Description:

Title: Analysis of Limit Cycle Oscillators Far Beyond the Weakly Perturbed Paradigm

Speaker: Dan Wilson, PhD, University of Tennessee 

Special Note: This online colloquium will be given at  https://northwestern.zoom.us/j/96768935311. The passcode is 795104.

Abstract:

Self-sustaining oscillatory behaviors are widely observed in the physical, chemical, and biological sciences. Art Winfree made great strides in characterizing the perturbed behavior of nonlinear limit cycles by representing these oscillatory dynamics in terms of an asymptotic phase. This so-called phase reduction has been used extensively in recent decades to successfully and elegantly characterize complicated patterns that emerge in groups of weakly interacting oscillators.

While standard phase reduction is useful in many situations, its applicability degrades as coupling strength increases often resulting in incorrect predictions about dynamical behavior. Currently, very few general reduction techniques exist that can be used to analyze oscillatory dynamics in response to arbitrary, large magnitude coupling and other large magnitude inputs.

In this presentation, I will discuss two recently developed reduced order modeling frameworks that can be used to understand the aggregate behaviors of coupled oscillators in regimes that extend far beyond the weakly perturbed paradigm. Both frameworks leverage the properties of isostable coordinates, which characterize level sets of the slowest decaying eigenmodes of the Koopman operator. Numerical illustrations show that the proposed methods accurately reflect synchronization and entrainment dynamics of coupled oscillators in situations where several other phase-amplitude reduction strategies fail. Applications related to neurological function and circadian physiology will be highlighted.

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