發(fā)布時(shí)間：2019-03-14 09:07

【摘要】：高維Kuramoto模型又稱為L(zhǎng)ohe模型,是一類非線性多個(gè)體系統(tǒng).所謂多個(gè)體系統(tǒng)是指由多個(gè)微分方程或差分方程按照一個(gè)圖的結(jié)構(gòu)耦合在一起所構(gòu)成的復(fù)雜動(dòng)態(tài)系統(tǒng),其動(dòng)態(tài)行為由個(gè)體的動(dòng)態(tài)方程及圖的結(jié)構(gòu)來決定.高維Kuramoto模型有很好的物理背景,它可以用來解釋互聯(lián)的多個(gè)量子振蕩器的同步現(xiàn)象.在已有的關(guān)于高維Kuramoto模型的結(jié)果中,大多數(shù)僅僅考慮無向圖的情形,而對(duì)于一般的有向圖的情形,高維Kuramoto模型方面的結(jié)果很少.本論文研究基于有向圖的高維Kuramoto模型的平衡點(diǎn)判定問題和同步問題.首先,對(duì)于平衡點(diǎn)判別問題,把已有結(jié)果中關(guān)于有向圖的強(qiáng)連通條件進(jìn)一步減弱為存在有向支撐樹的條件,得到了判別平衡點(diǎn)的充分必要條件.其次,關(guān)于同步問題,我們把無向圖情形的相關(guān)結(jié)果推廣到了有向圖的情形.具體來說包括:(1)當(dāng)有向圖是強(qiáng)連通并且所有個(gè)體的初始狀態(tài)都位于一個(gè)半球面時(shí),同步現(xiàn)象可以實(shí)現(xiàn);(2)當(dāng)有向圖有一個(gè)有向支撐樹并且第一個(gè)強(qiáng)連通分支僅包含一個(gè)頂點(diǎn)時(shí),初始狀態(tài)位于一個(gè)半球面的所有個(gè)體系統(tǒng)的狀態(tài)將實(shí)現(xiàn)同步;(3)當(dāng)僅僅假設(shè)有向圖有一個(gè)有向支撐樹時(shí),進(jìn)一步限制個(gè)體系統(tǒng)的初始狀態(tài)兩兩內(nèi)積大于零,則所有個(gè)體系統(tǒng)的狀態(tài)能夠?qū)崿F(xiàn)同步.本文綜合運(yùn)用了推廣的LaSalle不變集原理和數(shù)學(xué)歸納法,并對(duì)系統(tǒng)進(jìn)行了新的形式處理,所采用的證明方法與已有的方法明顯不同.
[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.
【學(xué)位授予單位】：南京師范大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類號(hào)】：O157.5

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