具有恒李雅普诺夫指数的类Colpitts混沌系统及其同步
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摘要
混沌作为一种复杂的非线性运动行为,在物理学、工程学、信息学、生物学和化学等领域得到了广泛的研究,围绕混沌而展开的应用研究越来越引起人们的重视,并成为混沌研究的前沿课题和发展方向之一。由于混沌内在的随机性、连续宽谱和对初值的极端敏感性等特点,使其特别适用于保密通信、信号处理和图像处理等方面,因此,混沌系统的构造与实现、混沌同步已成为混沌应用的关键技术。
Colpitts系统存在混沌现象并具有特殊的结构。本文在此基础上,提出了具有恒定的李雅普诺夫指数的类Colpitts混沌系统,建立了系统模型,分析了动力学特性。基于类Colpitts混沌系统方程,推广得到一族恒李雅普诺夫指数谱混沌系统。应用多种同步方法,对类Colpitts混沌系统展开同步研究,揭示了系统同步所具有的特殊性。对混沌系统的动力学行为与同步特征进行了理论证明、数值仿真,并基于实验仿真给出了混沌系统及其同步的实现电路。
论文的主要创新点和贡献归纳如下:
1.提出了具有恒李雅普诺夫指数的类Colpitts混沌系统,揭示了其特殊的信号演变规律,设计了实验电路在物理上实现了该混沌系统。
从混沌系统实现非线性作用的函数出发,将Colpitts混沌系统中的指数项用分段线性绝对值项来代替,发现了新的混沌吸引子。将混沌系统中的常数项参数与系数参数分离,发现常数项参数能线性调整系统各个状态变量的幅值,系数参数能对系统输出的某个状态变量进行倒相,在前述参数作用下状态变量演变发生变化的同时,系统保持恒定的李雅普诺夫指数谱。设计了模拟电路,在物理上实现了该混沌系统。
2.对具有恒李雅普诺夫指数的类Colpitts混沌系统进行了推广,提出了一族恒李雅普诺夫指数谱混沌系统,设计了可切换的实验电路实现了该族混沌系统。
在系统方程中添加线性项或者常数项,继而调整系统方程中的绝对值项,并引入新的绝对值项,对具有恒李雅普诺夫指数的类Colpitts混沌系统进行了推广研究。组合不同的线性项,系统可演变成性质类似而又相轨不同的子系统,调整绝对值项,系统将具有更为奇特的混沌吸引子,系统的状态变量演变幅值可被常数项线性调整而李雅普诺夫指数谱保持不变。设计了模拟可切换电路,通过跳线与开关的不同选择,在示波器上观测到了各个系统所产生的混沌吸引子。
3.对具有恒李雅普诺夫指数的类Colpitts混沌系统进行了同步研究,指出了同步系统所具有的状态调节灵活性,构建了同步体系,并基于模块化的设计思想,设计了同步电路。
应用反馈同步控制、广义同步和广义投影同步等方法对具有恒李雅普诺夫指数的类Colpitts混沌系统进行了同步研究,给出了同步所需要的控制器增益范围,构造了合适的驱动与响应系统,设计了合适的非线性反馈控制器,构建了同步体系,设计了同步电路。由于具有恒李雅普诺夫指数的类Colpitts混沌系统存在特殊的常数项参数与系数参数,因此,整个同步体系的状态变量具有幅度与相位的调节灵活性。在广义投影同步研究中,提出了具有恒李雅普诺夫指数的类Colpitts混沌系统的同结构和异结构两种广义投影同步方法,成功获得了内外两个幅度调节因子:比例因子使响应系统的状态变量信号任意比例于原驱动系统的状态变量信号,同步体系中常数项参数则线性调节同步体系中两个系统的状态变量之演变区间,使之同增或者同减。
Chaos, a well known complex nonlinear behavior, has applications in various fields such as physics, engineering, information, biology and chemistry. The research about applications of chaos has attracted increasing attention and has rapidly become one of the frontier directions. Because chaotic systems possess certain features, such as high randomicity, board spectra of its Fourier transform, and hyper sensitivity to initial conditions, the application of chaos can be found in secure communications, signal processing, image processing etc. Chaotic system construction and implementation as well as chaos synchronization have become the key processes in applying chaos.
