非扩散洛伦兹系统的周期轨道

混沌系统的奇怪吸引子是由无数条周期轨道稠密覆盖构成的,周期轨道是非线性动力系统中除不动点之外最简单的不变集,它不仅能够体现出混沌运动的所有特征,而且和系统振荡的产生与变化密切相关,因此分析复杂系统的动力学行为时获取周期轨道具有重要意义.本文系统地研究了非扩散洛伦兹系统一定拓扑长度以内的周期轨道,提出一种基于轨道的拓扑结构来建立一维符号动力学的新方法,通过变分法数值计算轨道显得很稳定.寻找轨道初始化时,两条轨道片段能够被用作基本的组成单元,基于整条轨道的结构进行拓扑分类的方式显得很有效.此外,讨论了周期轨道随着参数变化时的形变情况,为研究轨道的周期演化规律提供了新途径.本研究可为在其他类似的混沌体系中找到并且系统分类周期轨道提供一种可借鉴的方法.

Periodic orbits of diffusionless Lorenz system

The strange attractor of a chaotic system is composed of numerous periodic orbits densely covered. The periodic orbit is the simplest invariant set except for the fixed point in the nonlinear dynamic system, it not only reflects all the characteristics of the chaotic motion, but also is closely related to the amplitude generation and change of chaotic system. Therefore, it is of great significance to obtain the periodic orbits in order to analyze the dynamical behaviors of the complex system. In this paper, we study the periodic orbits of the diffusionless Lorenz equations which are derived in the limit of high Rayleigh and Prandtl numbers. A new approach to establishing one-dimensional symbolic dynamics is proposed, and the periodic orbits based on a topological structure are systematically calculated. We use the variational method to locate the cycles, which is proposed to explore the periodic orbits in high-dimensional chaotic systems. The method not only preserves the robustness characteristics of most of other methods, such as the Newton descent method and multipoint shooting method, but it also has the characteristics of fast convergence when the search process is close to the real cycle in practice. In order to apply the method, a rough loop guess must be made first based on the entire topology for the cycle to be searched, and then the variational algorithm will bring the initial loop guess to evolving toward the real periodic orbit in the system. In the calculations, the Newton descent method is used to achieve stability. Two cycles can be used as basic building blocks for initialization, searching for more complex cycles with multiple circuits around the two fixed points requires more delicate initial conditions; otherwise, it will probably lead to nonconvergence. We can initialize the loop guess for longer cycles constructed by cutting and gluing the short, known cycles. For this system, such a method yields quite a good systematic initial guess for longer cycles. Even if we deform the orbit manually into a closed loop, the variational method still shows its powerfulness for good convergence. The topological classification based on the entire orbital structure is shown to be effective. Furthermore, the deformation of periodic orbits with the change of parameters is discussed, which provides a route to the periods of cycles. The present research may provide a method of performing systematic calculation and classification of periodic orbits in other similar chaotic systems.