多翼多涡卷混沌系统生成、同步及其在保密通信中的应用
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摘要
非线性动力学,通常称为混沌理论,它改变了看待自然和社会系统动力学的传统科学思维方式,在过去的几十年里得到了蓬勃的发展。混沌是非线性系统的一种本质特征,是一个有界的不稳定的动力学行为,对初始条件敏感,包含无限不稳定周期运动。虽然它貌似随机,但是它发生在确定的条件下确定的非线性系统中。
对于许多工程应用来说,生成一个具有复杂拓扑结构如多翼或多涡卷吸引子的混沌系统,有时成为一个关键问题。在这一过程中,基于光滑和非光滑非线性函数的自治系统中产生多翼吸引子和基于非光滑非线性函数的自治系统中产生多涡卷吸引子是目前主要两种实现方法。在大多数物理和工程应用中,由于混沌的不可预测性,混沌通常被视为有害的和不受欢迎的。然而,近二十年来,随着对混沌控制和混沌同步等开创性工作的研究,混沌被证明可用于保密通信领域。通过研究复杂混沌系统生成和同步理论,本文主要的创新工作可概括如下：
(1)采用非线性的指数函数和双曲函数,构造了二个只有五项的三维指数型和双曲型混沌系统。当所有平衡点稳定的时候,系统能产生二翼混沌吸引子。与已发表的三维混沌系统相比,不仅系统项数要少一些,且在参数变化时,呈现出混沌的范围也大。基于相同理论,本文还提出了二个二翼指数型和双曲型超混沌系统。该二个系统都是自治的且只有一个平衡点,并系统在该平衡点处发生了Hopf分岔。
(2)本文提出了一组具有五个实平衡点的四维自治混沌系统。随着系统某个单一参数的变化,该组系统可以分别产生一翼、二翼、三翼和四翼吸引子,且吸引子在所有方向上都能表现出多翼形式。在三维单涡卷混沌Colpitts振荡器模型中引入两个分段线性三角波函数,构造了一个四维斜网格多涡卷超混沌系统。通过对一个三角波函数内部参数的调节,实现了多涡卷网格倾斜度的控制。同时,通过对两个三角波函数的零点配置来扩展相空间中指数2的鞍焦平衡点,使涡卷数量不但随着转折点数量的增加而增加,且能产生任意倍的乘积。
(3)本文研究了具有完全不确定参数的混沌系统投影同步。基于李雅普诺夫稳定性理论和Barbalat引理,设计了一个新的具有参数自适应律的控制器,利用该控制器分别实现了二个结构相同的双曲型和不相同混沌系统的渐进性和全局性完全同步和反同步。基于传统的投影同步技术,提出了一个新同步方法——完全切换改进型函数投影同步,驱动和响应系统的状态变量之间完全切换,且同步到一个标尺函数。该不确定标尺函数在完全切换改进型函数投影同步中可以有效地增强通信的保密性。最后,以五项指数型混沌系统为例,数值模拟验证了该控制方案的有效性和可行性。大多数同步方法在同步过程中需要构建合适的李雅普诺夫函数,在实际应用中,构建李雅普诺夫函数仍然是困难的。基于此,本文采用雅克比矩阵方法提出了一种扩展的同步控制机制,实现了二个混沌系统的同步,而且在同步过程中并未删除驱动系统的非线性项。最后,以双曲型超混沌系统为例验证了该方法的有效性。
(4)基于完全切换投影同步和指数型超混沌系统,提出了一种改进的混沌掩盖保密通信方案,在发送端和接收端的混沌系统中,分别引入有用信号和混沌信号的混合信号的反馈,提高同步精度,从而克服了传统的混沌掩盖的一些缺点,实现了有用信号的加密和恢复。其次,本文通过一个鲁棒的高阶滑模自适应控制器,研究了一个具有外界干扰的四翼混沌系统的保密通信机制。由参数调制理论和李雅普诺夫稳定性理论,实现了发射机与接收机之间的同步和保密通信,二个有用信号得以恢复。此外,通过所提出的自适应控制器,接收系统的增益可以连续可调、未知参数可以准确地识别以及外界干扰可以同时抑制。
Nonlinear dynamics, commonly called the chaos theory, changes the scientific way of looking at the dynamics of natural and social systems, which has been intensively studied over the past several decades. Chaos is a kind of characteristics of nonlinear systems, which is a bounded unstable dynamic behavior that exhibits sensitive dependence on initial conditions and includes infinite unstable periodic motions. Although it appears to be stochastic, it occurs in a deterministic nonlinear system under deterministic conditions.
