两类离散与连续系统的分支研究

发布时间：2018-11-17 10:19

【摘要】：动力系统是非线性科学领域重要的研究内容.历经庞家莱、李雅谱诺夫等大量学者的的研究、探索、发展和完善,动力系统已成为现代数学中的一个独立的、有趣的、具有很好的科研价值和应用前景的研究方向.其中,稳定性和分支是动力系统的动态行为研究领域里主要的的研究对象.稳定性是动力系统拓扑结构平衡性的一种体现,对它的深入探索能更好的展现丰富、复杂的系统动力学特征.分支是指系统的一些特性发生突变的现象伴随参数发生变化而经过临界值得过程中.本文借助规范型理论、中心流行定理、分支理论等全面的展示了两类离散和连续动力系统随参数变化时丰富的动力学行为.对离散时间动力系统的动态特征结构的研究,利用向前欧拉离散法详细的研究了一个离散的传染病模型的分支和混沌现象,通过中心流形定理和分支理论推理和证明了离散系统不动点的稳定性和在一定的参数条件下从不动处产生分支及分支闭轨的稳定性.最后,并借助数值模拟清晰地观察到系统出现了稳定的周期窗口、倍周期重叠、周期到混沌和混沌到稳定的周期窗口的动态行为的跳跃变化.对连续系统的研究,本文探究了一个延迟反馈控制混沌动力系统随分支参数的变化而产生的系统动力学行为的变化规律及反馈控制器对控制混沌的有效果性.利用中心流形、规范型和分支定理研究了参数变化时系统的稳定性及分支参数通过某一临界值时系统产生Hopf分支周期解和分支周期解的性质(稳定性、方向和振幅).利用数值方法进一步的得出混沌是可以控制的,延迟反馈控制可以诱发稳定的周期闭轨.
[Abstract]:Dynamic system is an important research content in the field of nonlinear science. After a lot of research, exploration, development and improvement, dynamic system has become an independent and interesting research direction with good research value and application prospect in modern mathematics. Among them, stability and bifurcation are the main research objects in the field of dynamic behavior of dynamic systems. Stability is a reflection of the equilibrium of dynamical system topological structure, and its deep exploration can better show the rich and complex dynamic characteristics of the system. Bifurcation refers to the phenomenon that some characteristics of the system change with the change of the parameters and pass through the critical value in the process of obtaining the critical value. In this paper, two kinds of discrete and continuous dynamical systems are presented with the help of normal form theory, central popular theorem, bifurcation theory and so on. In this paper, the bifurcation and chaos of a discrete infectious disease model are studied in detail by using forward Euler discrete method, which is used to study the dynamic characteristic structure of discrete time dynamic system. By means of the central manifold theorem and bifurcation theory, the stability of fixed point of discrete system and the stability of bifurcation and closed orbit generated from fixed point under certain parameter conditions are proved. Finally, with the help of numerical simulation, it is clearly observed that the system has stable periodic window, double period overlapping, period to chaos and chaotic to stable periodic window dynamic behavior jump change. For the study of continuous systems, the dynamic behavior of a delayed feedback chaotic dynamic system with the variation of bifurcation parameters and the effectiveness of the feedback controller in controlling chaos are studied in this paper. By using the central manifold, normal form and bifurcation theorem, we study the stability of the system and the properties (stability, direction and amplitude) of the Hopf bifurcation periodic solution and the bifurcation periodic solution (stability, direction and amplitude) when the bifurcation parameter passes through a certain critical value. By using numerical method, it is further concluded that chaos is controllable, and delay feedback control can induce stable periodic closed orbits.
【学位授予单位】：北方民族大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O19

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