动力系统曲率指数的分析研究

发布时间：2018-04-26 05:09

本文选题：动力系统 + 混沌　；参考：《广西大学》2017年硕士论文

【摘要】：在科技不断进步的今天,动力系统作为数学研究应用的一个热门领域,已经得到了前所未有的发展,渗透到了不同的自然科学和工程领域.混沌是典型确定的却不能够预测的非线性现象,作为非线性物理科学的一个重要分支,混沌动力学在上个世纪引起了众多学者的广泛关注,经过几十年的发展变化,混沌理论基本成型,主要包括分岔、同步、控制与反控制、混沌系统的识别及其应用.如何定量描述动力或混沌系统的代数或是几何特征是一项有用且重要的科研工作,比如Lyapunov指数、分形维数、熵等.通过对前人文献的大量阅读,本文基于Chen和Chang定义的一种新的命名为曲率指数的定量的方法来研究动力系统.(1)曲率指数被定义为动力系统的轨迹在时间演化过程中平均曲率的极限值,度量了系统的轨迹在进入吸引子时的弯曲程度,并以一个空间球体估计了吸引子的平均大小;N维系统就有N-1个曲率指数;给出了曲率指数的定义和计算方法,证明了三维空间下曲率指数的特殊形态.(2)研究了曲率指数在基本曲线和三维、四维、五维等动力系统中,对动力系统结构性变化的应用结果,表明一个或多个(维数2时)高阶的曲率指数可以分辨出系统结构变更时吸引子的拓扑结构变化.(3)研究了耦合混沌同步的问题.曲率指数能够有效地反映系统是否达到各种不同的耦合同步,并显示出系统耦合同步时耦合参数的阈值点.以耦合Rossler系统为例表明曲率指数的有效性.(4)通过模拟经典的动力系统实例并和著名的Lyapunov指数作对比,观察到这种曲率指数是非常可行的,对于动力系统的研究具有重要意义.
[Abstract]:With the development of science and technology, power system, as a hot field in mathematical research and application, has been developed unprecedented and penetrated into different fields of natural science and engineering. Chaos is a typical but not predictable nonlinear phenomenon. As an important branch of nonlinear physics science, chaos dynamics has attracted the attention of many scholars in the last century. Chaos theory is basically formed, including bifurcation, synchronization, control and anti-control, identification of chaotic system and its application. How to quantitatively describe algebraic or geometric features of dynamical or chaotic systems is a useful and important research work, such as Lyapunov exponent, fractal dimension, entropy and so on. By reading a lot of previous literature, In this paper, a new quantitative method named curvature exponent, defined by Chen and Chang, is used to study dynamic system. The degree of curvature of the trajectory of the system when it enters the attractor is measured, and the mean size of the attractor N dimensional system is estimated by a space sphere, and the definition and calculation method of the curvature exponent are given. It is proved that the special form of curvature exponent in three dimensional space is applied to the structural change of dynamic system in basic curve and in three dimensional, four dimensional and five dimensional dynamical systems. It is shown that one or more higher-order curvature exponents (dimension 2) can distinguish the topological changes of attractors when the system structure changes. Curvature exponents can effectively reflect whether the system achieves different coupling synchronization and shows the threshold points of coupling parameters. Taking coupled Rossler system as an example, the effectiveness of curvature exponent is demonstrated. 4) by simulating the classical dynamic system and comparing it with the famous Lyapunov exponent, it is observed that this curvature exponent is very feasible and has important significance for the study of dynamic system.
【学位授予单位】：广西大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O19

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