Generating a chaotic system with a more complicated topological structure such as multi-wing or multi-scroll attractors becomes a desirable task and sometimes a key issue for many engineering applications. In this endeavor, there are two major efforts as follows:based on generalizing smooth and nonsmooth nonlinear function autonomous systems with multi-wing attractor and generalizing nonsmooth nonlinear function autonomous systems with multi-scroll attractors. Meanwhile, based on linear sciences, chaos was often regarded as a harmful and undesirable characteristic for most physics and engineering applications due to its unpredictable nature. However, in the last two decades, and since the pioneer work of chaos control, and chaos synchronization, chaos proved to be of effective use for secure communications. Based on the complex chaos system modeling and synchronization theory, the main innovative works of this dissertation are as follows:
(1) In this dissertation we introduce a three-dimensional (3D) exponential-type chaotic system and a3D hyperbolic-type chaotic system with only five terms including one nonlinear term in the form of exponential function and hyperbolic function respectively. The two systems can generate a two-wing chaotic attractor when all of equilibria are stable. Compared with other3D chaotic systems, not only the terms of the two systems are less, but also the range of chaos is wider when the parameter varies. Based on the same method, this dissertation proposes two-wing exponential-type hyperchaotic system and hyperbolic-type hyperchaotic system equipped with a nonlinear term in the form of exponential function and hyperbolic function respectively. The two systems are autonomous with a unique equilibrium, especially, a Hopf bifurcation occurs at this equilibrium.
(2) In this dissertation, we propose a group of four-dimensional (4D) autonomous chaotic systems with five real equilibria. These systems can generate one-, two-, three-and four-wing attractors with variation of a single parameter, and the multi-wing type of the attractors can be displayed in all directions. In this dissertation we initiate a new approach for oblique grid multi-scroll hyperchaos generation. Through introducing two piecewise-linear triangular wave functions into a three-dimensional spiral chaotic Colpitts oscillator model, a4D grid multi-scroll hyperchaotic system is constructed. By adjusting a build-in parameter in a variable of one triangle wave function, the control of the gradient of the multi-scroll grid is achieved. Meanwhile, by deploying the zero points of the two triangular wave functions to extend the saddle-focus equilibrium points with index-2in phase space, the scroll numbers not only increase along with the number of turning points, but also can generate arbitrary multiples of products.
(3) Projective synchronization (PS) of two chaotic systems with fully uncertain parameters is investigated in this dissertation. Based on Lyapunov stability theory and Barbalat's lemma, a new adaptive controller with parameter update laws is designed to complete synchronization (CS) and antiphase synchronization (AS) between two chaotic systems asymptotically and globally, including two identical hyperbolic-type chaotic systems and two different chaotic systems. Based on PS technique, in this dissertation we also propose a new synchronization, complete switched modified function projective synchronization (CSMFPS), for two different chaotic systems, where the drive and response systems could be complete switched synchronized to a function matrix. The unpredictability of the function matrix in CSMFPS can additionally strengthen the security of communications. Finally, the five-term exponential-type chaotic system is taken for example and the corresponding numerical simulations are presented to verify the effectiveness and feasibility of the proposed control scheme. Most of synchronization methods mainly concerned the synchronization that needed to construct the appropriate Lyapunov functions. As is well known, the construction of the appropriate Lyapunov functions is still a difficult issue. An extended control scheme is presented by using Jacobin matrix method, which realizes synchronization of two different4D chaotic systems, and the nonlinear terms in the drive system are not removed. To illustrate the effectiveness of the proposed scheme, a numerical example based on the exponential-type hyperchaotic system is presented.
(4) A modified chaotic masking secure communication scheme applying complete switched projective synchronization of the exponential-type hyperchaotic system is proposed in this dissertation. In the chaotic systems of the sender and the receiver, the feedback of the mixed signal of the useful signal and chaotic signal is introduced, respectively. It can improve the precision of the synchronization, so as to overcome some shortcomings of the traditional chaos masking and realize the encryption and recovery of the useful signal. Secondly, another secure communication scheme based on a four-wing chaotic system with external disturbance via a convenient robust high-order sliding mode adaptative controller is also discussed in this dissertation. By parameter modulation theory and Lyapunov stability theory, synchronization and secure communication between transmitter and receiver is achieved and two message signals are recovered. In addition, the gains of the receiver system can be adjusted continually, the unknown parameters can be identified precisely and the external disturbance can be suppressed simultaneously by the proposed adaptative controller.
