A Symbolic Bifurcation Approach for Complex Deterministic Dynamics

Many significant advancements in deterministic nonlinear science, crucial for driving progress in cutting-edge engineering, rely heavily on a deeper understanding and practical application of complex theoretical elements borrowed from dynamical systems and bifurcation theory. To realize their full potential in practice, these advances require incorporating mathematical and simulation tools into powerful computing platforms, such as massively parallel and affordable graphics processing units. This project involves the development of new, algorithmically simple, yet efficient and generalizable mathematical approaches for analyzing bifurcations in high-dimensional systems that integrate into intelligent, comprehensive simulations. The goal is to enable quantitative and, more importantly, qualitative progress in higher-level studies of data-driven, detailed, and phenomenologically-reduced models with complex nonlinear dynamics. These models find diverse applications, ranging from engineering and meteorology to living systems, including neural networks. This project also involves interdisciplinary training and educational opportunities for graduate students, undergraduate students, and high school students, with a particular focus on involving under-represented students in STEM. The long-term goal of this project is two-fold: to further extend the applied theory of non-local bifurcations and to foster its broader applications. This is expected to lead to a better understanding and demonstration of the universality of the rules of complex dynamics across diverse systems. Homoclinic bifurcations are key to understanding the origin and fine organization structure of deterministic chaos in various systems, including diverse applications found in physics, neuroscience, and economics. A new symbolic approach will be developed which aims to reveal an array of homoclinic and heteroclinic bifurcations in typical systems with complex dynamics due to interactions of saddle equilibria. 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.