摘要
复杂拓扑结构混沌吸引子是环状多涡卷、嵌套多翅膀、网格多涡卷和多环面、多折叠环面、高维广义猫映射等多类在相空间中具有复杂拓扑结构的混沌吸引子的总称。与简单拓扑结构双涡卷或双翅膀混沌吸引子相比,环状多涡卷、嵌套多翅膀、网格多涡卷和多环面、多折叠环面等混沌吸引子的特点体现在具有十分复杂的拓扑吉构和更复杂的动力学行为。本学位论文重点研究了若干新型的复杂拓扑结构混沌及引子的生成、控制与同步及其相关的一些问题,主要内容分为以下几个方面：
1.随着对混沌研究的逐步深入,人们进一步发现了一些新型的多涡卷和多翅旁混沌系统,这些复杂拓扑结构混沌系统成为近年来非线性电路与系统领域中一个十分活跃的研究课题。本学位论文首先对几类典型复杂拓扑结构混沌吸引子的生成问题进行了深入研究,主要完成了以下三个方面的工作：
(1)提出了一个新的多涡卷超混沌Chua系统
尽管对于从三维Chua系统中产生多涡卷混沌吸引子进行了大量的研究,但是到目前为止从四维Chua系统产生多涡卷超混沌吸引子却鲜有报道。此外,构造四维超Chua系统有多种不同的方法,比如两个或两个以上三维蔡氏电路的耦合,π型电路的扩展,基于分解和基于结型场效应晶体管的方法,状态反馈控制法等。而状态反馈控制方法被认为是从已有的三维混沌系统产生不同的四维超混沌系统的好方法。鉴于上面已经完成的工作,本论文利用状态反馈控制,分别构造了带有光滑和分段光滑立方非线性项以及分段线性非线性项的四维Chua系统来产生多涡卷超混沌吸引子,并进一步研究了此系统的动力学行为,包括李亚普诺夫指数谱,分岔图和状态方程的解。此外,基于无量纲状态方程组和模块化电路设计,对构造的四维超Chua系统产生的双涡卷和三涡卷超混沌吸引子进行了电路设计。
(2)提出了一个新的多翅膀混沌系统
通过计算机模拟,对由改进的C-A投影同步得到的控制器进行修正,从而产生了一种特别的多翅膀混沌吸引子,并对受控Lorenz系统的一些基本的动力学性质进行了理论分析和数值模拟,研究结果表明此系统具有复杂有趣的混沌和超混沌行为。
(3)由多涡卷Chua系统演化得到了两个四维分数阶多涡卷超混沌Chua系统
在四维整数阶多涡卷超Chua系统的基础上进行演化得到了两个四维分数阶多涡卷超混沌Chua系统：带有光滑立方和分段光滑立方非线性项的四维分数阶超Chua系统以及带有分段线性非线性项的四维分数阶超Chua系统。通过使用分数阶微分理论和数值模拟,发现在阶数小于4的分数阶Chua系统中确实存在多涡卷超混沌吸引子。
2.在复杂拓扑结构混沌吸引子的控制与同步的研究方面,本学位论文完成了以下四个方面的工作：
(1)设计了分数阶混合反馈控制器,实现多涡卷Chua系统的混沌同步
分数阶控制器将整数阶系统转化为分数阶系统,通过提供稳定性增加系统的自由度,整数阶混合反馈控制器可以减少同步的时间,而本文中提到的分数阶混合反馈控制器充分结合了分数阶和整数阶控制器的优点。此外,利用此控制器实现了多涡卷Chua系统的混沌同步。
(2)提出了一种改进的C-A投影同步方法
当混沌系统具有一定程度的对称性时,就能达到完全同步-反同步(简称C-A同步)。基于线性分离法,对C-A同步进行改进达到C-A投影同步,也就是说,在Lorenz系统中驱动向量和响应向量达到按照比例因子变化的C-A同步。
(3)设计了统一的自适应控制器和参数更新规则,实现混沌连续时间系统的自适应全态混合函数投影时滞同步
对于投影同步,大部分研究成果集中研究的是常数比例因子,该方法基于李亚普诺夫稳定性理论和全态混合投影同步方法(FSHPS),提出了一种混沌连续时间系统的自适应全态混合函数投影时滞同步方案(简称FSHFPLS),并且设计了统一的自适应控制器和参数更新规则以达到按要求的比例函数变化的投影时滞同步。此外,还将LFRBM混沌系统的同步应用到保密通信中,数值模拟表明所提方案的有效性。
(4)改进的广义哈密尔顿系统方法的研究
通过李亚普诺夫稳定性理论和一些矩阵技术,建立一种以线性矩阵不等式(LMI)形式表达的新的充分判据,从而保证以指数收敛速度达到混沌同步。然后用此方法去对Lu系统和带有双曲正切函数的修正Chua电路的n涡卷混沌吸引子进行同步。
此外,还分别设计了分数阶积分型主动滑模控制器和线性控制器应用于响应系统实现了分数阶多涡卷超Chua系统的混沌同步。
3.研究了粒子群优化算法(Particle Swarm Optimization, PSO)作为一种进化计算技术的若干问题。本学位论文鉴于基本粒子群算法存在初始化过程的随机性以及容易陷入局部最优解的不足,对基本粒子群算法进行改进,利用混沌运动的遍历性,产生大量初始群体,从中择优出初始群体,并且在粒子群优化算法执行的过程中,对当前粒子个体产生混沌扰动,以使解跳出局部极值区间,最后用混沌粒子群算法对综合GM(1,1)参数优化模型的参数进行优化。
本学位论文的工作得到了禹思敏教授主持的国家自然科学基金(批准号：60871025,61172023)、教育部高等学校博士学科点(博导类)专项科研基金(批准号：20114420110003)、广东省自然科学基金(批准号：8151009001000060,S2011010001018)、广东省科技计划项目(批准号：2009B010800037)的资助。
