复杂混沌动力学网络系统的同步及其应用研究
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摘要
复杂性与复杂系统是21世纪的重点研究课题。复杂网络是描述和理解复杂系统的重要工具和方法,它将复杂系统高度概括为由相互作用的多个个体(节点)组成的网络系统。复杂动力学网络系统的同步是复杂网络理论中的一个重要研究课题,它以非线性动力学的研究,尤其是混沌及混沌同步控制的研究,为有效的理论基础和工具,并在保密通信、网络拥塞控制、调和振子生成、多智能体一致等领域有巨大的应用潜力。
本文在研究(分数阶)超混沌系统同步及其保密通信应用的基础上,依次对分数阶混沌动力学网络、时滞混沌动力学网络、时变混沌动力学网络展开研究,设计有效的控制器,以保证实现动力学网络系统的同步。本文的主要研究工作概括如下:
(1)研究了不确定超混沌系统的广义函数投影时滞同步及其在保密通信中的应用。考虑驱动系统与响应系统的参数完全未知和部分未知的情况,基于Lyapunov稳定性理论和自适应控制方法,本文分别提出了两种通用的自适应广义函数投影时滞同步方法,并给出了严格的理论证明;数值仿真结果验证了同步方法的有效性和鲁棒性,进一步讨论了比例因子和时滞对同步效果的影响。在上述研究基础上,基于超混沌系统的广义函数投影同步,并结合使用参数调制和混沌掩盖技术,设计了两种不同的超混沌保密通信方案,理论证明和数值仿真验证了超混沌保密通信方案的有效性和可行性。
(2)研究了分数阶超混沌系统的动力学特性与同步问题。首先,基于分数阶微积分理论和计算机模拟,分析了两个新的分数阶四维系统的动力学特性,并给出了系统产生超混沌的最低阶次。然后,基于分数阶系统稳定性理论,利用状态观测器方法、主动控制方法和系统耦合方法,提出了三种通用的分数阶混沌同步方法,并设计了一种基于分数阶超混沌广义投影同步和混沌掩盖的保密通信方案;与整数阶混沌保密通信相比,该保密通信方法具有更大的密钥空间和更高的安全性能。最后,基于分数阶系统稳定性理论,提出并理论证明了一种通用的实现具有未知参数的分数阶混沌系统同步和参数辨识方法,数值仿真结果证实了该同步方法的有效性。
(3)研究了分数阶复杂混沌动力学网络系统的同步问题。首先,利用非线性控制方法和双向耦合方法,分别研究了具有相同和不同拓扑结构的两个分数阶混沌动力学网络系统的外同步,并给出了实现外同步的充分条件。研究表明,分数阶次和反馈增益越大,网络外同步速度越快;具有相同拓扑结构和相同节点动力学系统的两个网络更容易达到外同步。其次,通过设计非线性控制器,实现了具有不同节点的分数阶混沌动力学网络系统的广义内同步,并得到了实现同步的充分条件。数值仿真结果表明,网络同步速度敏感依赖于分数阶次和反馈增益;对于相同的反馈增益,整数阶动力学网络系统的同步效果要远好于对应的分数阶动力学网络系统;当噪声和参数干扰存在时,利用所设计的控制器仍然可有效实现网络的广义同步。
(4)研究了时滞耦合的复杂混沌动力学网络系统的同步问题。首先,给出了一类无时滞耦合与时滞耦合并存的复杂动力学网络模型,节点内部可以是非线性耦合或线性耦合,网络可以是无向或有向的;接着,仅利用网络外部耦合配置矩阵的部分信息来设计控制器,实现了该类混沌动力学网络系统的指数同步,理论分析和数值仿真实例均证明了同步方法的有效性。其次,基于LaSalle不变集原理和自适应控制方法,研究了完全不同的两个时滞耦合复杂动力学网络系统的广义投影同步问题,设计了自适应控制器并给出了理论证明;数值仿真进一步表明了所给出的理论结果的正确性。
(5)研究了时变复杂混沌动力学网络系统的同步问题。首先,提出了一类自适应耦合的复杂动力学网络模型,网络由社团构成且具有时变的耦合强度,属于同一社团的节点彼此相同,否则,不相同。为使得自适应耦合的混沌动力学网络达到聚类同步,设计了局部控制器和耦合强度自适应律,并给出了相应的理论证明。分别以BA无标度网络和WS小世界网络为例,数值分析了网络拓扑结构、内部耦合矩阵、边重连概率、控制增益对网络同步的影响,并考虑了噪声干扰问题。研究表明,网络的聚类同步性能与上述要素密切相关,BA无标度网络比WS小世界网络更容易获得聚类同步,且提出的聚类同步方法具有一定的抗噪能力。其次,给出了一类时滞耦合的时变动力学网络模型,该网络模型具有时变的外部耦合矩阵,且包含时滞耦合项和不同的节点。并根据Barbalat引理和自适应控制方法,设计了自适应控制器和参数更新规则,使得具有未知参数的时滞耦合时变动力学网络系统达到外同步,同时辨识出系统参数。理论分析和数值仿真均证明了所提同步方法的正确性。
(6)基于牵制控制策略,研究了无时滞耦合与时滞耦合并存的复杂动力学网络系统的混合同步问题。基于LaSalle不变集原理和线性矩阵不等式,通过对网络中部分节点分别施加线性反馈控制器和自适应控制器,得到了实现网络混合同步的充分条件。数值仿真结果表明,仅使用单个控制器,即仅控制网络中的单个节点,即可实现动力学网络的混合同步;时滞越小,混合同步的性能越好;并且,在牵制控制策略下,使用自适应控制方法比使用线性反馈控制方法更易实现混合同步且实现成本更低。
Complexity and complex systems are the important topics in the 21st century. Complex networks are presently significant tools and methods to describe and understand the complex system, which highly summarize the complex system as the networks consisting of many interacting individuals or nodes. Synchronization of complex dynamical networks is one of significant topic in complex networks, which has great potential applications in secure communication, network congestion control, the generation of harmonic oscillator, multi-agent consensus, and so forth. The research on the nonlinear dynamics, especially on chaos and chaotic synchronization, are used as theoretical basis and tools for studying synchronization of complex networks.
In this dissertation, based on the research on synchronization of (fractional-order) hyperchaotic systems and its application in secure communication, different types of complex networks, i.e., the fractional-order chaotic dynamical networks, the general chaotic dynamical networks with delay coupling and the time-varying chaotic dynamical networks, are investigated. The effective controllers for complex dynamical networks are designed to guarantee to achieve the desired network synchronization. The main contributions of this dissertation are summarized as follows:
