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

这笔拨款将用于支持促进物理、化学和生物科学进步的研究，从而促进国家健康和繁荣。在过去的四分之一世纪里，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式增长。在这些复杂的高维系统中，模型约简是数学分析和后续主动控制策略实施的必要步骤。不幸的是，现有的还原策略不足以理解控制动力学行为质变的机制，特别是当潜在行为由系统非线性主导时。该奖项支持基础研究，这些研究将开发数学约简策略，可用于预测和设计高维非线性动力系统行为的期望变化。这些新方法将广泛应用于心脏电生理学、神经网络和流体流动等领域，从而为国家健康和安全带来益处。重要的研究成果将被纳入教育项目，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简框架来预测和设计高维非线性动力系统的分岔。等稳约简方法在保留分析可追溯性的同时明确地结合了优势系统非线性；因此，它复制了已建立的线性缩减策略无法复制的系统行为。作为本研究的一部分，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动力系统。此外，将创建数学框架来预测导致强耦合振荡器网络中自发同步开始的分岔。还将制定策略来推断实验应用中等稳约简方程的必要项，这将允许在基础模型方程不明确已知的系统中实施这些约简策略。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。该奖项反映了美国国家科学基金会的法定使命，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.