Dynamics Of Coupled Phase Oscillator System

In this thesis,the synchronization problem in the dynamical behavior of coupled phase oscillator system is considered.In the first part,we consider the transition from the non-synchronous state to the synchronous state for a class of coupled phase oscillators with shear term,and the effect of the non-uniform shear term on the synchronous behavior of the coupled phase-oscillator system is obtained.Firstly,using the improved Kuramoto model,the reduced equation of coupled phase oscillator system is obtained by means of OA manifold reduction method.Secondly,using the complex order parameter and the stability theory of the differential equation,the existence threshold of the shear term is obtained.At last,the corresponding wave diagram,phase diagram and the time-varying images of complex order parameters are given to verify the results.In the second part,considering the addition of pacemakers to the coupled phase oscillator system with shear terms,the synchronization behavior and synchronization rate of the system are studied.By using the gradient system and the (?)ojasiewicz inequality theory,we discuss the two cases where the natural frequency of the oscillator is the same or not same as the natural frequency of the pacemaker.Sufficient conditions for the initial phase,coupling strength,shear strength and pacemaker pacing strength of the coupled phase oscillator system in the phase synchronization or phase locking state can be achieved.Finally,it is proved that the coupled phase oscillator system can achieve synchronization at an exponential rate.