AIMR Math Group Seminar 2024/10/4
Date : October. 4th (Mon) 2024 16:00-17:30
Venue: AIMR, 2F Seminar room / Zoom
Speaker: Dr. Tomoharu Suda (Tokyo University of Science)
Title: Axiomatic theory of ordinary differential equations and its application to dynamical systems
Abstract:
Most of the continuous-time dynamical systems that appear in applications are generated by well-posed systems of ordinary differential equations. However, there are also important problems, such as the motion of objects with dry friction, where the uniqueness of the solution does not hold. Therefore, we would like to consider these as generalized dynamical systems and construct a theory for them. One approach to generalization of dynamical systems theory to such systems is an axiomatic one. In other words, instead of analytically considering specific differential equation problems, we formulate useful properties for developing dynamical systems theory as axioms and develop the theory synthetically.
In this talk, we generalize the classical theory of dynamical systems based on the axiomatic theory of ordinary differential equations proposed by Yorke. First, we introduce an overview of Yorke’s theory and then proceed to generalize topological equivalence. We will also introduce concepts such as invariant sets, limit sets, and invariant measures, and demonstrate how classical results in dynamical systems theory, such as the Poincaré recurrence theorem, can be generalized. The material presented here is primarily based on the following papers.
T. Suda, “Equivalence of topological dynamics without well-posedness “, Topology and its Applications 312, 108045, 2022.
T. Suda, “Dynamical properties in the axiomatic theory of ordinary differential equations “, Journal of Differential Equations 391, 1-24, 2024.
*Register here
*The seminar is co-hosted by MathCCS Seminar.
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