(1) Generalized function projective lag synchronization of uncertain hyperchaotic systems and its application in secure communication are studied. When the parameters of the drive and response systems are fully unknown or only the parameters of the response system are unknown, based on Lyapunov stability theory and the adaptive control method, two different universal adaptive generalized function projective lag synchronization schemes are proposed. Both rigorous theoretical proofs and numerical simulations are given to validate the effectiveness and robustness of the proposed synchronization methods. The influence of the scaling factor and time delay on the synchronization effect is further discussed. On the basis of the above studies, combining the parameter modulation and chaotic masking techniques, two distinct hyperchaotic secure communication schemes are proposed by applying generalized function projective synchronization of Chen hyperchaotic system. Corresponding theoretical proofs and numerical simulations demonstrate the validity and feasibility of the presented hyperchaotic secure communication schemes.
(2) The problem of the dynamical properties and synchronization of the fractional-order hyperchaotic systems is investigated. Firstly, based on the fractional calculus theory and computer simulations, the dynamical properties of two new fractional-order four-dimensional systems are analyzed. The lowest orders to have hyperchaos in two systems are also obtained, respectively. Secondly, based on the stability theory of the fractional-order systems, by using the state observer method, the active control method and the system coupling method, three general fractional-order chaos synchronization methods are presented. Furthermore, by applying generalized projective synchronization of the fractional-order hyperchaotic system and chaotic masking technique, a fractional-order hyperchaotic secure communication scheme is constructed, which has a larger key space and is more secure than the integer-order chaotic communication. Finally, a general method for parameter identification and synchronization of uncertain fractional-order chaotic systems is proposed and proved theoretically based on the stability theory of the fractional-order systems. The simulation results are performed to verify the validity of the presented method.
(3) Synchronization of the fractional-order complex chaotic dynamical networks is considered. Firstly, the outer synchronization between two fractional-order chaotic dynamical networks with identical or different topology is studied by applying the nonlinear control and bidirectional coupling methods. The sufficient criteria for the outer synchronization are derived analytically. Numerical results show that the larger the fractional order and the feedback gain, the faster is to achieve the outer synchronization; it is much easier to realize the outer synchronization between two networks with identical topology and uniform node dynamics. Secondly, the nonlinear controllers are designed to realize generalized synchronization of the fractional-order chaotic dynamical networks with distinct nodes, and some sufficient synchronization criteria are obtained. Simulation results show that the synchronization speed sensitively depends on both the fractional order and the feedback gain. For the same feedback gain, the synchronization effect of the integer-order dynamical network is much better than that of the fractional-order one. Furthermore, by the nonlinear controllers, the network synchronization can still be achieved effectively in presence of noise and parameter perturbations.
