【摘要】：本論文主要根據(jù)中心流形定理和分支理論,嚴(yán)格分析了一個(gè)五變量的胞內(nèi)鈣振蕩模型的分支動(dòng)態(tài),從理論上分析了所選模型的平衡點(diǎn)發(fā)生Hopf分支現(xiàn)象所需要滿足的條件并判斷發(fā)生的Hopf分支的類型。與此同時(shí),我們還進(jìn)行了數(shù)值模擬,得到了系統(tǒng)的平衡點(diǎn)和周期軌的分支圖,發(fā)現(xiàn)了此模型中存在3種不同形式的鈣振蕩現(xiàn)象,包括周期振蕩、擬周期振蕩和混沌。全文共分為四章：第一章為緒論,主要從非線性動(dòng)力學(xué)和鈣振蕩的研究與發(fā)展兩個(gè)方面對(duì)鈣振蕩的研究背景進(jìn)行了闡述。并且對(duì)文章中主要用到的兩個(gè)定理：中心流形定理和Hopf分支定理進(jìn)行了相對(duì)簡(jiǎn)單的闡述,同時(shí)也對(duì)混沌運(yùn)動(dòng)進(jìn)行了簡(jiǎn)要的說(shuō)明。第二章對(duì)一個(gè)以k3為分支參數(shù)的五變量的鈣振蕩模型進(jìn)行了研究分析,其中我們進(jìn)行的主要研究?jī)?nèi)容包含有：分析系統(tǒng)的平衡點(diǎn)的類型和穩(wěn)定性等性質(zhì)是否會(huì)隨著分支參數(shù)k3的改變而產(chǎn)生相應(yīng)的變化,理論分析的結(jié)果顯示系統(tǒng)的平衡點(diǎn)會(huì)因?yàn)榉种?shù)k3的改變而出現(xiàn)一次subcritical Hopf分支。之后我們還進(jìn)行了相應(yīng)的數(shù)值模擬,得到的數(shù)值模擬的結(jié)果可以檢驗(yàn)我們之前的理論分析的結(jié)果,并且我們還第一次得到了關(guān)于系統(tǒng)周期軌隨k3變化時(shí)的分支圖,發(fā)現(xiàn)系統(tǒng)周期性振蕩現(xiàn)象的消失和擬周期振蕩現(xiàn)象的產(chǎn)生是由于周期軌發(fā)生了環(huán)面分支,同時(shí)還發(fā)現(xiàn)系統(tǒng)本身存在著振蕩現(xiàn)象。第三章我們?nèi)匀灰郧懊娴诙轮械南到y(tǒng)作為我們的研究模型：選取了k5作為系統(tǒng)的分支參數(shù),對(duì)前面選取的一些參數(shù)的值作了改變,研究系統(tǒng)的平衡點(diǎn)的類型和穩(wěn)定性等性質(zhì)是否會(huì)隨著分支參數(shù)k5的改變而產(chǎn)生相應(yīng)的變化,理論分析的結(jié)果顯示系統(tǒng)的平衡點(diǎn)會(huì)隨著分支參數(shù)l5的改變出現(xiàn)兩次supercritical Hopf分支,而這兩次supercritical Hopf分支可以都可以使系統(tǒng)中的鈣離子濃度出現(xiàn)周期性的振蕩現(xiàn)象。之后我們也進(jìn)行了相應(yīng)的數(shù)值模擬,得到的數(shù)值模擬的結(jié)果同樣可以檢驗(yàn)我們之前的理論分析的結(jié)果,并且我們還第一次得到了系統(tǒng)的周期軌隨k5變化下的分支圖,發(fā)現(xiàn)系統(tǒng)周期性振蕩現(xiàn)象的消失和擬周期振蕩現(xiàn)象的產(chǎn)生是由于周期軌發(fā)生了環(huán)面分支,同時(shí)還發(fā)現(xiàn)在周期軌發(fā)生的兩次環(huán)面分支之間會(huì)有混沌現(xiàn)象出現(xiàn)。第四章是對(duì)全文工作所作出的一個(gè)概括性的總結(jié),同時(shí)也確定了未來(lái)研究工作的方向和重點(diǎn)。
[Abstract]:In this paper, based on the central manifold theorem and bifurcation theory, the bifurcation dynamics of a five-variable intracellular calcium oscillation model are strictly analyzed. The necessary conditions for the occurrence of Hopf bifurcation at the equilibrium point of the selected model are analyzed theoretically and the type of Hopf bifurcation occurring is determined. At the same time, numerical simulation is carried out, and the equilibrium point and the bifurcation diagram of the periodic orbit are obtained. It is found that there are three different forms of calcium oscillation in this model, including periodic oscillation, quasi-periodic oscillation and chaos. This paper is divided into four chapters: the first chapter is the introduction, mainly from the nonlinear dynamics and calcium oscillation research and development two aspects of calcium oscillation research background is described. Two main theorems used in this paper, the center manifold theorem and the Hopf bifurcation theorem, are described briefly, and the chaotic motion is also explained briefly. In the second chapter, a five-variable calcium oscillation model with K3 as a branch parameter is studied and analyzed. The main contents of our research include: whether the type and stability of the equilibrium point of the system will change with the change of the branch parameter K3. The results of theoretical analysis show that the equilibrium point of the system will have a subcritical Hopf bifurcation due to the change of the bifurcation parameter K3. Then we do the corresponding numerical simulation, and the results of the numerical simulation can test the results of our previous theoretical analysis, and we also get the branching diagram of the periodic orbit of the system with K3 for the first time. It is found that the disappearance of periodic oscillation and the occurrence of quasi-periodic oscillation are due to the toroidal bifurcation of the periodic orbit and the oscillation of the system itself. In the third chapter, we still take the system in the second chapter as our research model: we select K5 as the branch parameter of the system, and we change the value of some parameters. Whether the type and stability of the equilibrium point of the system will change with the change of the bifurcation parameter K5, the theoretical analysis results show that the equilibrium point of the system will appear twice supercritical Hopf bifurcation with the change of the branching parameter L5. Both supercritical Hopf branches can cause periodic oscillations of calcium concentration in the system. Then we also do the corresponding numerical simulation, and the results of the numerical simulation can also test the results of our previous theoretical analysis, and for the first time, we have obtained the bifurcation diagram of the system with the change of the periodic orbit with K5. It is found that the disappearance of periodic oscillation and the occurrence of quasi-periodic oscillation are due to the torus bifurcation of the periodic orbit and the chaos will occur between the two torus branches of the periodic orbit. The fourth chapter is a summary of the full text work, and also determines the direction and focus of future research work.
【學(xué)位授予單位】：北京化工大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2015
【分類號(hào)】：O19

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本文編號(hào)：2211353