High Order Collective Behaviors In Coupled Oscillators

Synchronization refers to the self-organized coherence in a system consisting of a large amount of interacting units.Such phenomena actually are ubiquitous in nature and human society.Theoretical methods of the synchronization transition have been developed in the past several decades.Among them,the Kuramoto model is the most classical.One central issue in the study of synchronization is the coherent state.So far,various coherent states have been identified in coupled oscillator systems,such as the synchronized state,the π state,the travelling wave state,the Chimera state,and the Bellerophon state(B state),etc.In particular,the B state reported recently was first observed in the frequency-weighted Kuramoto model with bimodal distribution,then in many other coupled phase oscillator systems.In such a state,oscillators form quantized,time-dependent coherent clusters,where their instantaneous frequencies are not locked,but the averaged frequency is locked.In this thesis,by both theoretical analysis and numerical simulations,the synchronization paths and the B states in the classical Kuramoto model with asymmetric bimodal frequency distribution,the Stuart–Landau oscillators with frequency-weighted coupling,and the non-globally coupled Kuramoto oscillators are investigated.The main results included as following:1.We studied the synchronization in classical Kuramoto model with asymmetric bimodal Lorentzian distributions.It is found that the phase transition paths and characteristics are related to the frequency distribution parameters.Also,we investigated the synchronization in the reduced Kuramoto system obtained by the Ott-Antonsen method,and the results are qualitatively consistent with the direct numerical simulations.Besides,we observed two types of travelling wave states and an asymmetric B state in the studied system,which have been fully characterized in detail.2.We studied the phase transition process of the frequency-weighted StuartLandau oscillator system,and found explosive synchronization and continuous transitions.The B state is also observed,indicating that such a state also exists in coupled oscillators with amplitude dynamics.In addition,we revealed novel transition behavior for coupled Stuart-Landau oscillators.It is shown that the regime of B state actually contains two stages.In the first stage the oscillators achieve chaotic phase synchronization while in the second stage oscillators form normal phase synchronization.3.We further observed B state in the non-globally coupled Kuramoto model,thus extended the study of such a high-order collective behavior to coupled oscillators in complex networks.Previously,the B state,as a high-order coherence,has been observed only in coupled phase oscillators.In this work,we further extended such study into the model of coupled Stuart-Landau oscillators which have amplitude dynamics,and the cases of asymmetric frequency distribution and non-globally coupled networks.The results suggest that the high-order coherent states similar to B state might generally exist in coupled oscillator systems.These investigations certainly provide us more understandings and insights of the collective behaviors in coupled oscillator systems.Due to the complexity of real systems,the high-order coherent states actually are ubiquitous.Therefore,it is very important and realistic to study such states in coupled oscillator systems.