| Popis: |
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. |
