날짜: 10월 28일 화요일

시각: 오후 5시

발표자: 윤욱

발표 제목: On the asymptotic patterns of Kuramoto type models

초록: The Kuramoto model describes the synchronization of coupled oscillators and it can be cast as a gradient flow on the Euclidean space. In this thesis, we study various Kuramoto-type dynamics. First, we study the asymptotic relaxation of Kuramoto oscillators in an analytic potential field. As nontrivial local potential fields, we consider two types: quadratic and linear-plus-oscillatory potentials. For each type of potential field, we provide sufficient frameworks ensuring the uniform boundedness of states, and then we use the gradient flow formulation to establish the existence of equilibrium and convergence toward them. In addition, we study a continuous transition from the discrete infinite Kuramoto model to the continuous counterpart in the whole time interval. The discrete infinite Kuramoto model corresponds to the discretization of the infinite Kuramoto model via the first-order Euler scheme. For this discrete model, we analyze the emergent behaviors and uniform stability with respect to initial data under a suitable framework. In  particular, for a homogeneous ensemble, we identify sufficient conditions for the existence of quasi-stationary state. For the continuous transition in a zero time-step limit, we provide an improved truncation error estimate by using the uniform stability and emergent dynamics. Moreover, we study the continuum Kuramoto model on the whole Euclidean space which corresponds to an integro-differential equation. Compared with the model on a finite lattice in a bounded region, we use the infinite Kuramoto lattice model to construct a sequence of approximate solutions. We provide a suitable set of conditions on the interaction kernel and initial data, depending on the natural frequency function, that ensure the emergent dynamics of this continuum Kuramoto model.