关于Kuramoto模型同步性的若干研究

发布时间：2018-05-15 02:34

本文选题：Kuramoto模型 + 同步　；参考：《哈尔滨工业大学》2017年硕士论文

【摘要】：同步现象普遍存在于复杂的生物系统中,比如萤火虫、神经元、心脏细胞等。为了能更好地解释生物系统的同步现象,人们试图从物理、生物、化学和社会科学等不同的方面进行研究。Kuramoto是研究同步问题较早的学者并给出了具体的数学模型,即Kuramoto模型(KM模型).在已有的文献中,人们对KM模型作了大量的研究,其中包括模型达到完全相位同步的充分条件、耦合强度对于同步的影响、一定条件下振子的收敛速率等。而对于平衡点的具体刻画以及稳定性分析还没有完备,故本文将会考虑KM模型的平衡点稳定性等相关问题。本文的主要工作如下:首先,对带有阻挫的完全图上的KM模型的稳定相锁解进行了探讨。研究发现,在恒等自然频率条件下,非零阻挫完全图上的KM模型的完全相位同步所对应的相锁状态是渐近稳定的,而振子均匀地放置在单位圆周上所对应的相锁状态是不稳定的。特别地,当振子个数N=2,3时,完全相位同步状态是系统的唯一渐近稳定相锁状态。其次,对环上的双向耦合KM模型进行了研究。分析了阻挫与环上的双向耦合KM模型的平衡点稳定性之间的关系。对零阻挫情形下的所有平衡点进行了描述,并对稳定平衡点和不稳定平衡点作了进一步的刻画。对于稳定的平衡点,估计了它的吸引域,并给出了收敛速率。
[Abstract]:Synchronization is common in complex biological systems, such as fireflies, neurons, heart cells and so on. In order to better explain the synchronization of biological systems, people try to study.Kuramoto from the different aspects of physical, biological, chemical and social sciences. The model, that is, the Kuramoto model (KM model). In the existing literature, people have done a lot of research on the KM model, including the sufficient conditions for the complete phase synchronization of the model, the effect of the coupling strength on synchronization, the convergence rate of the oscillator under certain conditions, and the specific characterization of the balance point and the stability analysis. Therefore, this paper will consider the equilibrium point stability of the KM model and other related problems. The main work of this paper is as follows: first, the stable phase locking of the KM model on the complete graph with frustration is discussed. It is found that, under the constant natural frequency, the phase locking corresponding to the complete phase synchronization of the KM model on the non zero block complete graph is found. The state is asymptotically stable, and the phase locking state corresponding to the oscillator uniformly on the unit circle is unstable. In particular, when the number of oscillators is N=2,3, the complete phase synchronization state is the only asymptotically stable phase lock state of the system. Secondly, the bidirectional coupling KM model on the ring is studied. The two-way coupling on the loop is analyzed. The relationship between the stability of the equilibrium point of the KM model is described. All equilibrium points in the case of zero setback are described, and the stable equilibrium point and the unstable equilibrium point are further depicted. For the stable equilibrium point, its attraction domain is estimated and the convergence rate is given.

【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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