混沌系统全局吸引集的新结果及对混沌控制与同步的应用

项目名称： 混沌系统全局吸引集的新结果及对混沌控制与同步的应用

项目编号： No.11426047

项目类型： 专项基金项目

立项/批准年度： 2015

项目学科： 数理科学和化学

项目作者： 张付臣

作者单位： 重庆工商大学

项目金额： 3万元

中文摘要： 俄罗斯学者Leonov和廖晓昕等人对于著名的Lorenz系统族的全局指数吸引集的得出，关键在于寻求合适的Lapunov函数，使该Lapunov函数关于系统正半轨线的导数在吸引域内广义负定，而在吸引域外广义正定。他们的方法成立的前提是Lorenz系统族的线性部分系数矩阵的主对角线的元素全为负数，而大部分混沌系统的线性部分系数矩阵的主对角元往往含有正数或零元素（如Chen系统和Lü、统一混沌系统等）。对于这种类型的混沌系统的全局指数吸引集问题，现在很少有结果。本项目主要研究了一类在原点的线性化矩阵的主对角线上的元素既有负数又有零元素的系统的全局指数吸引集和另一类在原点的线性化矩阵的主对角线上的元素既有负数又有正数的混沌系统的全局指数吸引集，推广了廖晓昕等人关于全局指数吸引集研究的结果。

中文关键词： 李雅普诺夫稳定性；广义李雅普诺夫函数；全局吸引集；混沌控制；混沌同步

英文摘要： The key to study the global exponential attractive sets of the Lorenz family of chaotic systems by G.A. Leonov and Liao is to research for the generalized posi -tive definite and radially unbounded Lyapunov functions. The derivative of the Lyapunov functions is generalized negative. And the elements of main diagonal of the Jacobian matrix of the Lorenz family of the chaotic systems at the origin are all negative. But for most dynamical systems, the elements of main diagonal of the Jacobian matrix of the system at the origin are not always negative, such as the Chen and the Lüystem. As far as we know, there is seldom results for papers to talk about the problem for this case. In this project, we will address the global exponential attractive sets for two classes of dynamical systems. The elements of main diagonal of the Jacobian matrix of the system at the origin are both negative and zero for one class of dynamical systems and the elements of main diagonal of the Jacobian matrix of the system at the origin are both negative and positive for another class of dynamical systems. We generalize Liao's results of the global attractive set.

英文关键词： Lyapunov stability；generalized Lyapunov functions；global attractive sets；chaos control；chaos synchronization