基于有向图的高维Kuramoto模型的同步
发布时间：2019-03-14 09:07
【摘要】：高维Kuramoto模型又称为Lohe模型,是一类非线性多个体系统.所谓多个体系统是指由多个微分方程或差分方程按照一个图的结构耦合在一起所构成的复杂动态系统,其动态行为由个体的动态方程及图的结构来决定.高维Kuramoto模型有很好的物理背景,它可以用来解释互联的多个量子振荡器的同步现象.在已有的关于高维Kuramoto模型的结果中,大多数仅仅考虑无向图的情形,而对于一般的有向图的情形,高维Kuramoto模型方面的结果很少.本论文研究基于有向图的高维Kuramoto模型的平衡点判定问题和同步问题.首先,对于平衡点判别问题,把已有结果中关于有向图的强连通条件进一步减弱为存在有向支撑树的条件,得到了判别平衡点的充分必要条件.其次,关于同步问题,我们把无向图情形的相关结果推广到了有向图的情形.具体来说包括:(1)当有向图是强连通并且所有个体的初始状态都位于一个半球面时,同步现象可以实现;(2)当有向图有一个有向支撑树并且第一个强连通分支仅包含一个顶点时,初始状态位于一个半球面的所有个体系统的状态将实现同步;(3)当仅仅假设有向图有一个有向支撑树时,进一步限制个体系统的初始状态两两内积大于零,则所有个体系统的状态能够实现同步.本文综合运用了推广的LaSalle不变集原理和数学归纳法,并对系统进行了新的形式处理,所采用的证明方法与已有的方法明显不同.
[Abstract]:The high-dimensional Kuramoto model, also known as the Lohe model, is a class of nonlinear multi-individual systems. The so-called multi-individual system is a complex dynamic system composed of several differential equations or difference equations coupled together according to the structure of a graph. The dynamic behavior of the system is determined by the dynamic equation of the individual and the structure of the graph. The high-dimensional Kuramoto model has a good physical background, which can be used to explain the synchronization of interconnected multiple quantum oscillators. Among the existing results on high-dimensional Kuramoto model, most of them only consider the case of undirected graph, but for the case of general directed graph, the result of high-dimensional Kuramoto model is very few. In this paper, we study the equilibrium point decision problem and synchronization problem of high-dimensional Kuramoto model based on directed graph. Firstly, for the problem of discriminating equilibrium points, the condition of strong connectivity for digraphs in the existing results is further weakened to the condition of the existence of directed support trees, and the necessary and sufficient conditions for distinguishing the equilibrium points are obtained. Secondly, for the synchronization problem, we extend the related results of the case of undirected graph to the case of directed graph. Specifically, it includes: (1) synchronization can be realized when the digraph is strongly connected and the initial states of all individuals are located on a hemispherical surface; (2) when a directed graph has a directed spanning tree and the first strongly connected branch contains only one vertex, the states of all individual systems with an initial state on a hemispherical surface will be synchronized; (3) if we only assume that a directed graph has a directed spanning tree, further limit the initial state of the individual system with a pairwise inner product greater than zero, then the states of all individual systems can be synchronized. In this paper, the generalized LaSalle invariant set principle and mathematical induction method are comprehensively used, and the system is treated in a new form. The method of proof is obviously different from the existing method.
【学位授予单位】：南京师范大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O157.5
[Abstract]:The high-dimensional Kuramoto model, also known as the Lohe model, is a class of nonlinear multi-individual systems. The so-called multi-individual system is a complex dynamic system composed of several differential equations or difference equations coupled together according to the structure of a graph. The dynamic behavior of the system is determined by the dynamic equation of the individual and the structure of the graph. The high-dimensional Kuramoto model has a good physical background, which can be used to explain the synchronization of interconnected multiple quantum oscillators. Among the existing results on high-dimensional Kuramoto model, most of them only consider the case of undirected graph, but for the case of general directed graph, the result of high-dimensional Kuramoto model is very few. In this paper, we study the equilibrium point decision problem and synchronization problem of high-dimensional Kuramoto model based on directed graph. Firstly, for the problem of discriminating equilibrium points, the condition of strong connectivity for digraphs in the existing results is further weakened to the condition of the existence of directed support trees, and the necessary and sufficient conditions for distinguishing the equilibrium points are obtained. Secondly, for the synchronization problem, we extend the related results of the case of undirected graph to the case of directed graph. Specifically, it includes: (1) synchronization can be realized when the digraph is strongly connected and the initial states of all individuals are located on a hemispherical surface; (2) when a directed graph has a directed spanning tree and the first strongly connected branch contains only one vertex, the states of all individual systems with an initial state on a hemispherical surface will be synchronized; (3) if we only assume that a directed graph has a directed spanning tree, further limit the initial state of the individual system with a pairwise inner product greater than zero, then the states of all individual systems can be synchronized. In this paper, the generalized LaSalle invariant set principle and mathematical induction method are comprehensively used, and the system is treated in a new form. The method of proof is obviously different from the existing method.
【学位授予单位】：南京师范大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O157.5
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