复杂动力学网络与混沌系统的控制与同步
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摘要
本文首先简单介绍了混沌和复杂网络的控制和同步研究的起源、现状和典型方法;然后研究了一些复杂动力学网络模型和混沌系统的控制与同步问题。内容包括单向耦合网络的追踪控制、自适应耦合网络的同步、自适应脉冲微扰法控制多个混沌系统从混沌运动到低周期运动、驱动无耦合R?ssler混沌系统达到脉冲同步、参数不匹配的统一混沌系统的滑模控制同步、脉冲方法实现混沌系统的广义同步及广义同步化流形的H?lder连续性。具体的研究工作概括如下:
(1)利用Lyapunov稳定性理论,针对单向耦合网络联接的Lorenz系统,设计一种追踪控制方案。只对网络系统的一个结点加入控制器,就实现了单向耦合网络联接的Lorenz系统的单个输出变量对任意给定参考信号的追踪。
(2)提出了关于平衡点的渐近稳定性一个新概念——平衡点的分量导出渐近稳定性;构造了一种自适应耦合联接的复杂网络。一般来说,要实现复杂动力学网络的同步化非常困难,往往需要网络的耦合强度系数非常大,特别是当网络的结点较多时。但我们利用著名的LaSalle不变原理,证明了只要自适应耦合强度系数为正数,这种自适应耦合的复杂动力网络就可以达到同步。值得指出的是这里只对网络中每一个作为结点的动力学系统的第一个状态方程作了自适应耦合,其余状态方程均没有任何形式的耦合。
(3)选择脉冲微扰控制下的系统的某个变量为输入变量,设计一种新的自适应控制器,对多个混沌系统进行脉冲微扰,引导这些系统从混沌运动到低周期运动,实现同时控制多个混沌系统到不同的周期态。当选择相同的输入变量实施脉冲微扰时,还可控制多个混沌系统达到周期态同步。
(4)选取适当的外部信号脉冲驱动两个无耦合的R?ssler混沌系统的参数,实现了控制混沌的目的。利用脉冲微分方程理论,从理论上证明了这种无耦合R?ssler混沌系统可以实现同步。注意到这里两个混沌系统不需要耦合即可实现同步,由于在实际中有些系统之间是很难甚至是不可能被耦合在一起的,因而这样的控制方法在实际操作中更实用、更易实现,期望这种新的控制方法能有更好的应用。
(5)采用滑模控制技术,设计了一种新的自适应滑模控制器,实现了参数不匹配的统一混沌系统的同步化。所作的理论分析和计算机仿真都有力地证明了这种新的控制方法的有效性。
(6)给出了两个完全不同的混沌动力系统实现脉冲广义同步的充分条件,并提出了一个通过构造响应系统实现其与驱动系统脉冲广义同步的一般方法。借助Lyapunov稳定性理论,证明了该脉冲广义同步的稳定性。
(7)从理论上严格证明了两个不同的混沌系统线性耦合时能实现广义同步化,在一定条件下广义同步化流形是H?lder连续的。采用的思想是Temam的无穷维动力系统的惯性流形理论的改进。在线性耦合下两个混沌系统具有吸收集和吸引子的基础上,通过定义一个函数类上的映射满足Schauder不动点定理,得到了具有不变性的广义同步化流形,还证明了线性耦合下两个混沌系统存在分数维的指数吸引子,指数吸引子与广义同步流形的交集具有指数吸引性。
本文对所有研究的问题给出了数值仿真例子,仿真结果均很好地验证了相应的理论分析结果。
Firstly in this dissertation, a brief introduction to the historical backgrounds of chaos and complex networks, research progress and typical methods for chaotic control and synchronization are given. Then this dissertation is devoted to the study on control and synchronization for some network models and chaotic systems. The research included tracking control of unidirectional coupling network, synchronization in complex networks with adaptive coupling, control chaos to low-period motion based on adaptive pulse perturbation, impulsive synchronization of uncoupled chaotic systems driven by an external signal, chaos synchronization induced by sliding-mode control for chaotic systems with parameter mismatches, generalized synchronization via impulsive control and H?lder continuity for generalized synchronization manifolds. The detail research results focuse mainly on the followings:
(1) Based on Lyapunov stability theory, a tracking controller is designed for the tracking control of Lorenz systems in unidirectional coupling network. A controller is put to only one node of this network, then it can be realized that some output signals of this network approach to any desired orbit.
