Collective Order in High-dimensional Chaos

1)隐藏相干和集体动力学：分析了耦合混沌系统的集体运动。特别地，研究了具有分布非线性的全局耦合映射的集体行为。结果表明，非均匀增强了集体动力学的规律性，这种有序的机制是由于内部分叉结构的形成，以及结构与平均场之间的自洽动力学。我们还回顾了胶着体动力学及其意义，并尝试用“集体李雅普诺夫指数”来刻画。2)胶着体有序是时空混沌：我们构造了一个由时空混沌产生的集体模式形成模型。这种空间模式被发现是由于自发的对称性破坏造成的。通过对相变的数值分析，我们表明用浑浊的粘性速度来解释这一现象在现象学上是有效的。3)耦合映象气体：内部引力动力学和元素的空间运动的干扰导致了一种新的有序类型，以及有序形成和崩溃之间的巡回性。4)随着元素的诞生和死亡的长期动力学：通过考虑自由度的变化，我们发现元素的分化导致了元素数量的增加。结果与细胞生物学的相关性也被讨论。

1) hidden coherence and collective dynamics : We have analyzed collective motion of coupled chaotic systems. In particular, collective behavior is studied in globally coupled maps with distributed nonlinearity. It is shown that the heterogeneity enhances regularity in the collective dynamics.The mechanism of this order is due to the formation of an internal bifurcation structure, and the self-consistent dynamics between the structures and the mean-field. We have also reviewed the colletcive dynamics and its significance, with some attempts of the characterization by "collective Lyapunove exponent".2) Colletcive order is spatiotemporal chaos : We have constrcuded a model of collective pattern formation arising from spatiotemporal chaos. The spatial pattern is found to be due to spontaneous symmetry breaking. Through numerical analysis of the transition, we have shown that the interpretation of the phenomenon by the turbluent visocity is valid phenomenologically.3) Coupled map gas : The interferrence of internal caotic dynamics and the spatial motion of elements leads to a novel type of order, and the itinerancy between the order formation and collapse.4) Long-term dynamics with the birth and death of elements : By allowing for the change of degrees of freedom, we have found that the differentiation of elements leads to the growth in number of elements. Relevance of the results to cell biology is also discussed.