混沌控制与同步的方法研究
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摘要
混沌控制与同步是非线性动力学研究的前沿课题。有些时候,混沌是有用的,比如混合过程、热传导;但混沌常常是有害的或不期望的,要对其进行控制。混沌控制不仅有学术价值还有实际应用的意义。近20年来,在混沌控制与同步研究领域,人们已经提出了各种各样的控制方法,但是至今还没有统一的控制方案,这值得进一步深入地探索。本文基于这样的目的,提出和发展了几种混沌控制与同步的方法。
简要回顾了混沌动力学研究的历史,介绍了三个常见的混沌的定义。着重介绍了混沌的两个基本特性——初值敏感性和运动轨道的复杂性,以及研究混沌的重要工具和方法,比如Lyapunov指数、相图、功率谱、拓扑熵、Poincare截面、奇怪吸引子等等。引进了几个稳定性理论,为以后的应用作了铺垫。
发现了两个线性耦合Lorenz混沌系统会发生完全同步与反同步共存的现象。在此基础上,找到了出现这种现象的内在原因,研究了出现这种同步现象时混沌系统的基本特性。借助线性稳定性理论、Lyapunov稳定性理论以及数值方法,探讨了线性耦合混沌系统产生完全同步与反同步共存的稳定性问题,给出了线性耦合混沌同步的充分条件。
研究了广义混沌同步问题。利用非线性反馈控制方法,给出了任意两个混沌系统实现广义同步的充分条件,并结合两个超混沌Chen系统的广义同步和两个非自治的统一混沌系统的广义同步对提出的控制方案进行了验证;对一个动力系统而言,有时希望调节系统运动的幅值,同时又不想改变系统运动的其它特性,比如吸引子的形状。在这种情况下,一般的控制方法很难做到。本文引进了一个变换,就可以得到一个反馈控制力。在该控制力的作用下,不管系统是周期的还是混沌的,均可以人为地改变系统运动的幅值,同时吸引子形状保持不变;对于系统具有不确定因素时,给出了任意两个混沌系统实现广义同步的自适应控制方法,并借助Lyapunov稳定性理论进行了严格的证明。数值模拟进一步验证了所给自适应控制方法的有效性。
研究两个耦合Duffing系统的分岔、混沌等动力学行为。对于弱非线性的情况,借助多尺度方法得到了耦合系统发生外共振和内共振情况下的近似解;对于强非线性的情况,利用相图、功率谱、分岔图和Poincare截面等数值手段分析了系统的分岔和混沌现象;随后对系统进行了幅值控制和混沌控制。
研究了由非线性力学模型推导出的Duffing系统和参数激励系统的混沌现象,并对所提到的几个混沌系统进行了滑模控制,让其可以追踪任意假设的目标信号。对力学系统进行了吸引子的控制:可以把吸引子的大小随意的扩大和压缩,而不改变其形状。引入了非理想控制的思想,可以实现动力系统的混沌控制与反控制,并结合数值例子给予了解释。
Chaos control and synchronization are the frontier problems in nonlinear dynamics research. Sometimes, chaos is useful, as in a mixing process, or in heat transfer, but often it is harmful or unwanted, and must be controlled. It is thus of great importance to control chaos in academic research and practical applications. Different techniques and methods have been proposed to achieve chaos control and synchronization during the last two decades, but not all these approaches are universally applicable for chaos control and other new techniques need be found. Based on this, the paper proposes and develops a few new methods for chaos control and synchronization.
The history of dynamic chaos is looked back on simply, and three definitions on chaos are introduced. Two fundamental properties of chaos (sensitive dependence on initial conditions and complex orbit structure) are detailed. Meanwhile, some important conceptions and techniques (Lyapunov exponents, phase portraits, frequency power spectra, the topological entropy, Poincare section, strange attractor) on chaos study are introduced. A few stability theoretics are proposed and prepared for the further research of chaos control and chaos synchronization.