对于许多工程应用来说,生成一个具有复杂拓扑结构如多翼或多涡卷吸引子的混沌系统,有时成为一个关键问题。在这一过程中,基于光滑和非光滑非线性函数的自治系统中产生多翼吸引子和基于非光滑非线性函数的自治系统中产生多涡卷吸引子是目前主要两种实现方法。在大多数物理和工程应用中,由于混沌的不可预测性,混沌通常被视为有害的和不受欢迎的。然而,近二十年来,随着对混沌控制和混沌同步等开创性工作的研究,混沌被证明可用于保密通信领域。通过研究复杂混沌系统生成和同步理论,本文主要的创新工作可概括如下：
(1)采用非线性的指数函数和双曲函数,构造了二个只有五项的三维指数型和双曲型混沌系统。当所有平衡点稳定的时候,系统能产生二翼混沌吸引子。与已发表的三维混沌系统相比,不仅系统项数要少一些,且在参数变化时,呈现出混沌的范围也大。基于相同理论,本文还提出了二个二翼指数型和双曲型超混沌系统。该二个系统都是自治的且只有一个平衡点,并系统在该平衡点处发生了Hopf分岔。
(2)本文提出了一组具有五个实平衡点的四维自治混沌系统。随着系统某个单一参数的变化,该组系统可以分别产生一翼、二翼、三翼和四翼吸引子,且吸引子在所有方向上都能表现出多翼形式。在三维单涡卷混沌Colpitts振荡器模型中引入两个分段线性三角波函数,构造了一个四维斜网格多涡卷超混沌系统。通过对一个三角波函数内部参数的调节,实现了多涡卷网格倾斜度的控制。同时,通过对两个三角波函数的零点配置来扩展相空间中指数2的鞍焦平衡点,使涡卷数量不但随着转折点数量的增加而增加,且能产生任意倍的乘积。
(3)本文研究了具有完全不确定参数的混沌系统投影同步。基于李雅普诺夫稳定性理论和Barbalat引理,设计了一个新的具有参数自适应律的控制器,利用该控制器分别实现了二个结构相同的双曲型和不相同混沌系统的渐进性和全局性完全同步和反同步。基于传统的投影同步技术,提出了一个新同步方法——完全切换改进型函数投影同步,驱动和响应系统的状态变量之间完全切换,且同步到一个标尺函数。该不确定标尺函数在完全切换改进型函数投影同步中可以有效地增强通信的保密性。最后,以五项指数型混沌系统为例,数值模拟验证了该控制方案的有效性和可行性。大多数同步方法在同步过程中需要构建合适的李雅普诺夫函数,在实际应用中,构建李雅普诺夫函数仍然是困难的。基于此,本文采用雅克比矩阵方法提出了一种扩展的同步控制机制,实现了二个混沌系统的同步,而且在同步过程中并未删除驱动系统的非线性项。最后,以双曲型超混沌系统为例验证了该方法的有效性。
(4)基于完全切换投影同步和指数型超混沌系统,提出了一种改进的混沌掩盖保密通信方案,在发送端和接收端的混沌系统中,分别引入有用信号和混沌信号的混合信号的反馈,提高同步精度,从而克服了传统的混沌掩盖的一些缺点,实现了有用信号的加密和恢复。其次,本文通过一个鲁棒的高阶滑模自适应控制器,研究了一个具有外界干扰的四翼混沌系统的保密通信机制。由参数调制理论和李雅普诺夫稳定性理论,实现了发射机与接收机之间的同步和保密通信,二个有用信号得以恢复。此外,通过所提出的自适应控制器,接收系统的增益可以连续可调、未知参数可以准确地识别以及外界干扰可以同时抑制。
Nonlinear dynamics, commonly called the chaos theory, changes the scientific way of looking at the dynamics of natural and social systems, which has been intensively studied over the past several decades. Chaos is a kind of characteristics of nonlinear systems, which is a bounded unstable dynamic behavior that exhibits sensitive dependence on initial conditions and includes infinite unstable periodic motions. Although it appears to be stochastic, it occurs in a deterministic nonlinear system under deterministic conditions.
