一个五变量钙振荡模型中的分支分析
发布时间：2018-08-29 13:33
【摘要】：本论文主要根据中心流形定理和分支理论,严格分析了一个五变量的胞内钙振荡模型的分支动态,从理论上分析了所选模型的平衡点发生Hopf分支现象所需要满足的条件并判断发生的Hopf分支的类型。与此同时,我们还进行了数值模拟,得到了系统的平衡点和周期轨的分支图,发现了此模型中存在3种不同形式的钙振荡现象,包括周期振荡、拟周期振荡和混沌。全文共分为四章：第一章为绪论,主要从非线性动力学和钙振荡的研究与发展两个方面对钙振荡的研究背景进行了阐述。并且对文章中主要用到的两个定理：中心流形定理和Hopf分支定理进行了相对简单的阐述,同时也对混沌运动进行了简要的说明。第二章对一个以k3为分支参数的五变量的钙振荡模型进行了研究分析,其中我们进行的主要研究内容包含有：分析系统的平衡点的类型和稳定性等性质是否会随着分支参数k3的改变而产生相应的变化,理论分析的结果显示系统的平衡点会因为分支参数k3的改变而出现一次subcritical Hopf分支。之后我们还进行了相应的数值模拟,得到的数值模拟的结果可以检验我们之前的理论分析的结果,并且我们还第一次得到了关于系统周期轨随k3变化时的分支图,发现系统周期性振荡现象的消失和拟周期振荡现象的产生是由于周期轨发生了环面分支,同时还发现系统本身存在着振荡现象。第三章我们仍然以前面第二章中的系统作为我们的研究模型：选取了k5作为系统的分支参数,对前面选取的一些参数的值作了改变,研究系统的平衡点的类型和稳定性等性质是否会随着分支参数k5的改变而产生相应的变化,理论分析的结果显示系统的平衡点会随着分支参数l5的改变出现两次supercritical Hopf分支,而这两次supercritical Hopf分支可以都可以使系统中的钙离子浓度出现周期性的振荡现象。之后我们也进行了相应的数值模拟,得到的数值模拟的结果同样可以检验我们之前的理论分析的结果,并且我们还第一次得到了系统的周期轨随k5变化下的分支图,发现系统周期性振荡现象的消失和拟周期振荡现象的产生是由于周期轨发生了环面分支,同时还发现在周期轨发生的两次环面分支之间会有混沌现象出现。第四章是对全文工作所作出的一个概括性的总结,同时也确定了未来研究工作的方向和重点。
[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.
【学位授予单位】：北京化工大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O19
本文编号：2211353
[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.
【学位授予单位】：北京化工大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O19
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