向量场的同宿轨分支及奇异吸引子的存在性研究
发布时间：2018-08-23 08:33
【摘要】：一直以来,混沌(Chaos)都是非线性科学研究的热点问题之一,而奇异吸引子则是反映系统混沌运动的典型特征,故而对奇异吸引子的产生机制,存在性条件,以及吸引子本身性质的探讨有着重要的意义。自从1963年,著名的Lorenz方程在数值求解被发现存在吸引子到1999年,首次利用规范形并结合计算机辅助证明Lorenz方程的确存在奇异吸引子,人们对奇异吸引子的分析和研究已逐步深入。其中,研究最多的是为模拟Lorenz方程的动力学行为而提出的几何Lorenz吸引子的模型。向量场的分支理论主要研究动力系统的轨道族的拓扑结构随参数变化所发生的变化及其变化规律。当存在鞍点的同宿轨时,系统是结构不稳定的,因此,同宿轨分支蕴含着丰富的动力学行为。在Lorenz模型中,对应于蝴蝶同宿的分支,两个对称的同宿轨道破裂时,每个回路产生一个周期轨道,并且这两个周期轨道的稳定流形与不稳定流形彼此横截相交。这种横截同宿现象导致了众多的复杂现象。本文对可能产生洛伦兹型吸引子的两种余维2的对称同宿轨分支：倾斜翻转分支和轨道翻转分支,分别进行了详细的研究分析。具体的,文中以含有鞍点的3维Cr(r≥3)对称系统X=f(X),X∈R3为研究对象,首先在鞍点平衡态附近将系统化为简单的、便于分析的形式,由此构造Poincare返回映射。根据得到的Poincare返回映射对倾斜翻转和轨道翻转情形的分支情况进行了讨论,并得到分支曲线图。随后讨论了这两种分支情形的开折中洛伦兹型吸引子的存在性问题。通过对几何洛伦兹模型的分析,以及一维洛伦兹映射的拓扑性质的讨论,并将二维Poincare返回映射约化到一维,最终得到了洛伦兹型吸引子的存在范围。
[Abstract]:Chaotic (Chaos) has always been one of the hot issues in nonlinear science, and singular attractor is a typical feature of chaotic motion of the system. It is of great significance to discuss the properties of the attractor itself. Since 1963, when the famous Lorenz equation was numerically solved, the existence of attractors was discovered. In 1999, the existence of singular attractors in Lorenz equation was proved for the first time by using canonical form and computer aid. The analysis and study of strange attractors have been gradually deepened. Among them, the geometric Lorenz attractor model proposed to simulate the dynamic behavior of Lorenz equation is the most studied. The bifurcation theory of vector field is mainly concerned with the variation of the topological structure of the orbital family of the dynamical system with the change of the parameters. When there is a saddle point homoclinic orbit, the system is structurally unstable, so the homoclinic orbit bifurcation contains abundant dynamic behavior. In the Lorenz model, when two symmetric homoclinic orbits break up, each loop produces a periodic orbit, and the stable manifold of the two periodic orbits intersects with the unstable manifold. This transversal homoclinic phenomenon leads to many complicated phenomena. In this paper, the symmetric homoclinic bifurcation of two kinds of codimension 2 which may produce Lorentz attractor is studied and analyzed in detail, that is, tilting overturning branch and orbit overturning branch. Specifically, in this paper, we take the 3-dimensional C r (r 鈮，
本文编号：2198498
[Abstract]:Chaotic (Chaos) has always been one of the hot issues in nonlinear science, and singular attractor is a typical feature of chaotic motion of the system. It is of great significance to discuss the properties of the attractor itself. Since 1963, when the famous Lorenz equation was numerically solved, the existence of attractors was discovered. In 1999, the existence of singular attractors in Lorenz equation was proved for the first time by using canonical form and computer aid. The analysis and study of strange attractors have been gradually deepened. Among them, the geometric Lorenz attractor model proposed to simulate the dynamic behavior of Lorenz equation is the most studied. The bifurcation theory of vector field is mainly concerned with the variation of the topological structure of the orbital family of the dynamical system with the change of the parameters. When there is a saddle point homoclinic orbit, the system is structurally unstable, so the homoclinic orbit bifurcation contains abundant dynamic behavior. In the Lorenz model, when two symmetric homoclinic orbits break up, each loop produces a periodic orbit, and the stable manifold of the two periodic orbits intersects with the unstable manifold. This transversal homoclinic phenomenon leads to many complicated phenomena. In this paper, the symmetric homoclinic bifurcation of two kinds of codimension 2 which may produce Lorentz attractor is studied and analyzed in detail, that is, tilting overturning branch and orbit overturning branch. Specifically, in this paper, we take the 3-dimensional C r (r 鈮，
本文编号：2198498
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