【作者】 李娜；

【导师】 刘辉昭；

【作者基本信息】 河北工业大学 ， 数学， 2019， 硕士

【摘要】 本文主要考虑耦合相位振子系统的动力学行为中的同步问题.第一部分,考虑一类具有剪切项的耦合相位振子系统由非同步态向同步态转变的问题,可以看到非均匀的剪切项对耦合相位振子系统同步行为有一定的影响.首先利用改进的Kuramoto模型,通过OA流形约化方法得到耦合相位振子系统的约化方程,其次利用复序参量以及微分方程的稳定性理论,得到剪切项强度存在标准差阈值,当剪切强度标准差超过该阈值时,该系统在任意耦合强度下不可能同步,最后利用MATLAB作出约化方程对应的波形图,相图以及复序参量关于时间变化的图像以此来进一步验证所得结果.第二部分,考虑在具有剪切项的耦合相位振子系统中加入起搏器的情况,研究该系统的同步行为以及同步速率问题.分别讨论了在振子的固有频率与起搏器的固有频率相同和不全相同两种情况下,利用梯度系统以及(?)ojasiewicz不等式理论,得到了要达到相位同步或者相位锁定状态的一些充分性条件,其中包括了耦合相位振子系统中振子初始相位,耦合强度,剪切强度以及起搏强度所满足的条件,最后证明了耦合相位振子系统能以指数的速率达到同步状态.更多还原

【Abstract】 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.更多还原