A Colpitts-like Chaotic System with Constant Lyapunov Exponents (CCSCLE) is proposed in this dissertation based on the chaos phenomenon found in Colpitts circuit which possesses special structure. The chaos model is constructed; its dynamical property is analyzed. Based on the equation of CCSCLE, and so a chaotic family with constant Lyapunov exponent spectrum is proposed. The synchronization of CCSCLE is studied by using various synchronization methods and the particularity is then expounded. Theory proving and numerical simulation on the dynamical behaviour and synchronization particularity of the chaotic system are achieved, and after experimental simulation, corresponding implemental chaotic circuits and synchronization circuit is designed. The main innovations and contributions are listed as follows:
1. A Colpitts-like chaotic system with constant Lyapunov exponent spectrum is proposed, its special signal variation law is revealed and it is implemented in physics after the experimental circuit is designed.
Based on the nonlinear function of chaos system, a novel chaotic attractor is found by using simple absolute term to substitute for exponent term in normalized chaotic Colpitts system. By differentiating the parameters of constant term and coefficient term, it is verified that the Lyapunov exponent spectrum remains constant when the amplitude of the system variables are modulated by the constant term and the phase of a certain variable is inversed by the coefficient term. The system is implemented in physics after the analog circuit is designed.
2. Followed with the extension of CCSCLE, a chaotic family with invariable Lyapunov exponent spectrum is proposed and which is also implemented successfully by an analog switchable circuit.
By introducing linear and constant terms in system equations of CCSCLE, and based on that, after further adjusting the absolute term and introducing a new absolute term in the dynamical equation, the extension system of CCSCLE is obtained. A class of subsystems with the same properties but different phase trajectories is achieved through different combinations of linear terms, and another novel chaotic attractor is found, all state variables can also be modified linearly by constant term while the Lyapunov exponent spectrum remains stable. An analog switchable circuit is proposed, by the choosing of a jump line or switch and all kinds of novel chaotic attractors generated in various systems are shown on an oscillograph.
3. Synchronization study on CCSCLE is conducted, the adjustable flexibility on state variables of synchronization system is pointed out, the synchronization system is constructed and based on modularization theory a synchronization circuit is designed.
Synchronization study using feedback control, generalized synchronization and generalized projective synchronization methods for CCSCLE are conducted. The proper controller gain region to realize synchronization is obtained, the proper driving and response systems are constructed and the appropriate nonlinear feedback controllers are designed, so then the synchronization systems are constructed, the synchronization circuits are also designed. There exists adjustable flexibility on amplitude and phase of state variables owing to the special parameters, i.e. constant term and coefficient of CCSCLE. Two generalized projective synchronization methods of CCSCLE with the same and different structure are studied, inner and outer amplitude adjusters are obtained, i.e., any scale signal of the driving system state variable can be obtained by adjusting scaling factor, and when the global linear amplitude adjuster is changed, the state variables of the synchronization system and the response system are modulated to increase and decrease in their own evolvement region synchronously in linearity.
