广义Lorenz系统族解的有界性研究

Ultimate boundedness of a chaotic dynamical system and its bound estimate is an important aspect of the qualitative theory of dynamical systems, which plays an important role in estimating the Lyapunov dimension of chaotic attractors and the Hausdorff dimension of the chaotic attractor, chaos control, chaos synchronization and so on. It has caused the wide attention of scholars from both at home and abroad.This project intends to study the boundedness solutions of the generalized Lorenz family of systems especially for the Lü system according to Lyapunov function stability theory and the optimization theory on the basis of Leonov, Liao, Yu and Nijmeijer. The approach for constructing the Lyapunov functions that applied to the former systems will not work for the generalized Lorenz family of systems. We will overcome this difficulty by adding a cross term to the Lyapunov functions of the generalized Lorenz family of systems. This result will contain the existing results as special cases. Also, we get a smaller boundedness solutions of the generalized Lorenz family of systems by using the idea of intersection in set theory.. The applicant has a solid foundation on the boundedness solutions of the generalized Lorenz family of systems and has published more than 20 papers in the international science citation index journals, such as 《Discrete and Continuous Dynamical Systems-B》、《Qualitative Theory of Dynamical Systems》、《Physical Review E》、《International Journal of Bifurcation and Chaos》 （the detailed information see the catalog of the published papers).

微分方程最终有界性研究是微分方程定性理论研究的一个重要方面，在混沌吸引子Lyapunov维数、Hausdorff维数估计、混沌控制、同步等方面有许多重要的应用。本项目拟在前人研究基础上，主要运用Lyapunov稳定性理论研究广义Lorenz系统族特别是Lü系统有界性，而此时前面学者构造的Lyapunov函数方法已经不再适用，我们将试图引进一个交叉项来构造广义Lyapunov函数族，此结果将推广和包含前面学者关于广义Lorenz系统族有界性研究的结果，并且将利用集合交集思想得到其界的一个较小估计。申请者本人已在动力系统专业期刊《Discrete and Continuous Dynamical Systems-B》、《Qualitative Theory of Dynamical Systems》、《Physical Review E》、《IJBC》等SCI期刊发表论文20余篇（详见目录)。

广义Lorenz系统族解的有界性是动力系统定性理论研究的一个重要方面，也是广义Lorenz系统族同步研究的理论基础，在广义Lorenz系统族的吸引子维数估计、混沌控制、混沌同步等方面有着重要的应用。本项目在前人研究基础之上，主要运用Lyapunov稳定性理论、最优化理论和迭代定理对广义Lorenz系统族标准型的有界性进行了一系列研究，得到了广义Lorenz系统族解的最终界，得到了著名Lorenz混沌系统解的最终界的新结果，推广了廖晓昕、郁培、 Nijmeijer 等人关于Lorenz系统有界性研究的结果。同时，对吕氏混沌系统解的有界性公开性问题进行了探讨，得到了吕氏混沌系统的系统参数满足一定条件时吕氏混沌系统的解最终有界的结论。本课题发表20余篇SCI论文。同时，本项目对广义Lorenz系统族解的有界性的研究方法可以推广到其它混沌系统。本项目的研究结果为广义Lorenz系统族的混沌控制、混沌的图像加密、混沌同步研究提供了重要的理论基础。