Kuramoto模型的同步与渐近行为

发布时间：2019-07-10 12:18

【摘要】：大量弱耦合振子的同步现象在自然系统中十分常见,被物理、生物、化学、经济学、社会学等许多科学领域的学者大量研究。在众多研究该现象的数学模型中,Kuramoto模型最受关注。对于这一模型的研究具有实际应用价值,例如心脏起博、智能电网等。本文主要研究带起搏器的对称连通的Kuramoto模型的同步行为以及随机离散Kuramoto模型的渐近行为。主要研究内容如下:首先,基于?ojasiewicz不等式,证明了带起搏器的对称连通图下Kuramoto模型在振子与起搏器自然频率相同的情况下:当振子与起搏器初始相位差在(-π2π,2)时,系统最终以指数速率达到完全相位同步;当振子与起搏器初始相位差在(-π,π)时,若起搏器对振子的耦合强度与振子间的耦合强度满足一定条件,系统最终以指数速率达到完全相位同步。随后,对于带起搏器的对称连通图下Kuramoto模型在振子与起搏器自然频率不全相同的情况,从初始相位与振子间耦合强度方面,给出了系统最终达到锁相状态的充分条件,并证明了在相应条件下,系统达到锁相状态的收敛速率为指数收敛。最后,基于大数定律给出随机离散Kuramoto模型的连续极限模型,并且应用Lyapunov中心极限定理证明了随机离散Kuramoto模型初值问题的解依概率收敛到连续极限模型初值问题的解。
[Abstract]:The synchronization of a large number of weakly coupled oscillators is very common in natural systems and has been studied by scholars in many scientific fields, such as physics, biology, chemistry, economics, sociology and so on. Among the many mathematical models that study this phenomenon, Kuramoto model has attracted the most attention. The research on this model has practical application value, such as cardiac booms, smart grid and so on. In this paper, the synchronization behavior of symmetric connected Kuramoto model with pacemaker and the asymptotic behavior of stochastic discrete Kuramoto model are studied. The main research contents are as follows: firstly, based on the ojasiewicz inequality, it is proved that the Kuramoto model with pacemaker has the same natural frequency as the pacemaker: when the initial phase difference between the oscillator and the pacemaker is (- 蟺 2 蟺, 2), the system finally achieves complete phase synchronization at an exponential rate; When the initial phase difference between the oscillator and the pacemaker is (- 蟺, 蟺), if the coupling strength of the pacemaker to the oscillator and the coupling strength between the oscillator meet certain conditions, the system finally achieves complete phase synchronization at the exponential rate. Then, for the case that the natural frequency of the oscillator and the pacemaker is not the same under the symmetric connected graph of the pacemaker, the sufficient conditions for the system to finally reach the phase-locked state are given from the aspect of the initial phase and the coupling strength between the pacemaker, and it is proved that under the corresponding conditions, the convergence rate of the system in the phase-locked state is exponential convergence. Finally, based on the law of large numbers, the continuous limit model of stochastic discrete Kuramoto model is given, and by using Lyapunov central limit theorem, it is proved that the solution of initial value problem of stochastic discrete Kuramoto model converges to the solution of initial value problem of continuous limit model according to probability.
【学位授予单位】：哈尔滨工业大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175

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