Generating a chaotic system with a more complicated topological structure such as multi-wing or multi-scroll attractors becomes a desirable task and sometimes a key issue for many engineering applications. In this endeavor, there are two major efforts as follows:based on generalizing smooth and nonsmooth nonlinear function autonomous systems with multi-wing attractor and generalizing nonsmooth nonlinear function autonomous systems with multi-scroll attractors. Meanwhile, based on linear sciences, chaos was often regarded as a harmful and undesirable characteristic for most physics and engineering applications due to its unpredictable nature. However, in the last two decades, and since the pioneer work of chaos control, and chaos synchronization, chaos proved to be of effective use for secure communications. Based on the complex chaos system modeling and synchronization theory, the main innovative works of this dissertation are as follows:
(1) In this dissertation we introduce a three-dimensional (3D) exponential-type chaotic system and a3D hyperbolic-type chaotic system with only five terms including one nonlinear term in the form of exponential function and hyperbolic function respectively. The two systems can generate a two-wing chaotic attractor when all of equilibria are stable. Compared with other3D chaotic systems, not only the terms of the two systems are less, but also the range of chaos is wider when the parameter varies. Based on the same method, this dissertation proposes two-wing exponential-type hyperchaotic system and hyperbolic-type hyperchaotic system equipped with a nonlinear term in the form of exponential function and hyperbolic function respectively. The two systems are autonomous with a unique equilibrium, especially, a Hopf bifurcation occurs at this equilibrium.
(2) In this dissertation, we propose a group of four-dimensional (4D) autonomous chaotic systems with five real equilibria. These systems can generate one-, two-, three-and four-wing attractors with variation of a single parameter, and the multi-wing type of the attractors can be displayed in all directions. In this dissertation we initiate a new approach for oblique grid multi-scroll hyperchaos generation. Through introducing two piecewise-linear triangular wave functions into a three-dimensional spiral chaotic Colpitts oscillator model, a4D grid multi-scroll hyperchaotic system is constructed. By adjusting a build-in parameter in a variable of one triangle wave function, the control of the gradient of the multi-scroll grid is achieved. Meanwhile, by deploying the zero points of the two triangular wave functions to extend the saddle-focus equilibrium points with index-2in phase space, the scroll numbers not only increase along with the number of turning points, but also can generate arbitrary multiples of products.
(3) Projective synchronization (PS) of two chaotic systems with fully uncertain parameters is investigated in this dissertation. Based on Lyapunov stability theory and Barbalat's lemma, a new adaptive controller with parameter update laws is designed to complete synchronization (CS) and antiphase synchronization (AS) between two chaotic systems asymptotically and globally, including two identical hyperbolic-type chaotic systems and two different chaotic systems. Based on PS technique, in this dissertation we also propose a new synchronization, complete switched modified function projective synchronization (CSMFPS), for two different chaotic systems, where the drive and response systems could be complete switched synchronized to a function matrix. The unpredictability of the function matrix in CSMFPS can additionally strengthen the security of communications. Finally, the five-term exponential-type chaotic system is taken for example and the corresponding numerical simulations are presented to verify the effectiveness and feasibility of the proposed control scheme. Most of synchronization methods mainly concerned the synchronization that needed to construct the appropriate Lyapunov functions. As is well known, the construction of the appropriate Lyapunov functions is still a difficult issue. An extended control scheme is presented by using Jacobin matrix method, which realizes synchronization of two different4D chaotic systems, and the nonlinear terms in the drive system are not removed. To illustrate the effectiveness of the proposed scheme, a numerical example based on the exponential-type hyperchaotic system is presented.
(4) A modified chaotic masking secure communication scheme applying complete switched projective synchronization of the exponential-type hyperchaotic system is proposed in this dissertation. In the chaotic systems of the sender and the receiver, the feedback of the mixed signal of the useful signal and chaotic signal is introduced, respectively. It can improve the precision of the synchronization, so as to overcome some shortcomings of the traditional chaos masking and realize the encryption and recovery of the useful signal. Secondly, another secure communication scheme based on a four-wing chaotic system with external disturbance via a convenient robust high-order sliding mode adaptative controller is also discussed in this dissertation. By parameter modulation theory and Lyapunov stability theory, synchronization and secure communication between transmitter and receiver is achieved and two message signals are recovered. In addition, the gains of the receiver system can be adjusted continually, the unknown parameters can be identified precisely and the external disturbance can be suppressed simultaneously by the proposed adaptative controller.
引文
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