Engineering Bifurcations in High-Dimensional Dynamical Systems Using Isostable Reduction Methods

这笔赠款将支持研究，将促进物理，化学和生物科学的进步，从而提高国家的健康和繁荣。 在过去的四分之一世纪中，超级计算能力的快速发展导致了计算模型的规模和复杂性的爆炸。 在这些复杂的高维系统中，模型降阶是数学分析和随后实施主动控制策略的必要的初步步骤。 不幸的是，现有的减少策略是装备不良，以了解机制的动力学行为的定性变化，特别是当基本的行为是由系统的非线性。 该奖项支持基础研究，将开发数学简化策略，可用于预测和设计高维非线性动力系统行为的预期变化。 这些新方法将用于各种应用，如心脏电生理学，神经网络和流体流动，从而有利于国家健康和安全。 重要的研究成果将被纳入教育计划，使来自代表性不足背景的学生受益。该项目的目标是研究如何使用新开发的等稳约化框架来预测和设计高维非线性动力系统中的分叉。 等稳降阶方法明确地结合了主导系统的非线性，同时保留了分析的易处理性;因此，它复制了完善的线性降阶策略无法复制的系统行为。 作为这项研究的一部分，将创建新的非反馈控制框架，可用于稳定任意高维的混沌和不稳定的动力系统。 此外，数学框架将被创建来预测分叉，导致在强耦合振荡器网络的自发同步的开始。 还将制定战略，以推断必要的条款，在实验应用中，这将允许实施这些减少战略的基础模型方程不明确已知的系统中的等稳简化方程。 该项目的主要应用将是心脏和神经电生理学模型，以寻求更好的疾病治疗方案。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This grant will support research that will promote progress in the physical, chemical, and biological sciences thereby enhancing national health and prosperity. Over the past quarter century rapid progress of supercomputing capabilities has resulted in an explosion in the size and complexity of computational models. Model reduction is an imperative preliminary step in the mathematical analysis and subsequent implementation of active control strategies in these complex and high-dimensional systems. Unfortunately, existing reduction strategies are ill-equipped to understand the mechanisms governing qualitative changes in dynamical behavior, particularly when the underlying behavior is dominated by system nonlinearities. This award supports fundamental research that will develop mathematical reduction strategies that can be used to anticipate and engineer desired changes in the behavior of high-dimensional, nonlinear dynamical systems. These new methods will find use in a wide variety of applications such as cardiac electrophysiology, neural networks, and fluid flows with resulting benefits to national health and security. Important research findings will be incorporated into educational programs that benefit students from underrepresented backgrounds.The goal of this project is to study how a newly developed isostable reduction framework can be used to predict and engineer bifurcations in high-dimensional, nonlinear dynamical systems. The isostable reduction approach explicitly incorporates dominant system nonlinearities while retaining analytical tractability; as such it replicates system behaviors that well-established linear reduction strategies cannot. As part of this research, novel nonfeedback control frameworks will be created that can be used to stabilize chaotic and unstable dynamical systems of arbitrarily high dimension. Additionally, mathematical frameworks will be created to anticipate bifurcations that lead to the onset of spontaneous synchronization in strongly coupled oscillator networks. Strategies will also be developed to infer the necessary terms of isostable reduced equations in experimental applications which will allow for the implementation of these reduction strategies in systems for which the underlying model equations are not explicitly known. The primary applications in this project will be to models of cardiac and neural electrophysiology in pursuit of better disease treatment options.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.