The co-existence of the complete synchronization and antisynchronization is found in two linear coupled Lorenz systems, and the inherent reason of the phenomenon is found and the basic property of the system that can display the co-existence phenomenon is investigated. The stability of synchronization in two linear coupled chaotic systems is studied using linear stability theory, the Lyapunov stability theory and the numerical method. Some sufficient conditions of global asymptotic synchronization are attained from rigorously mathematical theory.
Generalized synchronization phenomena are studied. Nonlinear feedback control methods are used to achieve generalized synchronization of two arbitrary chaotic systems. The two hyperchaotic Chen system and two non-autonomous unified chaotic systems are treated as numerical examples. the Numerical simulation results show the effectiveness and feasibility of the theoretical analysis. In terms of a dynamical system, the amplitude of its oscillator need to be modified to desired size, meantime the other properties, such as the shape of attractor, can be not changed. The usual control strategy can not work for it. The transformation is presented to obtain a new controller that achieve attractor control. The new controller may modify the size of attractor and make the shape of attractor invariably in spite of the motion of the system is period or chaotic. Based on Lyapunov stability theory, an adaptive control law is derived such that the trajectories of two arbitrary chaotic systems with unknown parameters asymptotically synchronized. Numerical simulations are provided for illustration and verification of the proposed method.
The dynamical behavior (example as bifurcation and chaos) of a system consisting of two coupled Duffing equations is investigated. In the situation of weakly nonlinearity, based on the method of multiple scales, approximate solutions of the coupled system are obtained both in the external and internal resonance cases. And in the situation of strongly nonlinearity, bifurcation and chaos behavior in the coupled system are found using numerical methods, for example, phase portraits, frequency power spectra, bifurcation diagrams and Poincare maps etal. Amplitude control and chaos control for uncertain chaotic systems are studied.
Chaos behavior of Duffing oscillators and parametrically excited oscillators that derived from nonlinear mechanical models is analyzed. A sliding control strategy is applied to drive the chaotic motion of the chaotic systems to any defined reference signal in spite of modelling errors, parametric variations and perturbing external forces. Attractor control of mechanical systems are investigated. Using the proposed control method in the paper, the size of attractors is modified, meanwhile its shape is not changed. Non-ideal control is presented to achieve chaos control or anticontrol, and a numerical example is used to illustrate the method.
简要回顾了混沌动力学研究的历史,介绍了三个常见的混沌的定义。着重介绍了混沌的两个基本特性——初值敏感性和运动轨道的复杂性,以及研究混沌的重要工具和方法,比如Lyapunov指数、相图、功率谱、拓扑熵、Poincare截面、奇怪吸引子等等。引进了几个稳定性理论,为以后的应用作了铺垫。
发现了两个线性耦合Lorenz混沌系统会发生完全同步与反同步共存的现象。在此基础上,找到了出现这种现象的内在原因,研究了出现这种同步现象时混沌系统的基本特性。借助线性稳定性理论、Lyapunov稳定性理论以及数值方法,探讨了线性耦合混沌系统产生完全同步与反同步共存的稳定性问题,给出了线性耦合混沌同步的充分条件。
研究了广义混沌同步问题。利用非线性反馈控制方法,给出了任意两个混沌系统实现广义同步的充分条件,并结合两个超混沌Chen系统的广义同步和两个非自治的统一混沌系统的广义同步对提出的控制方案进行了验证;对一个动力系统而言,有时希望调节系统运动的幅值,同时又不想改变系统运动的其它特性,比如吸引子的形状。在这种情况下,一般的控制方法很难做到。本文引进了一个变换,就可以得到一个反馈控制力。在该控制力的作用下,不管系统是周期的还是混沌的,均可以人为地改变系统运动的幅值,同时吸引子形状保持不变;对于系统具有不确定因素时,给出了任意两个混沌系统实现广义同步的自适应控制方法,并借助Lyapunov稳定性理论进行了严格的证明。数值模拟进一步验证了所给自适应控制方法的有效性。
研究两个耦合Duffing系统的分岔、混沌等动力学行为。对于弱非线性的情况,借助多尺度方法得到了耦合系统发生外共振和内共振情况下的近似解;对于强非线性的情况,利用相图、功率谱、分岔图和Poincare截面等数值手段分析了系统的分岔和混沌现象;随后对系统进行了幅值控制和混沌控制。
研究了由非线性力学模型推导出的Duffing系统和参数激励系统的混沌现象,并对所提到的几个混沌系统进行了滑模控制,让其可以追踪任意假设的目标信号。对力学系统进行了吸引子的控制:可以把吸引子的大小随意的扩大和压缩,而不改变其形状。引入了非理想控制的思想,可以实现动力系统的混沌控制与反控制,并结合数值例子给予了解释。