Ring multi-scroll, nesting multi-wing, grid multi-scroll and multi-torus, multi-folded torus, high-dimension generalized cat map and so on in phase space are called chaotic attractors with complex topological structure. Compared with double scroll or double wing attractors with simple topological structure, the above-mentioned chaotic attractors with complex topological structure have very complex topological structure and more complicated dynamical behavior. Generation, control, synchronization and some issues concerned of several new types of chaotic attractors with complex topological structure are studied in this dissertation. The main content is devided into the following aspects:
1. As the study on chaos becomes more in-depth, some new multi-scroll chaotic and multi-wing chaotic systems arises one after the other. In recent years, these systems become very active research focuses in nonlinear circuits and systems. First, generation of several typical chaotic attractors with complex topological structure are studied in this dissertation. The following three aspects are finished:
(1) A new multi-scroll hyperchaotic Chua system is proposed
Although many approaches for generating multi-scroll chaotic attractors from 3-D Chua system have been intensively studied, there are few detailed reports regarding the multi-scroll hyperchaotic attractors from 4-D Chua system up to now. In addition,4-D Hyperchaotic Chua system can be constructed so far by some different techniques, such as two or more 3-D coupled Chua circuits, expansion ofπtype circuit configuration, both decomposition-based and junction field-effect transistor-based approaches, state feedback control and so on. It is believed that state feedback control-based approach has some advantages for generating various 4-D hyperchaotic systems from many existing 3-D chaotic systems. Moving forward from the above-mentioned accomplished works, in this dissertation, a 4-D hyperchaotic Chua system via state feedback control is constructed, by introducing with both piecewise-linear nonlinearity and smooth and piecewise smooth cubic nonlinearity. Furthermore, dynamical behaviors of this hyperchaotic system are further investigated, including Lyapunov exponents spectrum. bifurcation diagram and solution of state equations. In addition, a circuit is designed for 4-D hyperchaotic Chua system such that the double-scroll and 3-scroll hyperchaotic attractors can be physically obtained.
