Transient spatiotemporal chaos in regular and complex networks

本研究探讨了网络节点上可激元间扩散耦合的规则网络和复杂网络的瞬态时空混沌。需要结合使用统计物理、动力系统、统计数学和数值建模的理论概念，以提高对这些具有类似稳态动力学的系统中的瞬态时空混沌的理解。本项目的目的是提供1)从线性稳定性分析中洞察反应-扩散网络的暂态和渐近时空混沌的主稳定性函数的判别性质；2) Gray-Scott反应-扩散网络统计坍缩特性的量化；3)辨识全局网络动力学渐近稳定性的非局部耦合的局部动力学和拓扑后果；4)常见数学坍缩性质的识别及其对非局部耦合的鲁棒性。5)基准计算，这将为反应扩散网络中瞬态时空混沌的泛化/分类提供第一步，作为瞬态时空混沌的更一般数学理论的基础。更广泛的影响包括关于自然系统崩溃过程的新见解，如生态学或分子生物学中的物种灭绝。教育方面的努力将侧重于在跨学科的环境中对不同的本科生和研究生进行研究培训，让学生了解不同职业的可能性。来自其他大学的学生将通过大学的REU（本科生研究经验）项目参与。研究结果将在一个网站上向非科学界的公众展示，并通过私人计划在阿拉斯加的推广活动进行展示。

This study explores transient spatiotemporal chaos on regular and complex networks with diffusive coupling between the excitable dynamical elements at network nodes. The combined use of theoretical concepts from statistical physics, dynamical systems, statistical mathematics, and numerical modeling is required to improve the understanding of transient spatiotemporal chaos in these systems with a similar steady state dynamics. The purpose of this project is to provide 1) insight from a linear stability analysis into the discriminating properties of the master stability function for transient and asymptotic spatiotemporal chaos for reaction-diffusion networks; 2) a quantification of statistical collapse properties in the Gray-Scott reaction-diffusion network; 3) an identification of local dynamical and topological consequences of nonlocal coupling for the asymptotic stability of the global network dynamics; 4) an identification of common mathematical collapse properties and their robustness against nonlocal coupling. 5) benchmark calculations that will provide a first step towards a generalization/classification of transient spatiotemporal chaos in reaction-diffusion networks as a basis for a more general mathematical theory on transient spatiotemporal chaos. Broader impacts include new insights regarding collapse processes in natural systems, like species extinction in ecology or molecular biology. The educational efforts will focus on research training of diverse undergraduate and graduate students in an interdisciplinary environment where students gain insight into possibilities for diverse careers. Students from other universities will be engaged through the University's REU (Research Experiences for Undergraduates) program. The research results will be presented to the non-scientific public on a Web site, and through the PI's outreach activities in Alaska.