Chaos control and synchronization are the frontier problems in nonlinear dynamics research. Sometimes, chaos is useful, as in a mixing process, or in heat transfer, but often it is harmful or unwanted, and must be controlled. It is thus of great importance to control chaos in academic research and practical applications. Different techniques and methods have been proposed to achieve chaos control and synchronization during the last two decades, but not all these approaches are universally applicable for chaos control and other new techniques need be found. Based on this, the paper proposes and develops a few new methods for chaos control and synchronization.
The history of dynamic chaos is looked back on simply, and three definitions on chaos are introduced. Two fundamental properties of chaos (sensitive dependence on initial conditions and complex orbit structure) are detailed. Meanwhile, some important conceptions and techniques (Lyapunov exponents, phase portraits, frequency power spectra, the topological entropy, Poincare section, strange attractor) on chaos study are introduced. A few stability theoretics are proposed and prepared for the further research of chaos control and chaos synchronization.
The co-existence of the complete synchronization and antisynchronization is found in two linear coupled Lorenz systems, and the inherent reason of the phenomenon is found and the basic property of the system that can display the co-existence phenomenon is investigated. The stability of synchronization in two linear coupled chaotic systems is studied using linear stability theory, the Lyapunov stability theory and the numerical method. Some sufficient conditions of global asymptotic synchronization are attained from rigorously mathematical theory.
Generalized synchronization phenomena are studied. Nonlinear feedback control methods are used to achieve generalized synchronization of two arbitrary chaotic systems. The two hyperchaotic Chen system and two non-autonomous unified chaotic systems are treated as numerical examples. the Numerical simulation results show the effectiveness and feasibility of the theoretical analysis. In terms of a dynamical system, the amplitude of its oscillator need to be modified to desired size, meantime the other properties, such as the shape of attractor, can be not changed. The usual control strategy can not work for it. The transformation is presented to obtain a new controller that achieve attractor control. The new controller may modify the size of attractor and make the shape of attractor invariably in spite of the motion of the system is period or chaotic. Based on Lyapunov stability theory, an adaptive control law is derived such that the trajectories of two arbitrary chaotic systems with unknown parameters asymptotically synchronized. Numerical simulations are provided for illustration and verification of the proposed method.
The dynamical behavior (example as bifurcation and chaos) of a system consisting of two coupled Duffing equations is investigated. In the situation of weakly nonlinearity, based on the method of multiple scales, approximate solutions of the coupled system are obtained both in the external and internal resonance cases. And in the situation of strongly nonlinearity, bifurcation and chaos behavior in the coupled system are found using numerical methods, for example, phase portraits, frequency power spectra, bifurcation diagrams and Poincare maps etal. Amplitude control and chaos control for uncertain chaotic systems are studied.
Chaos behavior of Duffing oscillators and parametrically excited oscillators that derived from nonlinear mechanical models is analyzed. A sliding control strategy is applied to drive the chaotic motion of the chaotic systems to any defined reference signal in spite of modelling errors, parametric variations and perturbing external forces. Attractor control of mechanical systems are investigated. Using the proposed control method in the paper, the size of attractors is modified, meanwhile its shape is not changed. Non-ideal control is presented to achieve chaos control or anticontrol, and a numerical example is used to illustrate the method.
引文
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