Applied Mathematics, University of Waterloo
Clustering Behavior in Networks with Time Delayed, Global, Circulant Coupling
Networks of coupled oscillators arise in a variety of areas. Clustering is a type of oscillatory network behavior where elements of a network segregate into groups. Elements within a group oscillate synchronously, while elements in different groups oscillate with a fixed phase difference. In this thesis, we study networks of N identical oscillators with time delayed, global circulant coupling with two approaches.
We first use the theory of weakly coupled oscillators to reduce the system
of delay differential equations to a phase model where the time delay
enters as a phase shift. We use the phase model to determine
model independent existence and stability results for symmetric
cluster solutions. We show that the presence of the time delay can lead to the coexistence of multiple stable clustering solutions.
We then perform stability and bifurcation analysis to the original system of delay differential equations with symmetry. We first study the existence of Hopf bifurcations induced by coupling time delay, and then use symmetric Hopf bifurcation theory to determine how these bifurcations lead to different patterns of symmetric cluster oscillations. We apply our results to two specific examples: a network of FitzHugh-Nagumo neurons with diffusive coupling and a network of Morris-Lecar neurons with synaptic coupling. In the case studies, we show how time delays in the coupling between neurons can give rise to switching between different stable cluster solutions, coexistence of multiple stable cluster solutions and solutions with multiple frequencies.