The Study On Chimera Dynamic Behavior Of Coupled Oscillator Systems

The collective dynamic behavior of coupled oscillator systems can simulate many group behaviors in nature,so it has received extensive attention from researchers.The synchronization is a common dynamic state in collective dynamic behavior,which can be divided into fully synchronized states and partially synchronized states.Chimera state is a special kind of partially synchronized states,which refers to the coexistence of coherent and incoherent oscillations in dynamic systems.This asymmetry phenomenon is also common in many real systems,such as the phenomenon of semi brain sleep in biology.Therefore,the study of chimera states has important practical significance.The initial research results show that the chimera state only shows the coexistence of coherent and incoherent and the discontinuity in space,and requires very strict initial conditions to produce.With the further research,the concept of chimera state extends to symmetry breaking,that is,the solution of the system does not have commutative symmetry.After the research on the definition of chimera states is basically complete,the different types of chimera states and the interaction between chimera states in different systems have become a new research focus in this field.In this thesis,we mainly studied the chimera state in the nonlocally coupled Brusselators system in the one-dimensional ring network and the nonlocally coupled phase oscillators system in the two-layer network.We found and explored the mechanism behind the two-frequency chimera state phenomenon in the nonlocally coupled Brusselators system,and further explained and verified the relationship between the multi-cluster chimera state and the coupling radius.Finally,we have explained the interaction between chimera states in the two-layer network.First,we investigate the two-frequency chimera state of nonlocally coupled Brusselators in a one-dimensional ring network,that is,the adjacent coherent domains in the chimera state have different average phase velocities.The Brusselators in the coherent domain with higher average phase velocity are roughly in phase,while the Brusselators in the coherent domain with lower average phase velocity are divided into two groups of antiphase.The coherent Brusselators actually perform quasi-periodic dynamics.The power spectrum results of the local mean field show that the dynamics of the local mean field in different types of coherent domains are dominated by different periodic components,which is the reason for the two different frequencies of the coherent Brusselators oscillation.We further give other solutions in the phase diagram of the two-frequency chimera state in the stability region of different parameter planes and in the larger parameter range outside the stability region.In the case of expanding the system size and extending the system evolution time,the twofrequency chimera state still exists,indicating that the two-frequency chimera state is not affected by the population size.Next,we study the nonlocally coupled Brusselators in the one-dimensional ring network,and find that the number of coherent clusters in the multi-cluster chimera state is independent of the coupling radius.There are two different kinds of chimera states when they are far from global coupling and close to global coupling.In the case of near global coupling,the proportion of oscillators in the coherent region of the chimera state is larger,and the oscillators in the coherent region are completely in phase,while in the case of far from global coupling,the proportion of oscillators in the coherent region of the chimera state is smaller than that in the incoherent region,and the oscillators in the coherent region are approximately in phase.Through linear stability analysis,we prove that the multistability of multi-clusters of chimera states is related to the traveling wave in the nonlocally coupled Brusselators when far from the global coupling.When it is close to the global coupling,the multi-stability of the chimera state is derived from the periodic bicluster dynamics in the globally coupled Brusselators.In addition to the chimera state in a one-dimensional ring network,we also studied the synchronization relationship of the chimera state in a nonlocal coupled phase oscillator in a two-layer network,we define five different types of typical chimera states according to the relative position and phase difference of the synchronization region and give their phase diagrams in the parameter plane.For the synchronous state between the two chimera states,we further analyze it by introducing the change of sub-Lyapunov exponent with the strength of interlayer coupling.In heterogeneous networks,due to the introduction of heterogeneity,it is not possible to define an obvious typical chimera state.Further increase the value of interlayer coupling strength,we find that there is a critical value of coupling strength,which makes the system enter the frequency synchronization state after exceeding this critical value.By further simplifying the model,we confirm the existence of this critical value.By further increasing the coupling strength,the system will be stable in two types of chimera states.Heterogeneity leads to a stronger dependence of the typical chimera state on the phase lag and interlayer coupling strength,and under the influence of heterogeneity,the phase synchronization between the two groups of phase oscillators can not be achieved.