(2) A new multi-wing chaotic system is proposed
The controller obtained by the improved C-A synchronization is modified by computer simulations to generate a special multi-wing chaotic attractor. By the theoretical analysis and numerical simulations, some basic dynamical properties of the controlled Lorenz system are investigated. Research results show that this system has complex and interesting chaotic and hyperchaotic dynamics.
(3) Two four-dimensional fractional-order multi-scroll hyperchaotic Chua systems are proposed
Two four-dimensional fractional-order multi-scroll hyperchaotic Chua systems are gained based on the four-dimensional integer-order multi-scroll hyperchaotic Chua systems, including a four-dimensional fractional-order hyperchaotic Chua system with both smooth and piecewise smooth cubic nonlinearity, and a four-dimensional hyperchaotic Chua system with piecewise-linear nonlinearity. By utilizing the fractional calculus theory and computer simulations, it is found that multi-scroll hyperchaotic attractor exists in this fractional-order Chua system with order less than 4.
2. On study of control and synchronization of chaotic attractors with complex topological structure, the following four aspects are finished in this dissertation:
(1) A fractional order hybrid feedback controller is designed to synchronize the multi-scroll Chua system
The fractional controller converts the system behavior with integer derivatives into a system with fractional derivatives. This increases the degree of freedom of the system by means of providing the stability. In addition, an integer hybrid feedback controller is used to decrease the time of synchronization. The proposed controller makes use of the good of both integer and fractional order controllers. The performance of the controller is shown via numerical simulation when it is used to synchronize the multi-scroll Chua system.
(2) An improved C-A projective synchronization scheme is proposed
The complete synchronization and anti-synchronization(C-A synchronization) can be achieved when chaotic system has some degree of symmetry. In this paper, C-A synchronization is improved based on the linear separation method to achieve the C-A projective synchronization, that is, the drive and response vectors synchronize up to the scaling factor in the Lorenz system.
(3) A unified adaptive controller and parameters update law are designed for achieved the adaptive full state hybrid function projective lag synchronization (FSHFPLS) in chaotic continuous-time system
For the projective synchronization, most of research efforts have concentrated on studying the constant scaling factor. Based on the Lyapunov stability theory and FSHPS, this scheme is investigated in chaotic continuous-time system, and a unified adaptive controller and parameters update law are designed for achieved the projective lag synchronization up to a desired scaling function. In addition, a scheme for secure communication is presented. Numerical simulations are performed to verify and illustrate the analytical results.
(4) An improved generalized Hamiltonian systems approach is proposed
By using the Lyapunov stability theory and some matrix techniques, a new sufficient criterion, formulated in the LMI form, is established. The new sufficient criterion can guarantee that chaotic synchronization is achieved at an exponential convergence rate. Then this improved approach is used to synchronize Lu system and n-scroll chaotic attractors in a modified Chua's circuit with hyperbolic tangent functions.
Besides, synchronization between the fractional-order multi-scroll hyperchaotic Chua systems is achieved by a fractional active sliding mode controller and a suitable linear controller applied to the response system.
3. Particle Swarm Optimization (PSO) which is an evolutionary computation is studied in this dissertation. Its research and application are not interrupted since it was put forward. In this dissertation, there are some problems of the randomness of initialization process and easy to fall into a local optimal solution when using PSO algorithm. In this paper, PSO algorithm is improved. The basic idea is:at the beginning, a great deal of initial population are produced by ergodicity of chaos, and then the initial particle swarm is preferred out; during the running time of PSO algorithm, to adjust the current particle by chaos so that the solution jumps out local convergence. In the end,Chaos PSO algorithm is proposed to optimize the integrated GM(1,1) parameter optimization model.
This work was supported by the National Natural Science Foundation of China under Grant Nos.60871025 and 61172023, the Specialized Research Foundation of Doctoral Subject Point of Education Ministry under Grant No.20114420110003, the Natural Science Foundation of Guangdong Province under Grant Nos. 8151009001000060 and S2011010001018, and the Science and Technology Program of Guangdong Province under Grant No.2009B010800037.
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