Collective Dynamics Behaviors Of Coupled Oscillator Systems

Coupled oscillators are widely used to study the collective behaviors of biological groups in nature.The synchronized states and chimera states are the typical collective behavior states.Chimera states in a oscillators system are the dynamic states that break the inherent symmetry of a system and are generally generated by non-locally coupling with special initial conditions.The chimera state is also part of the partially synchronized state.In subsequent studies,the chimera states were extended to show that symmetry breaking states can occur spontaneously in systems with symmetrically coupled oscillators in two indistinguishable sub-populations.Synchronized state is a symmetrical state,which is different to chimera state.In addition,incoherent state is often encountered and discussed in the study of synchronized state.In addition,there is also a special chimera state in 2-dimensional system,which is called spiral wave chimera state.The work of this paper focuses on the synchronized states and chimera states from many perspectives in different systems,such as different theoretical models,different coupling modes and ranges,different dimensional systems,different system parameters and so on.This paper strongly promotes the research of coupled systems,and finds some states in new systems,such as anti-phase chimera states,spiral wave states and so on.The contents of this paper are as follows:(1)The synchronized states in globally coupled Sakaguchi-Kuramoto system with bimodal natural frequency distribution.We use OA ansatz to simplify the global coupled phase oscillator into a low-dimensional coupled ordinary differential equation.The influence of different parameters on system dynamics is studied comprehensively.For symmetric bimodal frequency distribution,we analyze the linear stability of incoherent state and partially synchronized state.And different types of bifurcation between different dynamic states are identified.In addition,we also study the revived phenomenon of incoherent states with asymmetric bimodal frequency distribution,and it is found that the superposition of two natural frequencies with large difference in half widths determines the occurrence of this phenomenon.(2)Chimera states and synchronized states in a symmetric system composed of two interacting sub-populations of phase oscillators.We found three symmetry preserving states:incoherent state,in-phase synchronized state and anti-phase synchronized state,and three symmetry breaking states:in-phase chimera state,anti-chimera state and weak chimera state.We investigate the stability of these dynamical states in different parameter planes and the transformation between them.We find that the weak chimera state is a bridge between the in-phase and anti-phase chimera states.We also observe the existence of periodic two chimera state,chaotic chimera state and drift chimera state.(3)The relationship between chimera state and synchronized state in an asymmetric system consisting of two sub-populations of phase oscillators with different frequency distributions.We study the stability diagrams of chimera states and synchronized states on different parameter planes.Interestingly,by changing the parameters,chimera states will meet and evolve into synchronized states.The same state can be changed into the synchronized state or chimera state on different parameter paths.Asymmetry can transform pitchfork bifurcations into saddle-node bifurcations in symmetric systems.Saddle-node bifurcation blurs the difference between chimera state and synchronized state.(4)Spiral wave chimera states in a 2-dimensional non-locally coupled FitZhugh-Nagumo system.The rich dynamics properties of spiral wave chimera states are found numerically by means of numerical simulation.In addition to the outward and inward propagating single spiral wave chimera states,we also find the double spiral wave chimera state where two spiral wave chimera states rotate with each other and the breakup of spiral wave chimera states.We also find a transition between an ordinary spiral wave with a phase singularity at its center and a spiral wave chimera state.In conclusion,the collective dynamics behaviors of the coupled oscillators systems have been studied comprehensively,including synchronized state,chimera state and incoherent state.More importantly,our study deals with the phase transition problem and bifurcation details,which are fundamental concerns in the field of nonlinear dynamics.