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

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.<br/><br/>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.<br/><br/>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.