Dynamical systems and irregular gradient flows

Basic data for this project

Type of project: Own resources project
Duration: 01/10/2021 - 31/08/2025

Description

The central goal of this project is to study asymptotic properties for gradient flows (GFs) and related dynamical systems. In particular, we intend to establish a priori bounds and related regularity properties for solutions of GFs, we intend to study the behaviour of GFs near unstable critical regions, we intend to derive lower and upper bounds for attracting regions, and we intend to establish convergence speeds towards global attrators. Special attention will be given to GFs with irregularities (discontinuities) in the gradient and for such GFs we also intend to reveal sufficient conditions for existence, uniqueness, and flow properties in dependence of the given potential. We intend to accomplish the above goals by extending techniques and concepts from differential geometry to describe and study attracting and critical regions, by using tools from convex analysis such as subdifferentials and other generalized derivatives, as well as by employing concepts from real algebraic geometry to describe domains of attraction. In particular, we intend to generalize the center-stable manifold theorem from the theory of dynamical systems to the considered non-smooth setting. Beside finite dimensional GFs, we also study GFs in their associated infinite dimensional limits. The considered irregular GFs and related dynamical systems naturally arise, for example, in the context of molecular dynamics (to model the configuration of atoms along temporal evoluation) and machine learning (to model the training process of artificial neural networks).

Keywords: gradient flow; dynamical system; differential equation; a priori bounds; domain of attraction; critical points; center-stable manifold theorem; existence; uniqueness