Transient spatiotemporal chaos in regular and complex networks

本研究探讨规则与复杂网络上的暂态时空混沌，网络节点处的可激发动力学元素之间存在扩散耦合。从统计物理，动力系统，统计数学和数值模拟的理论概念的结合使用，需要提高这些系统中的瞬态时空混沌的理解与类似的稳态动力学。本项目的目的是：1）从线性稳定性分析中深入了解反应扩散网络的瞬态和渐近时空混沌的主稳定性函数的判别性质; 2）量化Gray-Scott反应扩散网络的统计崩溃性质; 3）非局部耦合对全局网络动力学渐近稳定性的局部动力学和拓扑后果的识别; 4）识别常见的数学崩溃性质及其对非局部耦合的鲁棒性。5)基准计算，这将提供第一步的概括/分类的瞬态时空混沌反应扩散网络的基础上瞬态时空混沌更一般的数学理论。更广泛的影响包括对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。教育工作将侧重于在跨学科环境中对不同本科生和研究生进行研究培训，让学生深入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）计划参与。研究结果将在网站上向非科学公众展示，并通过PI在阿拉斯加的推广活动。

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.