The course introduces abstract mathematical concepts and techniques that are crucial to the fundamental understanding of the behaviour of deterministic dynamical systems in metric spaces in general, finite dimensional Euclidean space and infinite dimensional Banach spaces in particular. This includes systems of nonlinear ordinary differential equations (ODEs), some types of partial differential equations (PDEs) and Volterra-type integral equations, from which we draw various examples, linked to applications. The course intends to provide a thorough background in abstract analysis for students in mathematics interested in continuing with a more detailed study in analysis of continuous-time dynamical systems defined by nonlinear evolutionary equations.
Some of the topics that will be addressed are:
(1) Preliminaries on analysis in metric spaces (Lipschitz mappings, measures of non-compactness, Hausdorff measure);
(2) Dynamical systems in metric spaces (semigroups of transformations, limit sets, attractors, map iterations and some fixed point theorems, existence of solutions to particular type of equations).
(3) Nonlinear operators (compact maps, inner- and outer superposition operators);
Assignments and oral exam
The course will be based on parts of several books. The reading material will be a combination of handouts and downloadable course notes. Participants need not purchase a book for the course.
Knowledge of the basics of Banach space theory is beneficial (e.g. Linear Analysis or another introductory course in Functional Analysis). Most concepts will be developed in self-contained fashion during the course however.
2 per week