【作者】 罗海庚；

【导师】 廖晓昕；

【作者基本信息】 华中科技大学 ， 控制理论与控制工程， 2007， 博士

【摘要】 本文针对一类特殊的Loenz型自治微分方程,提出了一个基于Lyapunov理论的辅助搜索方法。主要思想是通过构造满足一定条件的Lyapunov函数的导数,进而反推出系统方程的形式。和已有的混沌化方法比较,新方法简便可行,有一定的理论依据和系统性,大大缩小了在复杂数学模型中搜寻混沌行为的范围,并容易推广到高维系统。利用这个方法,发现了3个新的Lorenz型混沌系统。新系统都是三维二次自治耗散系统,其线性部分系数矩阵的主对角元都是负数,且有三个或四个非线性项。因此和已有的Lorenz型混沌系统比如Lorenz系统、Ro¨ssler系统、Chen系统、Lu¨系统和LC系统类似,但有明显的区别。利用平衡点的局部和全局稳定性分析、系统相轨迹、Lyapunov指数、Poincare′映象、分岔图等理论和数值方法详细分析了新混沌系统的动力学行为和特征。结果显示,第一个新系统具有不动点与极限环、不动点与混沌吸引子以及极限环与混沌吸引子共存的现象,第二个系统有一个三螺旋吸引子,而第三个系统具有复杂的多层锥形结构。因此新混沌系统的吸引子拓扑结构较为复杂。对于第一个和第二个系统,给出了全局指数吸引集,并进行了严格的数学证明,这个结论在一般混沌系统中是很难得到的。研究了新混沌系统和LC系统不稳定平衡点的镇定。讨论了基于线性和非线性反馈实现系统不稳定平衡点的指数镇定,得到了一系列简便的代数充分条件。然后针对参数不确定情形,研究了基于自适应方法的混沌控制问题,应用Lyapunov稳定性理论,给出了针对一般混沌系统平衡点自适应控制的渐近稳定的条件。并且研究了脉冲方法镇定系统不稳定平衡点的控制器设计策略,给出了系统稳定时脉冲间隔与控制器增益的关系。研究了新混沌系统和LC系统的同步。设计了非线性反馈控制器,保证两个具有不同初值的相同系统实现全局指数同步,得到了一系列简便的代数同步判据。当系统参数未知时,基于自适应控制策略,得到了系统全局渐近同步的代数充分条件和控制器设计方法;考虑了如果系统参数未知且是时变的,基于自适应控制和滑模控制思想以及参数变化有界但界未知的假设,设计了新的自适应滑模型控制器,实现了系统的鲁棒同步。讨论了几个时滞混沌系统动力学特性。利用超越特征方程讨论了时滞logistic模型平衡点与时滞相关和与时滞无关的局部稳定性判据,并针对时滞参数用中心流形定理和规范形理论确定了Hopf分岔的方向和首个Hopf分岔点处分岔周期解的稳定性。提出了一个新的四神经元时滞混沌神经网络新模型,基于Lyapunov稳定性理论和数值动力学分析方法分析了系统的动力学行为。最后介绍了一个简单中立型混沌神经网络,就我们所掌握的材料,直至目前尚未有中立型混沌系统的研究报道,由于理论分析和数值分析都有很大困难,这个系统还有待进一步深入研究。更多还原

【Abstract】 Based on Lyapunov theory, an new auxiliary search method is put forward for aspecial kind of Lorenz-type autonomous differential equations. The basic idea is toobtain the expected form of system equations, by means of constructing the derivativeof Lyapunov function which satisfies some determinate conditions. The new methodis very simple and convenient. The searching range is extremely reduced for findingchaotic behavior in complex mathematical model. Utilizing the new method, three newLorenz-type chaotical system is found. They are three dimensional quadratic dissipa-tive system with three or four quadratic terms and negative main diagonal elements ofcoefficient matrix in the linear part of the system. Therefore, they are similar to theexisting Lorenz-type systems, such as Lorenz system, Ro¨ssler system, Chen system, lu¨system and LC system, but explicitly different from them. The dynamical behaviors areinvestigated in terms of local and global stability analysis, phase trajectories, lyapunovexponents, Poincare′mapping and bifurcation diagrams. The results show there is co-existence phenomena of stable equilibrium, limit loop and chaotic attractor in the firstsystem, a 3-scroll chaotical attractor in the second, and a multilayer taper attractor in thethird. However, there are more complex chaotic attractors in the new systems. Further-more, the global attractive sets are found for the first and the second system, and strictmathematical proofs are given.Stabilization for unstable equilibria of the new systems and LC system is studied.Globally exponential stabilization of euilibria are discussed based on linear or nonlin-ear state feedback, and a series of simple algebraic criteria are obtained. The adaptivecontrol technique of the systems with unknown parameters is considered. Furthermore,by means of Lyapunov stability theory, an asymptotically stable criterion for a generalchaotical system is obtained. Finally, approach of controller design for chaos controlvia impulsive control technique is discussed, and the relation of impulse interval andcontroller gain is given if the controlled system is stable.The chaos synchronization of the new systems and LC system is investigated. Non-linear feedback controllers are presented to achieve globally exponential synchronization for two same chaotic systems with different initial conditions. Several algebraic suffi-cient conditions are developed. Adaptive synchronization strategy is introduced whenthe parameters of systems are all unknown. An adaptive controller is designed, whichcan guarantee globally asymptotic synchronization. If the unknown parameters varywith time, assuming that the varying parameters are bounded but the boundedness is un-known, the adaptive sliding mode type controller is designed, and globally asymptoticrobust synchronization is also realized.Dynamical behaviors of several time-delayed chaotical system is studied. Basedon transcendant characteristic equation associated with the Halanay inequality, the localstability criteria, delay independent or delay dependent, is obtained for the equilibria ofdelay logistic equation. The stability of bifurcation periodic solutions and the directionof Hopf bifurcation are determined by applying the normal form theory and the centermanifold theorem. Chaotic behavior of the parameterized Logistic differential systemswith a single delay is detected by numerical examples. A new chaotical time-delayrecurrent neural network with four neurons is put forward. Its dynamical propertyis investigated via both Lyapunov theory and numerical simulation method. Finally,a simple neutral chaotical system is presented. To our best knowledge, no neutralchaotical system is reported by far.更多还原