基于Lorenz型系统族的三维自治系统的复杂动力学研究

In 1998, S. Smale compiled a list of 18 problems in mathematics to be solved in the 21st century, known as Smale’s problems, the 14th issue of which is about study of the Lorenz system. Although Melnikov''s and Shil''nikov ''s methods are used for predicting the emergence of horseshoe chaos, it is a very difficult problem to determine the existence of a saddle-focus homoclinic or heteroclinic orbit in the three-dimensional system. Our numerical studies have shown that the existence of singularly degenerate heteroclinic cycles for a suitable choice of the parameters and Periodic solutions and chaotic attractors can be found when these cycles disappear, but the specific mechanism is not yet clear. In addition, our group has confirmed that the chaotic system can generate double-scroll chaotic attractors ina very wide parameter domain with only two stable equilibria. Based on the family of Lorenz-type systems, the project will mainly consider mechanism for the chaos and the geometric structure in the kind of chaotic system. These include: theoretical analysis and calculation of singularly degenerate heteroclinic cycles and the relationship between the chaotic attractors and the cycles; the investigation of complex dynamic behavior in chaotic system with only stable equilibria. This study will provide new ideas and new target for the reasonable classification of chaotic system in the autonomous system.

1998年，美国科学院院士S.Smale提出了21世纪18个数学问题，其中第14个问题就是关于Lorenz系统的研究。虽然Melnikov方法和Shil''nikov方法是经典混沌解析预测方法，但判定三维系统是否存在鞍焦型同宿或异宿轨是个相当困难的问题。本课题组已发现：奇异退化异宿环的消失与混沌吸引子产生有一定的密切关系，但具体调控机制尚不清楚。另外，本课题组已证实某些三维Lorenz-like系统有且仅有双曲稳定平衡点时，在一定参数区域内存在双曲渐进稳定不动点与混沌吸引子共存的结果。本项目将在此基础上，以与经典Lorenz系统密切相关的三维系统为研究对象，对其混沌吸引子产生的机理及吸引子几何结构进行研究。具体内容包括：理论分析与计算奇异退化异宿环的破裂与相继产生混沌吸引子之间的关系；分析有奇点但无不稳定奇点的混沌系统的复杂动力学。此项目将为有效地对混沌系统进行合理的分类提供新思路。

目前判定三维系统是否存在鞍焦型同宿或异宿轨是个相当困难的问题，同时确定混沌系统的界也是一个重要的理论问题。本项目通过构造广义的Lyapunov函数得到了Panchev系统（类三维Lorenz-like系统）的全局指数吸引集（三维圆柱估计），并将其结应用到混沌同步中；通过二次函数控制得到了有且仅有一个双曲稳定平衡点时的混沌系统，在一定参数区域内存在双曲渐进稳定不动点与混沌吸引子共存的结果，理论上分析了其Hopf分叉过程；继续深入研究三维Qi系统的混沌动力学行为，通过分析退化Hopf分叉得到其通向混沌的道路；在已有的结果基础之上，通过对非扩散Lorenz系统进行控制， 得到了四维无奇点的简单的超混沌系统，其超混沌吸引子不与奇点共存,进一步分析了该系统的动力学行为。