普遍存在且鲁棒性很强的伪吸引子。已知类型的伪吸引子（特别是在自由度较小的动力系统中）基本上是与盆地边界碰撞的吸引子的残余物，因此只能在参数空间中相当有限的区域观测到，而新发现的伪吸引子由于是与连续分布的动力系统中的吸引子相关联的鞍集，因此在参数空间中可以在很大的区域观测到。它对应于无穷多个自由度的极限。这里得到的结果将提供一个基本的思想来理解系统的行为，它不是完全无序的，尽管它本质上是不可约的大量的自由度，一类现象在所谓的复杂系统中是相当普遍的。除上述结果外，我们还发现了在具有少量自由度的动力系统中一些新的现象，这些现象也与间歇性或流动等慢动力学有关，并且具有一种鲁棒性，并且可能在具有某一类对称性的系统中观察到。结果表明，在复杂系统动力学领域还有许多有待探索的地方，同时也为分析此类系统的准稳定状态提供了一些线索。

We studied on structures in phase space of large dimensional dynamical systems, that attract and hold trajectories for a considerable duration while they are not completely stable. Such structures are assumed to play an essential role in the emergence of slow dynamics with long tail in temporal correlation. We adopted globally coupled chaotic map systems as the main working model, and as a result of the research, we uncovered a type of psudo attractors in large dimensional chaotic systems, that can appear quite commonly and robustly. The psudo attractors of known types (especially in the dynamical systems with small number of degrees of freedom are basically remnant of attractor that collide with its basin boundary, and therefore observed only in quite limited area in parameter space, on the other hand, the newly uncovered ones can ovserved in broad area in the parameter space, since they are saddle set that are associated to the attractors in the dynamical system of continuous distribution, which corresponds to a limit of infinitely many degrees of freedom. The result obtained here would supply a basic idea to understand the behavior of the systems which is not completely disordered in spite of its essentially irreducible large number of degrees of freedom, a class of phenomena that are quite ubiquitous in so called complex systems. In addition to the above mentioned result, we found some novel phenomena in dynamical systems with small number of degrees of freedom, that are also related to slow dynamics like intermittency or itinerancy, and have a kind of robustness, and possibly be observed in the systems with a certain class of symmetry. The results show that there still much to explore in this area, the dynamics of complex systems, while they provide some clue to analyze quasi-stable states in such systems.