The Dynamics In The System Of Nonlocally Coupled Phase Oscillators

A complex system composed of a large number of dynamic individuals has attracted many scholars to conduct research.From molecular biology to neuroscience,condensed-matter physics and the Internet,researchers are unravelling the structure of complex networks,studying how they evolve and function,and investigating the collective behavior they exhibite.The Kuramoto model is a classic model for studying complex systems.Since proposed by Kuramoto in 1975,the Kuramoto model has been widely used in the research fields,including physics,chemistry and biology.In the research works related to the Kuramoto model,by changing the coupling between the oscillators,adopting different natural frequency distributions,and introducing phase delay or time delay in the coupling term rich synchronization phenomena have been observed,such as fully synchronized state,chaotic phenomenon,chimera state,and so on.In this dissertation,we mainly study the effects of different natural frequency distributions on the dynamics processes in non-locally coupled phase oscillators.Firstly,we study a one-dimensional ring of phase oscillators with nonlocal coupling and a bimodal natural frequency distribution.As we have known,for a large system of phase oscillators whose natural frequencies are bimodal,the long-term dynamics evolves to one of three states:incoherence,partial synchrony,and a standing wave state when global coupling is considered.However,for non-local coupling phase oscillators,it is still an open question that how the biomodal frequency distribution effects the dynamics.Therefore,we study a one-dimensional ring of phase oscillators with nonlocal coupling and a bimodal natural frequency distribution.Our results show that,twisted standing waves and stationary twisted states appear successively with the increase of the coupling strength.In the continuum limit,we derive a low dimensional reduced equation using the Ott-Antonsen ansatz,which verifies the twisted states in the simulations of finite networks of oscillators.We also theoretically investigate the stationary twisted states and their stabilities by using the reduced equation.The theoretical results are consistent with the numerical simulations.Secondly,considering that the peak distance in the bimodal frequency distribution is narrow,the discontinuous transitions between different dynamical states are possible,we study a ring of non-locally coupled phase oscillators with the frequency distribution made up of two Lorentzians.Using the Ott-Antonsen ansatz,we derive a reduced model in the continuum limit.Based on the reduced model,we analyze the stability of the incoherent state and find the existence of multiple stability islands for the incoherent state depending on the parameters.Furthermore,we numerically simulate the reduced model and find a large number of twisted states resulting from the instabilities of the incoherent state with respect to different spatial modes.For some winding numbers,the stability region of the corresponding twisted state consists of two disjoint parameter regions,one for the intermediate coupling strength and the other for the strong coupling strength.Since the non-locally coupled phase oscillators is sensitive to initial conditions,the competition between different spatial modes leads to model evolution a more complex space-time model.We introduced several different types of space-time model.Finally,we summarize the whole work and list our future work.