几类复交替系统分形的控制与同步

发布时间：2018-04-29 12:52

本文选题：平面交替系统 + 广义交替系统　；参考：《山东大学》2016年博士论文

【摘要】：自然界中的许多演变过程可以利用离散动力系统建模进行研究,例如自然界中四季交替现象,经济学中的股票走势以及生物学中的肿瘤模型等等.值得注意的是,由于其演变过程的复杂性及其自身存在的多种相互作用,仅用一个动力系统来模拟演变过程是不准确的.因此,自然过程的演变应该由不同动力学系统的交替迭代来模拟.本文主要讨论几类交替系统的Julia集,研究其Julia集的控制与同步问题.1.平面交替Julia集的控制与同步讨论一种基本的平面交替系统,根据平面Julia集的基本性质及不动点理论,讨论了平面交替系统不动点的稳定性.在系统不动点可以计算的情况下,利用辅助反馈控制法和梯度控制法实现了平面交替Julia集的控制.由于非线性系统的复杂性,不动点往往很难求出,此时可以用最优控制法可以实现控制目的.根据平面交替Julia集同步的定义,利用梯度控制法和最优控制法,通过控制平面交替Julia集的轨迹同步,从而实现平面交替Julia集的同步.最后,用线性耦合法实现了平面交替Julia集的耦合.2.广义交替Julia集的控制与同步通过研究广义交替系统,给出了广义交替Julia集的定义及基本性质.通过对广义交替系统的不动点的稳定性分析,采用辅助反馈法,梯度控制法和最优控制法实现了对广义交替Julia集的控制.此外,利用梯度控制法和最优控制法,实现了不同广义交替Julia集的轨迹同步,从而实现了广义交替Julia集的同步.最后,用线性耦合法实现了广义交替Julia集的耦合.3.空间交替Julia集的控制与同步通过讨论空间交替系统,给出了空间交替Julia集的定义及基本性质.通过辅助反馈控制方法实现了空间交替Julia集的控制.引入了广义同步的概念,通过线性反馈法实现了不同空间交替Julia集的线性广义同步.最后,利用非线性反馈法实现了不同空间交替Julia集的非线性广义同步.4.一维复Logistic映射的Julia集的控制与同步以一维复Logistic映射为模型,讨论该映射在复平面C上迭代产生的Julia集,并利用梯度控制法,最优控制法和辅助参考反馈控制法实现了对一维Logistic映射的Julia集的控制.根据上广义交替Julia集的同步的概念,讨论了不同一维复Logistic映射的Julia集的同步,并且分别采用梯度控制法和最优控制法通过控制一维复Logistic映射的迭代轨道,实现了不同一维复Logistic映射的Julia集间的同步.综上所述,本文主要利用辅助反馈法,梯度控制法和最优控制法,实现了几类不同交替系统Julia集的控制与同步,丰富了分形理论,为更好地理解自然界中的非线性现象提供了理论基础.
[Abstract]:Many evolution processes in nature can be studied by using discrete dynamic system modeling, such as the alternation of four seasons in nature, stock trends in economics and tumor models in biology, and so on. It is worth noting that because of the complexity of its evolution process and its own existence of multiple interactions, it is not accurate to use only one dynamic system to simulate the evolution process. Therefore, the evolution of natural processes should be simulated by alternate iterations of different dynamic systems. In this paper, we mainly discuss the Julia sets of several classes of alternating systems, and study the control and synchronization of their Julia sets. The Control and synchronization of Planar alternating Julia sets A basic plane alternating system is discussed. According to the basic properties of plane Julia sets and the fixed point theory, the stability of fixed points of planar alternating Julia sets is discussed. When the fixed points of the system can be calculated, the auxiliary feedback control method and the gradient control method are used to realize the control of the plane alternating Julia sets. Because of the complexity of the nonlinear system, the fixed point is often difficult to find out, and the optimal control method can be used to achieve the control goal. According to the definition of plane alternating Julia set synchronization, by using gradient control method and optimal control method, the synchronization of plane alternating Julia sets is realized by controlling the trajectory synchronization of plane alternating Julia sets. Finally, the linear coupling method is used to realize the coupling. 2. 2 of the planar alternating Julia sets. Control and synchronization of Generalized alternating Julia sets the definition and basic properties of generalized alternating Julia sets are given by studying the generalized alternating systems. Based on the stability analysis of fixed points of generalized alternating systems, the control of generalized alternating Julia sets is realized by using auxiliary feedback method, gradient control method and optimal control method. In addition, the trajectory synchronization of different generalized alternating Julia sets is realized by using gradient control method and optimal control method, thus the synchronization of generalized alternating Julia sets is realized. Finally, the coupling. 3 of the generalized alternating Julia set is realized by linear coupling method. The control and synchronization of space alternating Julia sets the definition and basic properties of space alternating Julia sets are given by discussing the space alternating systems. The control of the space alternating Julia set is realized by the auxiliary feedback control method. The concept of generalized synchronization is introduced and the linear generalized synchronization of alternating Julia sets in different spaces is realized by linear feedback method. Finally, the nonlinear generalized synchronization of alternating Julia sets in different spaces is realized by nonlinear feedback method. 4. The control and synchronization of Julia sets of one-dimensional complex Logistic maps are modeled by one-dimensional complex Logistic mappings. The Julia sets generated by iterations on the complex plane C are discussed, and the gradient control method is used. The optimal control method and the auxiliary reference feedback control method are used to control the Julia set of one dimensional Logistic map. According to the concept of synchronization of upper generalized alternating Julia sets, the synchronization of Julia sets of different one-dimensional complex Logistic maps is discussed. The gradient control method and the optimal control method are used to control the iterative orbits of one-dimensional complex Logistic maps, respectively. Synchronization between Julia sets of different 1D complex Logistic maps is realized. To sum up, this paper mainly uses the auxiliary feedback method, gradient control method and optimal control method to realize the control and synchronization of the Julia sets of several different alternating systems, which enriches the fractal theory. It provides a theoretical basis for a better understanding of nonlinear phenomena in nature.
【学位授予单位】：山东大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O189;O231

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