摘要
混沌系统的有界性是动力系统中的一个重要概念,在研究奇点的唯一性、奇点的全局渐近稳定性、奇点的全局指数稳定性、吸引子的李雅普诺夫维数、吸引子的豪斯道夫维数、周期解的存在性、周期解的控制等方面有着重要的应用;然而根据作者所知由于高阶混沌系统代数结构的复杂性,对高阶混沌系统有界性的研究是一件困难的事情;基于以上原因,将研究来自于数学物理模型中一类高阶混沌系统和一类三维洛伦兹型混沌系统的有界性;基于李雅普诺夫稳定性理论,证明了两个混沌系统的解是最终有界的;创新点在于不仅证明了两类混沌系统是最终有界的,而且分别给出了两类混沌系统最终有界集的一族解析表达式;研究结果为混沌系统在工程中的应用和电路设计提供了理论依据。
The boundedness of a chaotic system is one of the fundamental concepts of dynamical systems,which plays an important role in investigating the uniqueness of singularity,global asymptotic stability,global exponential stability,estimating the Lyapunov dimension of attractors,the Hausdorff dimension of attractors,the existence of the periodic solution,its control and synchronization and so on. However,as far as the authors know,there are only a few papers dealing with bounds of high-order chaotic systems due to their complex algebraic structure. To sort this out,in this paper,we study the bounds of a high-order chaos system and a 3D Lorenz-like system arising in mathematical physics. Based on Lyapunov stability theory,we show that the solutions of two systems are bounded when t → ＋ ∞. The innovation of the paper is that we not only prove that two systems are globally bounded for all the parameters,but also give a family of mathematical expressions of ultimate bound sets of two systems with respect to its parameters. The results of this paper provide a theoretical basis for the application and circuit design of the chaotic system in engineerings.
作者
张付臣
陈睿
陈修素
李如意
曾偲
何毅章
LI Ru-yi3 , ZENG Cai3 , HE Yi-zhang3(1. College of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing 400067, China;2. Information Office, Chongqing Technology and Business University, Chongqing 400067, China; 3. Chengdu No. 7 Middle School, Chengdu 610041, Chin)
出处
《重庆工商大学学报（自然科学版）》
2018年第2期14-17,22,共5页
Journal of Chongqing Technology and Business University:Natural Science Edition
基金
国家自然科学基金(11501064)
中国博士后基金(2016M590850)
重庆市博士后科研项目特别资助项目(XM2017174)
关键词
高维混沌模型
奇点
稳定性
全局吸引域
high-order chaos model
singularity
stability
global domain of attraction
