几个高维混沌系统的奇异轨及其分岔

发布时间：2018-06-01 17:59

本文选题：类Lorenz系统 + 超混沌系统　；参考：《扬州大学》2016年博士论文

【摘要】：混沌,作为大自然中的一种分布广泛且具有复杂动力学的非线性现象,近年来受到了多个领域的科学家们和工程师们的普遍关注Lorenz系统——首个混沌数理模型——以及与之相关的类Lorenz系统族的探究极大地推动了混沌科学的发展.相比于低维混沌系统,高维混沌系统及其吸引子拥有更为复杂的动力学行为以及潜在的更广泛应用,成为近几年非线性科学的一个重要研究领域.基于L orenz型系统族的研究现状,本文不仅深入挖掘了已存在的混沌和超混沌系统的未被发现的动力学行为,而且提出并分析了两个新的超混沌系统(分别是四维和五维).确切地说,主要利用动力学理论和方法,诸如中心流型理论、规范型理论、分岔理论、投影法、Poincare紧致法、Lyapunov函数、数值仿真等,不仅讨论了这些系统的平衡点的分布,稳定性及其分岔等局部动力学行为,而且也研究了同宿异宿轨和奇异退化异宿环的存在性,探讨了存在无穷多个异宿轨的超混沌系统,超混沌吸引子与非孤立平衡点共存,奇异退化异宿环破裂产生超混沌吸引子等全局动力学行为.本文的主要研究工作组织如下.第一章介绍本文研究主题的一些背景知识和已经取得的最新进展.第二章简要概括混沌理论和相应的研究方法.第三章深入研究一个三维类Lorenz系统的未被研究的动力学行为.通过应用分岔理论,规范型定理,Lyapunov函数,投影法,Poincare紧致法等,呈现了其在参数空间内的局部和全局、有限和无限处的较完整的动力学行为.此外,数值仿真证实了相应的理论分析结果.第四章提出一个新的四维自治超混沌统-Lorenz型系统,它包含几个现有的系统作为特例.运用Routh-Hurwitz判别准则、中心流型理论和分岔理论,讨论了该系统的平衡点的稳定性,折分岔,叉形分岔和Hopf分岔等局部动力学行为.结合Lyapunov函数理论和α-极限集、ω-极限集的定义,严格证明了该系统在特定的参数范围内仅存两条异宿轨而不存在同宿轨.此外,也给出了异宿轨不存在的结果.特别是,数值仿真发现该系统奇异退化异宿环破裂时不会产生超混沌吸引子.第五章深入挖掘了复Lorenz系统的未被探究的动力学行为,诸如所有环形平衡点的非双曲性,奇异退化异宿环附近的超混沌吸引子的不存在性和无穷多个环形异宿于原点和环形平衡点的异宿轨的存在性等.第六章在Shimizu-Morioka系统基础上构造了一个新的五维自治超混沌系统.结合理论分析和数值仿真,发现该系统存在如下有趣且独特的动力学行为：1.存在椭圆抛物型和双蓝抛物型的平衡点；2.超混沌吸引子和非孤立平衡点共存；3.存在奇异退化异宿环分岔出的超混沌吸引子;4.存在Cantor集型的参数空间中的无穷多个椭圆抛物型和双曲抛物型的异宿轨.
[Abstract]:Chaos, as a widely distributed and complex dynamic nonlinear phenomenon in nature, In recent years, scientists and engineers in many fields have paid close attention to the research of Lorenz system, the first chaotic mathematical model, and the related Lorenz system family, which has greatly promoted the development of chaotic science. Compared with low-dimensional chaotic systems, high-dimensional chaotic systems and their attractors have more complex dynamic behaviors and potential wider applications, and have become an important research field of nonlinear science in recent years. Based on the research status of L orenz type systems, this paper not only excavates the undiscovered dynamical behaviors of the existing chaotic and hyperchaotic systems, but also proposes and analyzes two new hyperchaotic systems (four and five dimensions respectively). To be exact, the dynamic theory and methods, such as center flow theory, normal form theory, bifurcation theory, projection method Poincare compactness method Lyapunov function, numerical simulation and so on, are used to discuss not only the distribution of equilibrium points of these systems, but also the distribution of equilibrium points. Local dynamical behaviors such as stability and bifurcation are also studied. The existence of homoclinic heteroclinic orbits and singular degenerate heterotropic rings is also studied. The hyperchaotic systems with infinite heteroclinic orbits are studied. The hyperchaotic attractors coexist with non-isolated equilibrium points. The global dynamical behavior such as hyperchaotic attractor is produced by the rupture of the singular degenerate heteroclinic ring. The main work of this paper is organized as follows. The first chapter introduces some background knowledge and the latest progress of this paper. The second chapter briefly summarizes the chaos theory and the corresponding research methods. In chapter 3, the unstudied dynamic behavior of a three-dimensional Lorenz-like system is studied. By applying bifurcation theory, normal form theorem and Lyapunov function, the projection method and Poincare compactness method, the local and global, finite and infinite dynamic behaviors of the system are presented. In addition, the corresponding theoretical analysis results are verified by numerical simulation. In chapter 4, a new four-dimensional autonomous hyperchaotic system-Lorenz type system is proposed, which contains several existing systems as special cases. By using the Routh-Hurwitz criterion, the central flow theory and the bifurcation theory, the local dynamic behaviors of the system such as the stability of the equilibrium point, the folding bifurcation, the fork bifurcation and the Hopf bifurcation are discussed. Based on the Lyapunov function theory and the definition of 伪 -limit set and 蠅 -limit set, it is strictly proved that there are only two heteroclinic orbits and no homoclinic orbits in the given parameter range. In addition, the results of nonexistence of heteroclinic orbit are also given. In particular, numerical simulations show that the hyperchaotic attractor will not be produced when the singular degenerate heteroclinic ring breaks down. In chapter 5, the unexplored dynamical behaviors of complex Lorenz systems, such as the nonhyperbolic properties of all annular equilibrium points, are explored in depth. The nonexistence of hyperchaotic attractors near singular degenerate heteroclinic rings and the existence of heteroclinic orbits of infinite annular heteroclinic at origin and annular equilibrium points etc. In chapter 6, a new five dimensional autonomous hyperchaotic system is constructed on the basis of Shimizu-Morioka system. Combined with theoretical analysis and numerical simulation, it is found that the system has the following interesting and unique dynamic behavior: 1. The equilibrium point of elliptical parabolic type and double blue parabolic type is 2. Hyperchaotic attractors coexist with non-isolated equilibrium points. The hyperchaotic attractor 4 with singular degenerate heteroclinic ring bifurcation. There are infinite elliptic parabolic and hyperbolic parabolic heteroclinic orbits in parameter space of Cantor set type.
【学位授予单位】：扬州大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O415.5

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