摘要
随着混沌控制和反控制研究的日益深入,出现了各种新的混沌吸引子和混沌控制方法,由于混沌反控制在离散系统中的研究已经较为完善,但是在连续系统中的研究还远未达到成熟,如何实现利用反馈控制方法产生新的连续混沌吸引子,以及对连续混沌系统提出更有效的控制策略仍是人们普遍关注的问题之一。本课题目的是用混沌反控制的思想和方法构造出几类新的连续混沌系统,为混沌理论与实际应用研究提供更多的连续混沌模型。另外,研究了混沌控制和同步的一些控制策略,并结合大量的混沌系统进行了数值验证,为混沌控制与同步方法的实现做了一些很有意义的尝试。新的混沌吸引子的发现,将有助于我们加强对混沌现象的认识,丰富了混沌模型。同时,新的控制与同步方法将充实混沌控制与同步已有的方法。针对这些问题,本文做了以下几个方面的研究工作。
首先,给出了本课题研究的目的和意义,指明了本课题的主要研究内容和创新点。
其次,对混沌控制与同步作了综述性介绍。介绍了混沌学的发展；并着重介绍了连续混沌系统的研究现状和混沌控制与同步的研究现状。
接着,利用混沌反控制方法构造了五类新的连续混沌系统,分析了它们基本的动力学特性。构造的五类新连续混沌系统它们分别有如下特点：第一类连续混沌系统是通过反馈控制Lorenz系统得到的一个混沌系统,结果分析表明它仍属于Lorenz系统族；第二类连续混沌系统是通过反馈控制Lorenz系统族得到的3个混沌系统,结果分析表明它们不再属于Lorenz系统族；第三类连续混沌系统的表达式仅具有5项,它比已存在的具有6项及6项以上的三维混沌系统的表达式更简单；第四类连续混沌系统的表达式仅含有一个参数,但在这个参数变化时能产生多种涡卷形式的混沌吸引子；第五类连续混沌系统能产生4-涡卷混沌吸引子,但和已存在的2-涡卷或者4-涡卷混沌吸引子相比,第五类连续混沌系统产生的新吸引子是由瞬时混沌2-涡卷和混沌2-涡卷吸引子组成的。
紧接着,讨论了混沌系统的控制问题。首先,提出错位控制的一般方法,将系统的混沌轨迹控制到系统的平衡点和极限环；其次,提出降维控制的方法,将系统的混沌轨迹控制到系统的任意点和任意周期。该章以实例对所提的两种控制方法用数值仿真验证其有效性。当然,这两种控制方法可适用于讨论其它混沌系统的控制问题,从而说明了该分析方法的普适性。
再接着,讨论了混沌系统的同步问题。首先,提出一般混合错位投影同步方法,该方法包括完全错位同步、错位反同步、错位投影同步。并且所讨论的混沌系统可以是不同的混沌系统甚至是不同维数的混沌系统；其次,提出混沌系统的异双边匹配自适应同步方法,该方法包含了文献中已有的等同双边匹配自适应同步方法；接着,在讨论混沌系统的延迟同步中提出了混沌系统的自适应同步策略,在混沌系统同步的过程中,不仅混沌系统的参数可以被识别,而且延迟时间也能被识别；紧接着,讨论了混沌系统的混合同步,该方法包含了已有的全维同步和部分维同步,并且通过设计控制器,在同一个混沌系统中可以同时存在控制与同步现象；再接着,提出了混沌系统的自适应比例函数投影同步,该方法包含了文献中已有的函数投影同步和一般投影同步,在混沌系统同步的过程中,不仅参数可以被识别,而且比例函数也能被识别；最后给出了混沌系统同步性质的一个注记,讨论了混沌同步中的一个现象。在讨论混沌系统的同步问题中,用大量的实例对所提的同步方法验证其有效性。当然,这一部分所讨论的同步方法同样可适用于讨论其它低维或者高维混沌系统的同步问题,从而说明了这些分析方法的普适性。
最后,基于混沌系统提出了一个新的复杂动力学网络模型,由于该动力学网络模型增加了节点间的联系,因此它能较好的模拟现实复杂网络。基于LaSalle's不变集原理,给出了这个新动力学网络的模型几个自适应同步准则。
With the chaos control and anti-control increasing, and the emergence of new chaotic attractors and chaos control methods, chaos anti-control study in the discrete system has been better researched. But the research in a continuous system is far from reach maturity. How to generate the new continuous chaotic attractor using feedback control method, and present more effective control strategy for the continuous chaotic system, which is one of the issues of common concern. Purpose of this subject apply anti-control of thoughts and methods to construct several new continue chaotic system, which provide more continuous chaotic model for the practical application of chaos theory. In addition, studies of chaotic control and synchronization of some control strategies, combined with a large number of chaotic systems, numerical simulations are given to show the effectiveness of this method. The new chaotic attractors will help us enhance the understanding of chaos. Meanwhile, the new control and synchronization method will enrich the existing chaos control and synchronization methods. The paper made the following aspects of the research.
