本文選題：類Lorenz系統(tǒng) + 超混沌系統(tǒng)�。� 參考：《揚州大學(xué)》2016年博士論文

【摘要】：混沌,作為大自然中的一種分布廣泛且具有復(fù)雜動力學(xué)的非線性現(xiàn)象,近年來受到了多個領(lǐng)域的科學(xué)家們和工程師們的普遍關(guān)注Lorenz系統(tǒng)——首個混沌數(shù)理模型——以及與之相關(guān)的類Lorenz系統(tǒng)族的探究極大地推動了混沌科學(xué)的發(fā)展.相比于低維混沌系統(tǒng),高維混沌系統(tǒng)及其吸引子擁有更為復(fù)雜的動力學(xué)行為以及潛在的更廣泛應(yīng)用,成為近幾年非線性科學(xué)的一個重要研究領(lǐng)域.基于L orenz型系統(tǒng)族的研究現(xiàn)狀,本文不僅深入挖掘了已存在的混沌和超混沌系統(tǒng)的未被發(fā)現(xiàn)的動力學(xué)行為,而且提出并分析了兩個新的超混沌系統(tǒng)(分別是四維和五維).確切地說,主要利用動力學(xué)理論和方法,諸如中心流型理論、規(guī)范型理論、分岔理論、投影法、Poincare緊致法、Lyapunov函數(shù)、數(shù)值仿真等,不僅討論了這些系統(tǒng)的平衡點的分布,穩(wěn)定性及其分岔等局部動力學(xué)行為,而且也研究了同宿異宿軌和奇異退化異宿環(huán)的存在性,探討了存在無窮多個異宿軌的超混沌系統(tǒng),超混沌吸引子與非孤立平衡點共存,奇異退化異宿環(huán)破裂產(chǎn)生超混沌吸引子等全局動力學(xué)行為.本文的主要研究工作組織如下.第一章介紹本文研究主題的一些背景知識和已經(jīng)取得的最新進展.第二章簡要概括混沌理論和相應(yīng)的研究方法.第三章深入研究一個三維類Lorenz系統(tǒng)的未被研究的動力學(xué)行為.通過應(yīng)用分岔理論,規(guī)范型定理,Lyapunov函數(shù),投影法,Poincare緊致法等,呈現(xiàn)了其在參數(shù)空間內(nèi)的局部和全局、有限和無限處的較完整的動力學(xué)行為.此外,數(shù)值仿真證實了相應(yīng)的理論分析結(jié)果.第四章提出一個新的四維自治超混沌統(tǒng)-Lorenz型系統(tǒng),它包含幾個現(xiàn)有的系統(tǒng)作為特例.運用Routh-Hurwitz判別準則、中心流型理論和分岔理論,討論了該系統(tǒng)的平衡點的穩(wěn)定性,折分岔,叉形分岔和Hopf分岔等局部動力學(xué)行為.結(jié)合Lyapunov函數(shù)理論和α-極限集、ω-極限集的定義,嚴格證明了該系統(tǒng)在特定的參數(shù)范圍內(nèi)僅存兩條異宿軌而不存在同宿軌.此外,也給出了異宿軌不存在的結(jié)果.特別是,數(shù)值仿真發(fā)現(xiàn)該系統(tǒng)奇異退化異宿環(huán)破裂時不會產(chǎn)生超混沌吸引子.第五章深入挖掘了復(fù)Lorenz系統(tǒng)的未被探究的動力學(xué)行為,諸如所有環(huán)形平衡點的非雙曲性,奇異退化異宿環(huán)附近的超混沌吸引子的不存在性和無窮多個環(huán)形異宿于原點和環(huán)形平衡點的異宿軌的存在性等.第六章在Shimizu-Morioka系統(tǒng)基礎(chǔ)上構(gòu)造了一個新的五維自治超混沌系統(tǒng).結(jié)合理論分析和數(shù)值仿真,發(fā)現(xiàn)該系統(tǒng)存在如下有趣且獨特的動力學(xué)行為：1.存在橢圓拋物型和雙藍拋物型的平衡點；2.超混沌吸引子和非孤立平衡點共存；3.存在奇異退化異宿環(huán)分岔出的超混沌吸引子;4.存在Cantor集型的參數(shù)空間中的無窮多個橢圓拋物型和雙曲拋物型的異宿軌.
[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 偽 -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.
【學(xué)位授予單位】：揚州大學(xué)
【學(xué)位級別】：博士
【學(xué)位授予年份】：2016
【分類號】：O415.5

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