(4) We study synchronization of complex chaotic dynamical networks with delayed coupling. Firstly, a general complex dynamical network model with non-delayed and delayed coupling is proposed. The nodes in the network may be coupled nonlinearly or linearly. And the network can be directed or undirected. Only partial information of the coupling configuration matrix is used to design the controllers to achieve the exponential synchronization of the dynamical networks with non-delayed and delayed coupling. Both theoretical analysis and numerical simulations have validated the effectiveness of the synchronization method. Secondly, based on the LaSalle invariant principle and adaptive control method, generalized projective synchronization between two completely different complex dynamical networks with delayed coupling can be obtained by constructing the adaptive controllers. Corresponding theoretical proofs are also given. Numerical simulations further demonstrate the validity of the theoretical results.
(5) Synchronization of the time-varying complex chaotic dynamical networks is investigated. A general complex dynamical network model with adaptive coupling strengths and community structure is firstly proposed. The local dynamics of individual nodes in each community are identical, while those of any pair of nodes in different communities are diverse. The local controllers and the adaptive law for the coupling strengths are designed to achieve cluster synchronization in the adaptive dynamical networks. We take the BA scale-free network and the WS small-world network as the examples to do the simulations. Numeric evidences show that the synchronization performance is sensitively affected by the network topological structure, the inner-coupling matrix, the rewiring probability and the control gain; the BA scale-free network is much easier to achieve cluster synchronization than the WS small-world network; the presented scheme is robust against noise. Secondly, we present a general delayed dynamical network model with different nodes and time-varying coupling configuration matrix. The adaptive controllers and corresponding parameter update rule are constructed to achieve the outer synchronization and parameter identification of uncertain time-varying delayed dynamical networks. Theoretical analysis and numerical simulations have verified the effectiveness of the proposed scheme.
(6) The hybrid synchronization problem of two coupled complex dynamical networks with non-delayed and delayed coupling is investigated by the pinning control strategy. Based on the LaSalle invariance principle and linear matrix inequality technique, we obtain some sufficient synchronization conditions by applying the simple linear feedback controllers and adaptive controllers to a part of nodes, respectively. Numerical results show that the hybrid synchronization of two coupled networks can be realized by a single controller; the synchronization effect turns better with the decrease of time delay; the hybrid synchronization is easier to realize with low cost by the adaptive control method than that by the linear feedback control approach.
本文在研究(分数阶)超混沌系统同步及其保密通信应用的基础上,依次对分数阶混沌动力学网络、时滞混沌动力学网络、时变混沌动力学网络展开研究,设计有效的控制器,以保证实现动力学网络系统的同步。本文的主要研究工作概括如下:
(1)研究了不确定超混沌系统的广义函数投影时滞同步及其在保密通信中的应用。考虑驱动系统与响应系统的参数完全未知和部分未知的情况,基于Lyapunov稳定性理论和自适应控制方法,本文分别提出了两种通用的自适应广义函数投影时滞同步方法,并给出了严格的理论证明;数值仿真结果验证了同步方法的有效性和鲁棒性,进一步讨论了比例因子和时滞对同步效果的影响。在上述研究基础上,基于超混沌系统的广义函数投影同步,并结合使用参数调制和混沌掩盖技术,设计了两种不同的超混沌保密通信方案,理论证明和数值仿真验证了超混沌保密通信方案的有效性和可行性。
(2)研究了分数阶超混沌系统的动力学特性与同步问题。首先,基于分数阶微积分理论和计算机模拟,分析了两个新的分数阶四维系统的动力学特性,并给出了系统产生超混沌的最低阶次。然后,基于分数阶系统稳定性理论,利用状态观测器方法、主动控制方法和系统耦合方法,提出了三种通用的分数阶混沌同步方法,并设计了一种基于分数阶超混沌广义投影同步和混沌掩盖的保密通信方案;与整数阶混沌保密通信相比,该保密通信方法具有更大的密钥空间和更高的安全性能。最后,基于分数阶系统稳定性理论,提出并理论证明了一种通用的实现具有未知参数的分数阶混沌系统同步和参数辨识方法,数值仿真结果证实了该同步方法的有效性。