First, after introducing the motive and the significance of chaos control and synchronization, then, we gave out the main contents and the innovations of this study.
Second, chaos control and synchronization were summarized in this paper. We introduced the definition and characteristics of chaos, chaos control and synchronization features, and focused on research situation of the control and synchronization.
Next, we gain five new chaotic systems by using the method of chaotification, and some basic dynamical properties are studied. The five new chaotic systems possess the following features. The first new chaotic system is gained by feedback controlling Lorenz chaotic system, according to the definition of generalized Lorenz system, and the system still belong to generalized Lorenz systems. The second new chaotic system is gained by feedback controlling Lorenz systems, according to the definition of generalized Lorenz system, and the system does not belong to generalized Lorenz systems. The third new chaotic system has five terms, in comparison with those of the existing six-term or seven-term chaotic attractors, the new attractor is simpler and fewer terms. The fourth new chaotic system is a single parameter chaotic system, which can generate a complex 4-scroll chaotic attractor, a 3-scroll chaotic attractor, and two 2- scroll chaotic attractors with variation of single parameter. The fifth new chaotic system is four-scroll chaotic attractor in three-dimensional autonomous system, in comparison with those of the existing four-scroll chaotic attractors, the novel chaotic attractor can generate four scrolls two of which are transient chaotic and the other two of which are chaotic.
Then, we discuss control of the new chaotic system. Firstly, we present hybrid dislocated control method for stabilizing chaos to unstable equilibrium and limit cycle. Secondly, the control method of dimension reduced is presented. And the trajectories of a chaotic system can be controlled to approach arbitrary points or arbitrary target periodic orbits by the control method of dimension reduced.
Furthermore, we discuss synchronization of the new chaotic system. First, we present hybrid general hybrid projective dislocated synchronization, which includes complete dislocated synchronization, dislocated anti-synchronization and projective dislocated synchronization as its special item. The drive and response systems discussed in this paper can be strictly different dynamical systems (including different dimensional systems). Second, we investigate the different bidirectionally coupled chaotic systems, which includes identical bidirectionally coupled synchronization. Third, we investigate the problem of adaptive lag synchronization and parameters adaptive lag identification of chaotic systems. In comparison with those of the existing parameters identification schemes, the unknown parameters are identified by adaptive lag laws and the delay-time is also identified. Fourth, we investigate hybrid synchronization, in comparison with those of the existing synchronization methods, the hybrid synchronization includes full-order, reduced-order synchronization and the modified projective synchronization. What's more, the control, complete synchronization and anti-synchronization can coexist in the same system. Fifth, we investigate the problem of adaptive synchronization of chaotic systems with adaptive scaling function. In comparison with those of the existing scaling function synchronization, the scaling function is also identified. Sixth, we discuss a note on synchronization quality of chaotic system.
Finally, we investigate the synchronization of a general complex dynamical network with non-derivative and derivative coupling. Based on LaSalle's invariance principle, adaptive synchronization criteria are obtained.
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