December 01, 2023, ESB 2001

A common way to study coupled oscillators is to reduce them to a system of phase oscillators, where each oscillator is reduced to a single variable. In this talk, I will describe some recent approaches toward expanding phase oscillators to include the effect of motion that deviates from the limit cycle. I will use the normal form for the Hopf bifurcation as a case study to show (1) how noise affects the expected frequency of am oscillator only when amplitude is included; (2) amplitude uncovers some bifurcations in coupled phase oscillators when these higher order terms are included; (3) amplitude allows one to extend the spatial domains that allow for stable rotating waves. Most of what I describe in this example will hold for general oscillators using the ideas of isostable reduction.

Bard Ermentrout received his BA and MA (mathematics) at Johns Hopkins University and his PhD in biophysics & theoretical biology at the University of Chicago in 1979. He is the author of over 200 papers in math, biology, physics, and neuroscience, including XPPAUT software for the simulation and analysis of dynamical systems, and the books, Simulating, Analyzing, and Animating Dynamical Systems (2002) and, with David Terman, Mathematical Foundations of Neuroscience (2010). Professor Ermentrout is a Sloan Fellow and a SIAM Fellow, and received the Mathematical Neuroscience Prize in 2015.