(3)研究了分数阶复杂混沌动力学网络系统的同步问题。首先,利用非线性控制方法和双向耦合方法,分别研究了具有相同和不同拓扑结构的两个分数阶混沌动力学网络系统的外同步,并给出了实现外同步的充分条件。研究表明,分数阶次和反馈增益越大,网络外同步速度越快;具有相同拓扑结构和相同节点动力学系统的两个网络更容易达到外同步。其次,通过设计非线性控制器,实现了具有不同节点的分数阶混沌动力学网络系统的广义内同步,并得到了实现同步的充分条件。数值仿真结果表明,网络同步速度敏感依赖于分数阶次和反馈增益;对于相同的反馈增益,整数阶动力学网络系统的同步效果要远好于对应的分数阶动力学网络系统;当噪声和参数干扰存在时,利用所设计的控制器仍然可有效实现网络的广义同步。
(4)研究了时滞耦合的复杂混沌动力学网络系统的同步问题。首先,给出了一类无时滞耦合与时滞耦合并存的复杂动力学网络模型,节点内部可以是非线性耦合或线性耦合,网络可以是无向或有向的;接着,仅利用网络外部耦合配置矩阵的部分信息来设计控制器,实现了该类混沌动力学网络系统的指数同步,理论分析和数值仿真实例均证明了同步方法的有效性。其次,基于LaSalle不变集原理和自适应控制方法,研究了完全不同的两个时滞耦合复杂动力学网络系统的广义投影同步问题,设计了自适应控制器并给出了理论证明;数值仿真进一步表明了所给出的理论结果的正确性。
(5)研究了时变复杂混沌动力学网络系统的同步问题。首先,提出了一类自适应耦合的复杂动力学网络模型,网络由社团构成且具有时变的耦合强度,属于同一社团的节点彼此相同,否则,不相同。为使得自适应耦合的混沌动力学网络达到聚类同步,设计了局部控制器和耦合强度自适应律,并给出了相应的理论证明。分别以BA无标度网络和WS小世界网络为例,数值分析了网络拓扑结构、内部耦合矩阵、边重连概率、控制增益对网络同步的影响,并考虑了噪声干扰问题。研究表明,网络的聚类同步性能与上述要素密切相关,BA无标度网络比WS小世界网络更容易获得聚类同步,且提出的聚类同步方法具有一定的抗噪能力。其次,给出了一类时滞耦合的时变动力学网络模型,该网络模型具有时变的外部耦合矩阵,且包含时滞耦合项和不同的节点。并根据Barbalat引理和自适应控制方法,设计了自适应控制器和参数更新规则,使得具有未知参数的时滞耦合时变动力学网络系统达到外同步,同时辨识出系统参数。理论分析和数值仿真均证明了所提同步方法的正确性。
(6)基于牵制控制策略,研究了无时滞耦合与时滞耦合并存的复杂动力学网络系统的混合同步问题。基于LaSalle不变集原理和线性矩阵不等式,通过对网络中部分节点分别施加线性反馈控制器和自适应控制器,得到了实现网络混合同步的充分条件。数值仿真结果表明,仅使用单个控制器,即仅控制网络中的单个节点,即可实现动力学网络的混合同步;时滞越小,混合同步的性能越好;并且,在牵制控制策略下,使用自适应控制方法比使用线性反馈控制方法更易实现混合同步且实现成本更低。
Complexity and complex systems are the important topics in the 21st century. Complex networks are presently significant tools and methods to describe and understand the complex system, which highly summarize the complex system as the networks consisting of many interacting individuals or nodes. Synchronization of complex dynamical networks is one of significant topic in complex networks, which has great potential applications in secure communication, network congestion control, the generation of harmonic oscillator, multi-agent consensus, and so forth. The research on the nonlinear dynamics, especially on chaos and chaotic synchronization, are used as theoretical basis and tools for studying synchronization of complex networks.
In this dissertation, based on the research on synchronization of (fractional-order) hyperchaotic systems and its application in secure communication, different types of complex networks, i.e., the fractional-order chaotic dynamical networks, the general chaotic dynamical networks with delay coupling and the time-varying chaotic dynamical networks, are investigated. The effective controllers for complex dynamical networks are designed to guarantee to achieve the desired network synchronization. The main contributions of this dissertation are summarized as follows:
(1) Generalized function projective lag synchronization of uncertain hyperchaotic systems and its application in secure communication are studied. When the parameters of the drive and response systems are fully unknown or only the parameters of the response system are unknown, based on Lyapunov stability theory and the adaptive control method, two different universal adaptive generalized function projective lag synchronization schemes are proposed. Both rigorous theoretical proofs and numerical simulations are given to validate the effectiveness and robustness of the proposed synchronization methods. The influence of the scaling factor and time delay on the synchronization effect is further discussed. On the basis of the above studies, combining the parameter modulation and chaotic masking techniques, two distinct hyperchaotic secure communication schemes are proposed by applying generalized function projective synchronization of Chen hyperchaotic system. Corresponding theoretical proofs and numerical simulations demonstrate the validity and feasibility of the presented hyperchaotic secure communication schemes.
