Particle and kinetic descriptions of Kuramoto ensemble in the presence of adaptive couplings and noises

Abstract

In this thesis, we investigate the particle and kinetic models of the Kuramoto-type model with adaptive coupling strengths and noise environment. In the classical Kuramoto model, the pairwise coupling strengths are assumed to be uniform and the oscillators are always supposed to be deterministic. For more realistic description of many-body systems in the real world, however, the need for improving the classical model arises. In this regard, we first introduce the Kuramoto-type models where the coupling strengths among the oscillators depend on the pairs and time. Then, we consider the behaviour of the Kuramoto oscillator ensemble when the number of oscillators goes to infinity($N \to \infty$) in the presence of phase lag and Gaussian noise. Finally, we study the swarming model on a unit sphere that can be regarded as a variation of the classical Kuramoto model.