【摘要】：隨著研究工作的進(jìn)行,很多種不同的簇發(fā)形式被提出,但是大部分簇發(fā)現(xiàn)象都只含有一個(gè)或者兩個(gè)振蕩分支,很少見多振蕩分支的簇發(fā)現(xiàn)象;而且像這樣的簇發(fā)現(xiàn)象大部分都是在高階連續(xù)系統(tǒng)中發(fā)現(xiàn)的,有學(xué)者就提出了采用離散系統(tǒng)去模擬這種簡(jiǎn)單的簇發(fā)現(xiàn)象,主要是由于離散系統(tǒng)在數(shù)值仿真的時(shí)候比連續(xù)系統(tǒng)更方便,可以在短時(shí)間內(nèi)得到大量的數(shù)據(jù)。但是現(xiàn)有的離散模型也不能夠模擬更為復(fù)雜的多分支簇發(fā)振蕩現(xiàn)象,所以有必要進(jìn)一步研究關(guān)于離散系統(tǒng)中的多分支簇發(fā)振蕩現(xiàn)象及其產(chǎn)生機(jī)理。本文以一個(gè)正弦映射為研究對(duì)象,通過(guò)研究一維正弦系統(tǒng)的動(dòng)力學(xué)特征研究發(fā)現(xiàn)其具有初值敏感性、倍周期分岔和對(duì)稱性破缺分岔等非線性現(xiàn)象,但是這些性質(zhì)不足以產(chǎn)生多分支簇發(fā)振蕩現(xiàn)象。為了采用離散系統(tǒng)模擬多分支簇發(fā)振蕩現(xiàn)象,我們進(jìn)一步采用正弦映射和一個(gè)三次方映射,通過(guò)非線性耦合方式組合成一個(gè)二維的離散系統(tǒng),通過(guò)分析發(fā)現(xiàn)此系統(tǒng)中存在多種不同的分岔特性,但是此系統(tǒng)為單一尺度系統(tǒng),也不能夠產(chǎn)生多分支簇發(fā)振蕩現(xiàn)象。根據(jù)以上分析本文最后構(gòu)造了三維離散系統(tǒng),通過(guò)對(duì)其平衡點(diǎn)的穩(wěn)定性分析,得到了其快子系統(tǒng)發(fā)生Fold分岔和Neimarker-sacker分岔的參數(shù)條件。通過(guò)數(shù)值計(jì)算得到了系統(tǒng)在三種不同參數(shù)條件下,呈現(xiàn)三種不同的快慢簇發(fā)振蕩現(xiàn)象,發(fā)現(xiàn)此簇發(fā)振蕩由多個(gè)快慢振蕩的分支所組成,且每個(gè)分支由其獨(dú)特的簇發(fā)振蕩現(xiàn)象,進(jìn)一步采用Rinzel快慢分析法給出了三種不同的多分支快慢簇發(fā)振蕩現(xiàn)象及其相應(yīng)分支的產(chǎn)生機(jī)理。根據(jù)三種不同快慢振蕩現(xiàn)象的產(chǎn)生機(jī)理,文中對(duì)這三種簇發(fā)振蕩現(xiàn)象進(jìn)行了分類。最后設(shè)計(jì)了一組相應(yīng)的電路實(shí)驗(yàn)得到了三種不同快慢振蕩現(xiàn)象的相圖,從而驗(yàn)證了理論分析與數(shù)值計(jì)算的正確性。
[Abstract]:With the development of the research work, many different clusters have been proposed, but most of the cluster phenomena only contain one or two oscillatory branches, and it is rare to find clusters with multiple oscillatory branches. Moreover, most of the cluster discovery images like this are found in high-order continuous systems. Some scholars have proposed to use discrete systems to simulate this simple cluster discovery image. The main reason is that the discrete system is more convenient than the continuous system in numerical simulation, and a large amount of data can be obtained in a short time. However, the existing discrete models can not simulate the more complex multi-branching cluster oscillation phenomenon, so it is necessary to further study the multi-branch cluster oscillation phenomenon and its mechanism in discrete systems. In this paper, a sinusoidal mapping is studied. By studying the dynamic characteristics of one dimensional sinusoidal system, it is found that it has some nonlinear phenomena, such as initial value sensitivity, period doubling bifurcation, symmetry breaking bifurcation and so on. However, these properties are not sufficient to produce multi-branch cluster oscillation. In order to simulate the multi-branching cluster oscillations with discrete systems, we further use sinusoidal maps and a cubic map to form a two-dimensional discrete system by nonlinear coupling. It is found that there are many different bifurcation characteristics in this system, but the system is a single scale system and can not produce multi-branch cluster oscillation. Based on the above analysis, a three dimensional discrete system is constructed. By analyzing the stability of its equilibrium point, the parameter conditions for Fold bifurcation and Neimarker-sacker bifurcation in its fast subsystem are obtained. By numerical calculation, three different fast and slow cluster oscillations are obtained under three different parameter conditions. It is found that the cluster oscillation is composed of several fast and slow oscillating branches, and each branch is composed of its unique cluster oscillation phenomenon. Furthermore, three different multi-branch fast and slow cluster oscillations and their corresponding branching mechanisms are given by using the Rinzel fast and slow analysis method. According to the generation mechanism of three different fast and slow oscillation phenomena, the three cluster oscillation phenomena are classified in this paper. Finally, a set of corresponding circuit experiments are designed to obtain three phase diagrams of different fast and slow oscillation phenomena, which verifies the correctness of theoretical analysis and numerical calculation.
【學(xué)位授予單位】：電子科技大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2015
【分類號(hào)】：O415.5;TB30
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本文編號(hào)：2390595