(2) A new concept about the asymptotic stability of balance point, component leading asymptotic stability, is presented. A new kind model of complex networks with adaptive coupling is constructed. Generally it is very difficult to realize synchronization for some complex networks. In order to synchronize, the coupling strength coefficient of networks has to be very large, especially when the number of coupled nodes is larger. However we proved by using the well-known LaSalle invariance principle, that the state of such a complex network can synchronize as long as the coupling strength coefficient is positive. Moreover it is noted that there is a couple only between the first state equations for each node of the network.
(3) A new controller is designed, which input variable comes from some state variable of a pulse perturbed chaotic system, to direct many chaotic systems towards low-period motions via adaptive pulse perturbing. Many chaotic systems can be controlled to be different periodic orbits simultaneously. When the same input state variable of the controller is used, many chaotic systems can be controlled to be different periodic synchronizations.
(4) In oreder to synchronize two uncoupled systems, a new method is designed that is to drive impulsively the parameters of two uncoupled R?ssler systems via using external chaotic signal. By using the existing results of impulsive control theory, it is proved theoretically that chaotic and periodic synchronizations can be implemented. The key point is that there is no couple between two chaotic systems, in practice, sometimes it is difficult even impossible to couple two chaotic systems. So it is more useful and easier to realize in practice, and hope have better application.
(5) Using the sliding mode control technique, a new adaptively control law is established which realizes the synchronization of two unified chaotic systems with parameter mismatches. Both theoretical analysis and illustrative examples have been presented to verify the validity of the developed control scheme.
(6) A sufficient condition for the occurrence of impulsively generalized synchronization of two completely different dynamical systems is given. Moreover a method of constructing a response system to achieve impulsively generalized synchronization with drive system is presented. By using Lyapunov theory, the stability of this impulsively generalized synchronization is demonstrated.
(7) It is also proved theoretically that generalized synchronization can occur in two linear coupled dynamical systems, and the generalized synchronization manifolds are H?lder continuous under certain conditions. By using Temam’s amended interial manifolds theory. Under the presumption that two linear coupled systems have attractor and the basin of attraction, the existence of generalized synchronization manifold can be attained by Schauder’s Fixed-Point Theorem via defining a map over a functional category. This kind of manifold has positive invariant property. It is also proved that there is an exponential attractor with a fraction number dimension. The intersection set of a H?lder continuous inertial manifold and an exponential attractor has exponential attraction.
For all the above theoretical results, we proposed numerical simulations which verity the corresponding theoretical results.
(1)利用Lyapunov稳定性理论,针对单向耦合网络联接的Lorenz系统,设计一种追踪控制方案。只对网络系统的一个结点加入控制器,就实现了单向耦合网络联接的Lorenz系统的单个输出变量对任意给定参考信号的追踪。
(2)提出了关于平衡点的渐近稳定性一个新概念——平衡点的分量导出渐近稳定性;构造了一种自适应耦合联接的复杂网络。一般来说,要实现复杂动力学网络的同步化非常困难,往往需要网络的耦合强度系数非常大,特别是当网络的结点较多时。但我们利用著名的LaSalle不变原理,证明了只要自适应耦合强度系数为正数,这种自适应耦合的复杂动力网络就可以达到同步。值得指出的是这里只对网络中每一个作为结点的动力学系统的第一个状态方程作了自适应耦合,其余状态方程均没有任何形式的耦合。
(3)选择脉冲微扰控制下的系统的某个变量为输入变量,设计一种新的自适应控制器,对多个混沌系统进行脉冲微扰,引导这些系统从混沌运动到低周期运动,实现同时控制多个混沌系统到不同的周期态。当选择相同的输入变量实施脉冲微扰时,还可控制多个混沌系统达到周期态同步。
(4)选取适当的外部信号脉冲驱动两个无耦合的R?ssler混沌系统的参数,实现了控制混沌的目的。利用脉冲微分方程理论,从理论上证明了这种无耦合R?ssler混沌系统可以实现同步。注意到这里两个混沌系统不需要耦合即可实现同步,由于在实际中有些系统之间是很难甚至是不可能被耦合在一起的,因而这样的控制方法在实际操作中更实用、更易实现,期望这种新的控制方法能有更好的应用。
(5)采用滑模控制技术,设计了一种新的自适应滑模控制器,实现了参数不匹配的统一混沌系统的同步化。所作的理论分析和计算机仿真都有力地证明了这种新的控制方法的有效性。
(6)给出了两个完全不同的混沌动力系统实现脉冲广义同步的充分条件,并提出了一个通过构造响应系统实现其与驱动系统脉冲广义同步的一般方法。借助Lyapunov稳定性理论,证明了该脉冲广义同步的稳定性。
(7)从理论上严格证明了两个不同的混沌系统线性耦合时能实现广义同步化,在一定条件下广义同步化流形是H?lder连续的。采用的思想是Temam的无穷维动力系统的惯性流形理论的改进。在线性耦合下两个混沌系统具有吸收集和吸引子的基础上,通过定义一个函数类上的映射满足Schauder不动点定理,得到了具有不变性的广义同步化流形,还证明了线性耦合下两个混沌系统存在分数维的指数吸引子,指数吸引子与广义同步流形的交集具有指数吸引性。
本文对所有研究的问题给出了数值仿真例子,仿真结果均很好地验证了相应的理论分析结果。