(2) The problem of the dynamical properties and synchronization of the fractional-order hyperchaotic systems is investigated. Firstly, based on the fractional calculus theory and computer simulations, the dynamical properties of two new fractional-order four-dimensional systems are analyzed. The lowest orders to have hyperchaos in two systems are also obtained, respectively. Secondly, based on the stability theory of the fractional-order systems, by using the state observer method, the active control method and the system coupling method, three general fractional-order chaos synchronization methods are presented. Furthermore, by applying generalized projective synchronization of the fractional-order hyperchaotic system and chaotic masking technique, a fractional-order hyperchaotic secure communication scheme is constructed, which has a larger key space and is more secure than the integer-order chaotic communication. Finally, a general method for parameter identification and synchronization of uncertain fractional-order chaotic systems is proposed and proved theoretically based on the stability theory of the fractional-order systems. The simulation results are performed to verify the validity of the presented method.
(3) Synchronization of the fractional-order complex chaotic dynamical networks is considered. Firstly, the outer synchronization between two fractional-order chaotic dynamical networks with identical or different topology is studied by applying the nonlinear control and bidirectional coupling methods. The sufficient criteria for the outer synchronization are derived analytically. Numerical results show that the larger the fractional order and the feedback gain, the faster is to achieve the outer synchronization; it is much easier to realize the outer synchronization between two networks with identical topology and uniform node dynamics. Secondly, the nonlinear controllers are designed to realize generalized synchronization of the fractional-order chaotic dynamical networks with distinct nodes, and some sufficient synchronization criteria are obtained. Simulation results show that the synchronization speed sensitively depends on both the fractional order and the feedback gain. For the same feedback gain, the synchronization effect of the integer-order dynamical network is much better than that of the fractional-order one. Furthermore, by the nonlinear controllers, the network synchronization can still be achieved effectively in presence of noise and parameter perturbations.
(4) We study synchronization of complex chaotic dynamical networks with delayed coupling. Firstly, a general complex dynamical network model with non-delayed and delayed coupling is proposed. The nodes in the network may be coupled nonlinearly or linearly. And the network can be directed or undirected. Only partial information of the coupling configuration matrix is used to design the controllers to achieve the exponential synchronization of the dynamical networks with non-delayed and delayed coupling. Both theoretical analysis and numerical simulations have validated the effectiveness of the synchronization method. Secondly, based on the LaSalle invariant principle and adaptive control method, generalized projective synchronization between two completely different complex dynamical networks with delayed coupling can be obtained by constructing the adaptive controllers. Corresponding theoretical proofs are also given. Numerical simulations further demonstrate the validity of the theoretical results.
(5) Synchronization of the time-varying complex chaotic dynamical networks is investigated. A general complex dynamical network model with adaptive coupling strengths and community structure is firstly proposed. The local dynamics of individual nodes in each community are identical, while those of any pair of nodes in different communities are diverse. The local controllers and the adaptive law for the coupling strengths are designed to achieve cluster synchronization in the adaptive dynamical networks. We take the BA scale-free network and the WS small-world network as the examples to do the simulations. Numeric evidences show that the synchronization performance is sensitively affected by the network topological structure, the inner-coupling matrix, the rewiring probability and the control gain; the BA scale-free network is much easier to achieve cluster synchronization than the WS small-world network; the presented scheme is robust against noise. Secondly, we present a general delayed dynamical network model with different nodes and time-varying coupling configuration matrix. The adaptive controllers and corresponding parameter update rule are constructed to achieve the outer synchronization and parameter identification of uncertain time-varying delayed dynamical networks. Theoretical analysis and numerical simulations have verified the effectiveness of the proposed scheme.
(6) The hybrid synchronization problem of two coupled complex dynamical networks with non-delayed and delayed coupling is investigated by the pinning control strategy. Based on the LaSalle invariance principle and linear matrix inequality technique, we obtain some sufficient synchronization conditions by applying the simple linear feedback controllers and adaptive controllers to a part of nodes, respectively. Numerical results show that the hybrid synchronization of two coupled networks can be realized by a single controller; the synchronization effect turns better with the decrease of time delay; the hybrid synchronization is easier to realize with low cost by the adaptive control method than that by the linear feedback control approach.
引文
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