Collective Behavior In Frequency Weighted Coupled Oscillators

Synchronization refers to achieving the same macroscopic rhythm through co-operation and competition in a system consisting of large amount of elements.Such phenomena extensively exist in nature and human society.Theoretically,Kuramoto model is the prototype to study synchronization.The investigations over the last four decades have shown that in most cases the synchronization transitions in Kuramoto-type systems are continuous(second-order),and the formed coherent states are sta-tionary.Recently,explosive synchronization(first-order transition)has become a hot topic in the frontiers of nonlinear dynamics and complex systems.In this thesis,through numerical simulations and theoretical analysis,we system-atically investigated two models:the frequency-weighted Kuramoto model and the frequency-weighted Stuart-Landau oscillators.The main results including:1.We found that the synchronization in the frequency-weighted Kuramoto model with bimodal distribution could convert from the first-order to the second-order,and we analytically solved the critical coupling strength for the forward synchronization transition;2.We investigated the synchronization with the increase of the coupling strength in the frequency-weighted Kuramoto model with uniform distribution,and analytically solved the critical coupling strength for the forward synchronization transition;3.We first identified a quantized,multi-clustered,and time-dependent high order coherent state-Bellerophon state.In such a state,the oscillators in the system form multiple discrete coherent clusters.Inside each cluster,the instantaneous phases and instantaneous frequencies of oscillators are not locked,but their average frequencies lock to a constant so that to form a staircase structure for the average frequencies of different coherent clusters;4.We also observed Bellerophon state in the frequency-weighted Kuramoto mod-el with biased Lorentzian distribution,as well as in the Kuramoto model with both conformists and contrarians;5.We studied the explosive oscillation death in the frequency-weighted Stuart-Landau oscillators,characterized three different scenarios;6.We analytically solved the critical coupling strength for the backward synchro-nization transition in the frequency-weighted Stuart-Landau oscillators,showing it is independent of the specific frequency distribution.The above results suggest that the high order coherent state similar to Bellerophon may generally exist in coupled oscillators,and explosive transition could also occur in coupled oscillators with amplitude dynamics.Such insights not only deepen our understanding of the collective behaviors in coupled oscillators,but also pave a way for further studies on explosive synchronization in real networks.