Model Reduction for the Kuramoto-Sakaguchi Model: analyzing the effect of non-entrained rogue oscillators

Abstract

The Kuramoto-Sakaguchi model is a paradigmatic model of coupled oscillator system which displays collective behaviour. This thesis is concerned with better understanding of the model through construction of lowerdimensional reduced models that are more tractable for analysis. The role of non-entrained rogue oscillators on the synchronized oscillators is highlighted. After reviewing traditional analysis via mean-field theory in the thermodynamic limit of infinitely many oscillators, we proceed to construct reduced models for finite-size systems, where we investigate on how the effects of rogue oscillators should be incorporated. We first describe the rogue oscillators’ effect via averaging, leading to a closed deterministic system that involves the synchronized oscillators only. We perform model reduction analysis on the system via the collective coordinate framework. It is demonstrated that inclusion of the effect of rogue oscillators is crucial for obtaining an accurate description of the system. A new non-linear ansatz is introduced which significantly improves the accuracy of the reduced system, both for finite-size systems and in the thermodynamic limit. We then analyze the fluctuation of rogue oscillator’s effect around their mean, by constructing stochastic process approximations. It is demonstrated that utilizing an Ornstein- Uhlenbeck process leads to stochastic reduced model that can capture the fluctuations exhibited in the full model. This thesis also adds to the mean-field theory analysis for Kuramoto-like model by performing meanfield analysis on the Kuramoto-Sakaguchi model with uniform intrinsic frequency distribution, which reviews that for a non-zero phase-offset parameter, the system exhibits an intricate transition to synchronization, with first-order transition to partial synchronization followed by a second-order transition to global synchronization.