【作者】 邓奎彪；

【导师】 禹思敏；

【作者基本信息】 广东工业大学 ， 控制理论与控制工程， 2013， 博士

【摘要】 混沌是非线性系统中存在的一种运动形式,它是在确定性中出现的一种貌似不规则、内在随机的运动、既普遍存在又极其复杂的现象。混沌系统的最大特点就是系统演化对初始条件敏感性,因此混沌系统的未来行为是短期可预测长期不可预测的。混沌运动模糊了确定性运动和随机运动的界限,它具有不可重复性,局部不稳定而整体稳定。在混沌的研究中,混沌系统的有界性是前沿的研究课题之一。混沌系统的最终有界在混沌的定性行为的研究中有着重要的作用,然而估计一个动力系统的全局指数吸引集和最终有界是一项比较困难的任务。一个混沌系统是有界的,意味着混沌吸引子在相空间是有界的,并且所估计的界在混沌控制、同步及其应用中有非常重要的作用。本学位论文研究混沌系统的界、分岔、控制与同步若干问题,主要内容分为以下几个方面：1.研究了修正超混沌Lu系统的Hopf分岔。首先,得到Hopf分岔存在的条件。然后,用规范形理论分析了Hopf分岔的类型、分岔的方向、分岔的近似周期解。最后,用数值仿真也验证了理论分析的正确性。2.统一混沌系统在参数已知和参数未知情况下的同步问题。基于LaSalle不变集原理和线性矩阵不等式,当参数已知时,用一个带有自适应增益的线性反馈控制器实现了两个相同的统一混系统的同步,其次,用一个自适应控制器实现了参数未知时统一混沌系的同步,最后用仿真结果验证了文中的理论分析。3.利用Shilnikov定理构造一个含有平方项的三维混沌系统,且系统有两个平衡点。构造的过程表明该混沌具有Smale马蹄(同宿轨混沌),满足同宿轨道Shilnikov定理条件,数值仿真验证了该方法的有效性。最后,用待定系数法证明系统中存在Smale马蹄,因而是Shilnikov意义下的混沌。4.动态输出反馈的方法来控制一类混沌系统。利用李亚普诺夫稳定理论,结合线性矩阵不等式优化方法,分析了系统的稳定性。数值仿真验证了设计方法的可行性。5.针对一类参数随时间变化,具有外部噪声干扰,不确定混沌系统,用鲁棒自适应方法实现了异结构混沌系统的同步和参数辨识。利用李亚普诺夫稳定理论,设计了鲁棒自适应控制器和参数自适应律。最后以Lorenz系统和Chen系统为例子进行仿真验证,数值仿真验证了所设计方法具有鲁棒性和有效性。6.引入了一个典型自治混沌系统,该系统包含2个系统变量乘积的非线性函数项。当系统参数变化时,系统属于广义Lorenz系统,又能够不属于广义Lorenz系统.用分岔图,Poincare截面图和Lyapunov指数对典型系统进行了分析。最后,对典型系统的界进行了估计,给出当参数a,b,c, d全为正数时界的表达式。7.运用参数法给出了一个三维自治系统在满足一定条件下两变量上界的估计并证明,在此基础上给出了该系统三个变量的一个上界估计定理并给予了证明,给出了能产生不稳定的鞍焦平衡点参数的条件,有一个正的李氏指数,从而构造出一个混沌系统。8.构造了一个具有光滑四次函数的自治混沌系统,系统中包含两个系统变量乘积的非线性函数和三个实平衡点。讨论了平衡点的稳定性、用Lyapunov维数、Poincare截面图、Lyapunov指数谱和分岔图分析了系统的动力学特性。最后,基于拓扑马蹄映射理论,证明自治混沌系统存在拓扑马蹄。9.提出了一个新的混沌系统,其基本的动力学性质,用平衡点的稳定性、Lyapunov维数、Poincare截面图、Lyapunov指数谱和分岔图进行详尽的分析。然后用脉冲控制将新的系统稳定到原点。本学位的论文工作得到了禹思敏教授主持的主持的国家自然科学基金(批准号：61172023)和广东省自然科学基金(批准号：S2011010001028)资助。更多还原

