吕金虎
论文题目：一个统一混沌系统及其研究
作者简介：吕金虎，男，1974年09月出生，2000年09月师从于中国科学院数学与系统科学研究院张锁春教授，于2002年08月获博士学位。
摘
要
本论文提出了一个新的混沌系统，它是介于著名的Lorenz系统和Chen系统之间的临界系统。这个新系统在Lorenz系统和Chen系统之间架起了桥梁。同时，我们提出了一个单参数的统一混沌系统——Lorenz系统族，它包含Lorenz系统和Chen系统作为参数区间的两个极端系统，临界系统作为一个过渡系统。我们还发现了Chen吸引子和临界吸引子的复合结构，即它们都是由两个简单的混沌吸引子——左半吸引子和右半吸引子——经过一个镜象映射相互融合而成。本文对Chen系统，临界系统和统一混沌系统进行了深入的研究，包括它们的基本动力学行为、周期窗口、局部分岔、复合结构以及它们的控制与同步问题。基于线性耦合多振子之间的互同步问题的研究，本文提出了研究这一类问题的新方法——模式分解法，并应用上述方法深入研究了两个及三个线性耦合的恒等Lorenz系统、Chen系统和临界系统之间的互同步问题。首次从理论上给出了上述三个典型系统混沌同步的几个严格的充分条件。本文还提出了一类新的简单的转换分段线性反控制器，它能够使一个简单的线性系统在三维空间中同时产生n个，或一个具有多吸引域融合的混沌吸引子。以上是本论文的主要创新点，当然作为完整的博士论文，自然还包含有本领域内研究的历史背景，发展现状和已取得的成果介绍，以及对今后进一步研究的展望。
Abstract
This paper produces a new chaotic system, which is a critical system and bridges the gap between the Lorenz and Chen systems. At the same time, we introduces a unified chaotic system with single parameter, called Lorenz system families, which contains the Lorenz and the Chen systems as two dual systems at the two extremes of its parameter spectrum, and critical system as a transition system. We have also found the compound structures of the Chen and critical attractors, which are both obtained by merging together two simple attractors —— the left and right attractors after performing a mirror operation. This paper further investigates the basic dynamic behaviors, periodic windows, local bifurcations, compound structure and the relative problems of chaos control and synchronization for the Chen, critical and unified systems. Based on the study of the chaos synchronization between linearly coupled multi-systems, this paper presents a new method —— mode decomposition for analyzing the stability of synchronization solution. Using this method, this paper further investigates the chaos synchronization between two and three linearly coupled Lorenz, Chen and critical systems. Also, some sufficient conditions for chaos synchronization are gained from rigorously mathematical analysis for the first time. Finally, this paper introduces several new chaos generators, switching piecewise-linear controllers, which can create simultaneously n, or one with multiple merged basins of attraction chaotic attractors from a 3-D linear system within a wide range of parameter values. The main innovative points of this paper are listed above. As an integral thesis, it also contains the historical background, research progress, relative results in this field, and the prospect for the future study.
