Seminar Mathematics - Dr. Hildeberto Jardón

Title: On the geometric theory of dynamical systems with multiple time scales:
challenges and perspectives.

Abstract:

There are many complex phenomena in nature that display a combination of
slow and fast dynamics.  One example are enzymatic reactions, where
usually enzyme concentrations change at much faster rates than other
substrate concentrations.  Another example is found in neurons, or other
excitable cells, where the action potential has a much faster behaviour
than that of the ion pumps.

In the context of Ordinary Differential Equations (ODEs), models that
capture slow-fast dynamics are known as singularly perturbed ODEs.  One
of the mathematical approaches to analyze slow-fast dynamics is
Geometric Singular Perturbation Theory (GSPT),  which during the past 20
years has seen a rapid increase not only in its theoretical
methodologies, but also in its far-reaching applications.

In this talk I will review the basics of GSPT. Afterwards, by use of an
example involving consensus networks with slow adaptation, I will
illustrate the usefulness of GSPT. Finally, we shall digress on some
open problems on the theory of dynamical systems with multiple time
scales that are motivated by applications.