Structure preserving schemes for the continuum Kuramoto model: phase transitions

The construction of numerical schemes for the Kuramoto model is challenging due to the structural properties of the system which are essential in order to capture the correct physical behavior, like the description of stationary states and phase transitions. Additional difficulties are represented by the high dimensionality of the problem in presence of multiple frequencies. In this paper, we develop numerical methods which are capable to preserve these structural properties of the Kuramoto equation in the presence of diffusion and to solve efficiently the multiple frequencies case. The novel schemes are then used to numerically investigate the phase transitions in the case of identical and non identical oscillators

Emergent dynamics of the Kuramoto ensemble under the effect of inertia

We study the emergent collective behaviors for an ensemble of identical Kuramoto oscillators under the effect of inertia. In the absence of inertial effects, it is well known that the generic initial Kuramoto ensemble relaxes to the phase-locked states asymptotically (emergence of complete synchronization) in a large coupling regime. Similarly, even for the presence of inertial effects, similar collective behaviors are observed numerically for generic initial configurations in a large coupling strength regime. However, this phenomenon has not been verified analytically in full generality yet, although there are several partial results in some restricted set of initial configurations. In this paper, we present several improved complete synchronization estimates for the Kuramoto ensemble with inertia in two frameworks for a finite system. Our improved frameworks describe the emergence of phase-locked states and its structure. Additionally, we show that as the number of oscillators tends to infinity, the Kuramoto ensemble with infinite size can be approximated by the corresponding kinetic mean-field model uniformly in time. Moreover, we also establish the global existence of measure-valued solutions for the Kuramoto equation and its large-time asymptotics

Structure preserving schemes for the continuum Kuramoto model: Phase transitions

The construction of numerical schemes for the Kuramoto model is challenging due to the structural properties of the system which are essential in order to capture the correct physical behavior, like the description of stationary states and phase transitions. Additional difficulties are represented by the high dimensionality of the problem in presence of multiple frequencies. In this paper, we develop numerical methods which are capable to preserve these structural properties of the Kuramoto equation in the presence of diffusion and to solve efficiently the multiple frequencies case. The novel schemes are then used to numerically investigate the phase transitions in the case of identical and nonidentical oscillators

The Kuramoto model: A simple paradigm for synchronization phenomena

Synchronization phenomena in large populations of interacting elements are the subject of intense research efforts in physical, biological, chemical, and social systems. A successful approach to the problem of synchronization consists of modeling each member of the population as a phase oscillator. In this review, synchronization is analyzed in one of the most representative models of coupled phase oscillators, the Kuramoto model. A rigorous mathematical treatment, specific numerical methods, and many variations and extensions of the original model that have appeared in the last few years are presented. Relevant applications of the model in different contexts are also included

쿠라모토 진동자들의 동역학에서 일어나는 문제들에 대한 고찰

학위논문 (박사)-- 서울대학교 대학원 : 수리과학부, 2014. 2. 하승열.In this thesis, we study several problems on the ensemble of Kuramoto oscillators. We present the nonlinear stability of the phase-locked states using a robust $\ell_1$-metric as a Lyapunov functional. We show that the phase-locked states are congruent each other in the sense that one phase-locked state is the simply translation of the other and phase-shift is the difference of averaged initial phases. We also show the contraction property for measure valued solutions of the kinetic Kuramoto model. We next consider the effect of interaction frustration on the complete synchronization of Kuramoto oscillators. In general, interaction frustration hinders the formation of complete frequency synchronization. For more quantitative estimates, we considered three Kuramoto-type models. Our first model is for an ensemble of Kuramoto oscillators with uniform interaction frustration. Our second model is, as a special case of the first model, a mixture of two identical Kuramoto oscillator groups with distinct natural frequencies. Our third model is like the Kuramoto model for identical oscillators on the bipartite graph. Finally, we investigate the intricate interplay between the inertia and frustration in an ensemble of Kuramoto oscillators. We cannot apply the explicit macro-micro decomposition to reduce the dynamics of initial phases to that of fluctuations. However, we can still derive second-order differential inequalities for the phase or frequency diameters so that the second-order Gronwall inequality method still works well. Moreover, both the analytical and numerical studies demonstrate this fact.Abstract 1 Introduction 2 Preliminaries 2.1 The Kuramoto model 2.2 The kinetic Kuramoto equation 3 Nonlinear stability 3.1 Orbital stability of phase-locked states 3.2 Stability estimate of the kinetic Kuramoto equation 3.2.1 Alternative formulation of the KKE 3.2.2 Strict contractivity in the Wasserstein distance 4 Kuramoto type models with frustration 4.1 Kuramoto model with frustration 4.2 Synchronization estimate for Model A 4.2.1 Existence of a trapping region 4.2.2 Entrance to the exponential stability regime 4.2.3 Relaxation estimate 4.3 Synchronization estimates for Model B 4.3.1 Existence of a trapping region 4.3.2 From mixed stage to segregated stage 4.3.3 Formation of the two-point cluster configuration 4.4 Synchronization estimates for Model C 4.4.1 Existence of a trapping region 4.4.2 Relaxation estimate 4.5 Numerical examples 4.5.1 Model A 4.5.2 Model B 4.5.3 Model C 5 Kuramoto model with inertia and frustration 5.1 The Kuramoto model with inertia 5.2 Synchronization estimate: identical oscillators 5.2.1 Notations 5.2.2 Complete synchronization 5.3 Synchronization estimate: non-identical oscillators 5.3.1 A small inertia regime 5.3.2 A large inertia regime 5.4 Numerical simulations 5.4.1 Identical oscillators 5.4.2 Nonidentical oscillators 6 Conclusion and future works 6.1 Conclusion 6.2 Future works Abstract (in Korean) Acknowledgement (in Korean)Docto

The mathematics behind chimera states

Chimera states are self-organized spatiotemporal patterns of coexisting coherence and incoherence. We give an overview of the main mathematical methods used in studies of chimera states, focusing on chimera states in spatially extended coupled oscillator systems. We discuss the continuum limit approach to these states, Ott--Antonsen manifold reduction, finite size chimera states, control of chimera states and the influence of system design on the type of chimera state that is observed