Electronic Thesis and Dissertation Repository

Thesis Format

Integrated Article


Master of Science




Muller, Lyle.

2nd Supervisor

Minac, Jan.



The Kuramoto Model (KM) is a nonlinear model widely used to model synchrony in a network of oscillators – from the synchrony of the flashing fireflies to the hand clapping in an auditorium. Recently, a modification of the KM (complex-valued KM) was introduced with an analytical solution expressed in terms of a matrix exponential, and consequentially, its eigensystem. Remarkably, the analytical KM and the original KM bear significant similarities, even with phase lag introduced, despite being determined by distinct systems. We found that this approach gives a geometric perspective of synchronization phenomena in terms of complex eigenmodes, which in turn offers a unified geometry for synchrony, chimera states, and waves in nonlinear oscillator networks. These insights are presented in Chapter 2 of this thesis. This surprising connection between the eigenspectrum of the adjacency matrix of a ring graph and its Kuramoto dynamics invites the question: what is the eigenspectrum of joins of circulant matrices? We answered this question in Chapter 3 of this thesis.

Summary for Lay Audience

The story of the synchronous fireflies popularized the study of synchrony. Fireflies flash in disorder at first and slowly synchronize with each other to light up the forest in which they reside. The fireflies are examples of oscillators, with some way to communicate with each other, described via an adjacency matrix. Each oscillator possesses a phase, which indicates the state of the oscillator over time. The central mathematical model used to study synchrony is called the Kuramoto Model (KM), which is determined by simple, elegant, and nonlinear defining equations. The KM allows for the numerical investigation into synchrony, among other sophisticated dynamics such as the partial synchronous dynamics called chimera states. In this thesis, we extend the discussion by introducing a novel analytical and geometric framework to study synchrony, bridging our current understanding of the dynamics that the oscillators exhibit with how the oscillators are connected. With this bridge, or the eigenspectrum of the adjacency matrix, established in Chapter 2, this thesis presents further descriptions of the eigenspectrum when the oscillators are in a particular multilayered formation called the joins of circulant matrices.