Date of Award

8-11-2020

Degree Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics and Statistics

First Advisor

Igor Belykh

Second Advisor

Vladimir Bondarenko

Third Advisor

Yaroslav Molkov

Fourth Advisor

Michael Stewart

Abstract

Modeling cooperative dynamics using networks of phase oscillators is a common practice for a wide spectrum of biological networks. Patterns of synchronized clusters are some of the most prevalent instances of such cooperative behavior, manifesting themselves in ways similar to groups of neurons firing together during epileptic seizures or Parkinson's tremors. Despite significant interest, the emergence and hysteretic transitions between stable clusters in oscillator networks have still not been fully understood. In particular, the celebrated Kuramoto model of phase oscillators is known to exhibit multiple spatio-temporal patterns, including co-existing clusters of synchrony and chimera states in which some oscillators form a synchronous cluster, while the others oscillate asynchronously. Rigorous analysis of the stability of clusters and chimeras in the finite-size Kuramoto model has proven to be challenging, and most existing results are numerical. In this thesis, we contribute toward the rigorous understanding of the emergence of stable clusters in networks of identical Kuramoto oscillators with inertia. We first study the co-existence of stable patterns of synchrony in two coupled populations of identical Kuramoto oscillators with inertia. The two populations have different sizes and can split into two clusters where the oscillators synchronize within a cluster while there is a phase shift between the dynamics of the two clusters. Due to the presence of inertia, which increases the dimensionality of the oscillator dynamics, this phase shift can oscillate, inducing a breathing cluster pattern. We derive analytical conditions for the co-existence of stable two-cluster patterns with constant and oscillating phase shifts. We then study the emergence of stable clusters of synchrony with complex inter-cluster dynamics in a three-population network of identical Kuramoto oscillators with inertia. We extend the results of the bistability of synchronized clusters in the two-population network and demonstrate that the addition of a third population can induce chaotic inter-cluster dynamics. This effect can be captured by the old adage "two is company, three is a crowd'' which suggests that the delicate dynamics of a romantic relationship may be destabilized by the addition of a third party, leading to chaos. Through rigorous analysis and numerics, we demonstrate that the inter-cluster phase shifts can stably co-exist and exhibit different forms of chaotic behavior, including oscillatory, rotatory, and mixed-mode oscillations. We also discuss the implications of our results for predicting the emergence of chimeras and solitary states in real-world biological networks.

DOI

https://doi.org/10.57709/18665743

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