NSF Award
2308225
The main goal of this project is to study graphs of general dynamical systems, both analytically and numerically. The outcome of this work will represent an important advance in the understanding of fundamental aspects of dynamical systems. It will provide a unifying setting and a set of tools applicable, in particular, to any dynamical system, from one-dimensional discrete-time systems, such as the logistic maps, to infinite-dimensional continuous-time ones, such as the Belousov–Zhabotinsky chemical reaction. This project also aims at the creation of a group of graduate and undergraduate students at Howard University working on the numerical analysis of the qualitative dynamics of the systems, together with the lead investigator. Moreover, within this project, the investigator will write a monograph on the logistic map, with help from participating students. This monograph will finally collect the most important results on the logistic map in a single place and will be aimed at applied readers, emphasizing readability over formal elegance. In order to make it available to the widest audience possible, the monograph will be released freely in “open source” online format. The project will be fully developed and undertaken at Howard University, a Historically Black Research University. <br/><br/><br/>The main goals of this project are: <br/>1. Investigating the graph of several finite-dimensional and infinite-dimensional dynamical systems, including but not limited to the following: unimodal maps, multimodal maps, Lorenz map, forced dumped pendulum, Newhouse maps, semilinear parabolic PDEs.<br/>2. Investigating the general properties of graphs themselves, including the types of possible appearance/disappearance of nodes in parametric families, the conditions for the graph to be connected, alternate definitions of nodes and edges.<br/> Moreover, this project will include a comprehensive study of several concepts of recurrence, such as chain-recurrence, strong chain-recurrence and Auslander’s generalized recurrence, and developing a system of axioms that will work as a framework for all kinds of recurrence and from which it will be possible to prove general properties of graphs of dynamical systems. The numerical results will be achieved by using refined versions of the codes developed and used to numerically study the logistic map and the Lorenz system. New code will be developed to study the infinite dimensional systems corresponding to the semilinear parabolic PDEs mentioned in the previous point, coming from several important models of chemical reaction-diffusion phenomena.<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.
