Collective Order in High-dimensional Chaos

1)隐相干和集体动力学：我们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，非均匀性增强了集体动力学的规则性，这种有序性的形成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。我们还回顾了集体动力学及其意义，并尝试用“集体Lyapunove指数”来表征。2）集体秩序是时空混沌：我们建立了一个由时空混沌产生的集体图案形成模型。发现这种空间图案是由于自发对称性破缺造成的。通过数值分析，我们证明了用湍流粘性解释这一现象的唯象性是有效的。3）耦合映象气体：内部混沌动力学和元素空间运动的干涉导致了一种新的有序类型，以及有序形成和无序之间的巡回性。4）元素生与死的长期动力学：通过考虑自由度的变化，我们发现元素的分化导致元素数量的增长。还讨论了结果与细胞生物学的相关性。

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.