Firstly in this dissertation, a brief introduction to the historical backgrounds of chaos and complex networks, research progress and typical methods for chaotic control and synchronization are given. Then this dissertation is devoted to the study on control and synchronization for some network models and chaotic systems. The research included tracking control of unidirectional coupling network, synchronization in complex networks with adaptive coupling, control chaos to low-period motion based on adaptive pulse perturbation, impulsive synchronization of uncoupled chaotic systems driven by an external signal, chaos synchronization induced by sliding-mode control for chaotic systems with parameter mismatches, generalized synchronization via impulsive control and H?lder continuity for generalized synchronization manifolds. The detail research results focuse mainly on the followings:
(1) Based on Lyapunov stability theory, a tracking controller is designed for the tracking control of Lorenz systems in unidirectional coupling network. A controller is put to only one node of this network, then it can be realized that some output signals of this network approach to any desired orbit.
(2) A new concept about the asymptotic stability of balance point, component leading asymptotic stability, is presented. A new kind model of complex networks with adaptive coupling is constructed. Generally it is very difficult to realize synchronization for some complex networks. In order to synchronize, the coupling strength coefficient of networks has to be very large, especially when the number of coupled nodes is larger. However we proved by using the well-known LaSalle invariance principle, that the state of such a complex network can synchronize as long as the coupling strength coefficient is positive. Moreover it is noted that there is a couple only between the first state equations for each node of the network.
(3) A new controller is designed, which input variable comes from some state variable of a pulse perturbed chaotic system, to direct many chaotic systems towards low-period motions via adaptive pulse perturbing. Many chaotic systems can be controlled to be different periodic orbits simultaneously. When the same input state variable of the controller is used, many chaotic systems can be controlled to be different periodic synchronizations.
(4) In oreder to synchronize two uncoupled systems, a new method is designed that is to drive impulsively the parameters of two uncoupled R?ssler systems via using external chaotic signal. By using the existing results of impulsive control theory, it is proved theoretically that chaotic and periodic synchronizations can be implemented. The key point is that there is no couple between two chaotic systems, in practice, sometimes it is difficult even impossible to couple two chaotic systems. So it is more useful and easier to realize in practice, and hope have better application.
(5) Using the sliding mode control technique, a new adaptively control law is established which realizes the synchronization of two unified chaotic systems with parameter mismatches. Both theoretical analysis and illustrative examples have been presented to verify the validity of the developed control scheme.
(6) A sufficient condition for the occurrence of impulsively generalized synchronization of two completely different dynamical systems is given. Moreover a method of constructing a response system to achieve impulsively generalized synchronization with drive system is presented. By using Lyapunov theory, the stability of this impulsively generalized synchronization is demonstrated.
(7) It is also proved theoretically that generalized synchronization can occur in two linear coupled dynamical systems, and the generalized synchronization manifolds are H?lder continuous under certain conditions. By using Temam’s amended interial manifolds theory. Under the presumption that two linear coupled systems have attractor and the basin of attraction, the existence of generalized synchronization manifold can be attained by Schauder’s Fixed-Point Theorem via defining a map over a functional category. This kind of manifold has positive invariant property. It is also proved that there is an exponential attractor with a fraction number dimension. The intersection set of a H?lder continuous inertial manifold and an exponential attractor has exponential attraction.
For all the above theoretical results, we proposed numerical simulations which verity the corresponding theoretical results.
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