【Abstract】 Chaos is an existing phenomenon in nonlinear systems, which arises from deterministic system in a random and non-reguar fashion. It is ubiquitous and complex. The most reamarkable chaotic character is that chaotic systems exhibit high sensitivity to initial conditions. In other words, a nonlinear system called chaotic system is unpredictable. Chao breaks the rigorous division between deterministic systems and random systems, which is un-repeatable, local un-stable and whole stable. In the study on chaos, The bound of chaotic systems have become front research topics, extremely challenging. The ultimate bound of a chaotic system is important for the investigation of the qualitative behavior of a chaotic system, however, estimating the globally exponentially attractive and ultimate bound set for a dynamic system is a quite challenging task in general. A chaotic system is bounded, in the sense that its chaotic attractor is bounded in the phase space, and estimate of its bound is important in chaos control, chaos synchronization and applications. On the one hand, Hopf bifurcation of a hyperchaotic system is studied in this dissertation, on the other hand, bound, control, synchronization and some issues concerned of several novel chaotic systems are investigated in this paper.The main content is divided into the following aspects:1. The Hopf bifurcation of a new modified hyperchaotic Lii system is investigated. Firstly, A detailed set of conditions are derived, which guarantee the existence of the Hopf bifurcation. Furthermore, by applying the normal form theory, the direction and type of the Hopf bifurcation, and the approximate expressions of bifurcating periodic solutions and their periods are determined. In addition, numerical simulation results supporting the theoretical analysis are given.2. Two novel synchronization criterions of the unified chaotic systems with known or unknown parameters is investigated. Based on LaSalle’s invariance principle and linear matrix inequality (LMI) formulation, firstly, a simple linear feedback controller with the updated gain is obtain to make the state of two identical unified chaotic systems asymptotically synchronized, secondly, a simple adaptive controller is proposed for synchronization of the unified chaotic systems with unknown parameters. Finally, simulation results are given to verify the theoretical analysis.3. Based on the Shilnikov theorem, a new three-dimensional square chaotic system, which has tow equilibrium poits. There is hyperbolic saddle focus. The formation mechanism shows that this chaotic system has Smale horseshoes (homoclinic chaos). Numerical simulations are given to show the effectiveness of the theoretical results. Finally, Smale horseshoses has been found in the system using undetermined-coefficient method, and it is chaotic in the sense of Shilnikov.4. In this work, a dynamic outputs feedback controller for a class for a class of chaotic systems is developed for the first time. For stability, a well-known Lyapunov stability theorem combining with linear matrix inequality optimizaion is utilized. A numerical simulation is presented to show the feasibility of the proposed scheme.5. A robust adaptive control is proposed to realize chaos synchronization and parameters identification between two different chaotic systems with uncertainties, bounded time-varying unknown parameters, and noise perturbation. Based on Lyapunov stability theory, a robust control law and a parameters identification scheme are presented. Finally, numerical esults are presented for the Lorenz, Chen systems. Numerical simulations are given to demonstrate the robustness and efficiency of the proposed method.6. A typical autonomous chaotic system is firstly presented in this paper which contain two smooth quadratic terms of systemic variables. When the parameters of system are changed, the typical system belongs to or does not belong to the generalized Lorenz system. Basic dynamical properties of the typical system are studied in terms of numerical simulation, Lyapunov exponent spectrum, bifurcation diagrams and Poincare section diagrams. At last, the ultimate bound of typical system is estimated and the expression of the bound is derived for all the positive values of its parameters a,b,c,d.7. At first, under the certain condition, the theorem about an upper bound estimate of two variables was given by parameterization and proved. Secondly, the theorem about and upper bound estimate of three variables was proposed. Then, the condition of generating unstable saddle focus equilibrium points is presented. Therefore, a chaotic system was constructed.8. A novel chaotic system which contains two smooth biquadratic terms of systemic variables and three real equilibriums is presented. The characters of three equilibriums are discussed. The dynamic properties of the novel system are investigated via theoretical analysis, numerical simulation, Lyapunov dimension, Poincare diagrams, Lyapunov exponent spectrum and bifurcation diagrams. Finally, in the light of topological horseshoe map theory, the paper analyses the existence of topological horseshoe in the presented autonomous chaotic system.9. A chaotic system is presented and the dynamic properties of the novel system are investigated in term of Lyapunov dimension, Poincare diagrams, Lyapunov exponent spectrum and bifurcation diagrams. At last, the impulsive control of the system is global asymptotical stable at origin.更多还原