Colpitts系统存在混沌现象并具有特殊的结构。本文在此基础上,提出了具有恒定的李雅普诺夫指数的类Colpitts混沌系统,建立了系统模型,分析了动力学特性。基于类Colpitts混沌系统方程,推广得到一族恒李雅普诺夫指数谱混沌系统。应用多种同步方法,对类Colpitts混沌系统展开同步研究,揭示了系统同步所具有的特殊性。对混沌系统的动力学行为与同步特征进行了理论证明、数值仿真,并基于实验仿真给出了混沌系统及其同步的实现电路。
论文的主要创新点和贡献归纳如下:
1.提出了具有恒李雅普诺夫指数的类Colpitts混沌系统,揭示了其特殊的信号演变规律,设计了实验电路在物理上实现了该混沌系统。
从混沌系统实现非线性作用的函数出发,将Colpitts混沌系统中的指数项用分段线性绝对值项来代替,发现了新的混沌吸引子。将混沌系统中的常数项参数与系数参数分离,发现常数项参数能线性调整系统各个状态变量的幅值,系数参数能对系统输出的某个状态变量进行倒相,在前述参数作用下状态变量演变发生变化的同时,系统保持恒定的李雅普诺夫指数谱。设计了模拟电路,在物理上实现了该混沌系统。
2.对具有恒李雅普诺夫指数的类Colpitts混沌系统进行了推广,提出了一族恒李雅普诺夫指数谱混沌系统,设计了可切换的实验电路实现了该族混沌系统。
在系统方程中添加线性项或者常数项,继而调整系统方程中的绝对值项,并引入新的绝对值项,对具有恒李雅普诺夫指数的类Colpitts混沌系统进行了推广研究。组合不同的线性项,系统可演变成性质类似而又相轨不同的子系统,调整绝对值项,系统将具有更为奇特的混沌吸引子,系统的状态变量演变幅值可被常数项线性调整而李雅普诺夫指数谱保持不变。设计了模拟可切换电路,通过跳线与开关的不同选择,在示波器上观测到了各个系统所产生的混沌吸引子。
3.对具有恒李雅普诺夫指数的类Colpitts混沌系统进行了同步研究,指出了同步系统所具有的状态调节灵活性,构建了同步体系,并基于模块化的设计思想,设计了同步电路。
应用反馈同步控制、广义同步和广义投影同步等方法对具有恒李雅普诺夫指数的类Colpitts混沌系统进行了同步研究,给出了同步所需要的控制器增益范围,构造了合适的驱动与响应系统,设计了合适的非线性反馈控制器,构建了同步体系,设计了同步电路。由于具有恒李雅普诺夫指数的类Colpitts混沌系统存在特殊的常数项参数与系数参数,因此,整个同步体系的状态变量具有幅度与相位的调节灵活性。在广义投影同步研究中,提出了具有恒李雅普诺夫指数的类Colpitts混沌系统的同结构和异结构两种广义投影同步方法,成功获得了内外两个幅度调节因子:比例因子使响应系统的状态变量信号任意比例于原驱动系统的状态变量信号,同步体系中常数项参数则线性调节同步体系中两个系统的状态变量之演变区间,使之同增或者同减。
Chaos, a well known complex nonlinear behavior, has applications in various fields such as physics, engineering, information, biology and chemistry. The research about applications of chaos has attracted increasing attention and has rapidly become one of the frontier directions. Because chaotic systems possess certain features, such as high randomicity, board spectra of its Fourier transform, and hyper sensitivity to initial conditions, the application of chaos can be found in secure communications, signal processing, image processing etc. Chaotic system construction and implementation as well as chaos synchronization have become the key processes in applying chaos.
A Colpitts-like Chaotic System with Constant Lyapunov Exponents (CCSCLE) is proposed in this dissertation based on the chaos phenomenon found in Colpitts circuit which possesses special structure. The chaos model is constructed; its dynamical property is analyzed. Based on the equation of CCSCLE, and so a chaotic family with constant Lyapunov exponent spectrum is proposed. The synchronization of CCSCLE is studied by using various synchronization methods and the particularity is then expounded. Theory proving and numerical simulation on the dynamical behaviour and synchronization particularity of the chaotic system are achieved, and after experimental simulation, corresponding implemental chaotic circuits and synchronization circuit is designed. The main innovations and contributions are listed as follows:
1. A Colpitts-like chaotic system with constant Lyapunov exponent spectrum is proposed, its special signal variation law is revealed and it is implemented in physics after the experimental circuit is designed.
Based on the nonlinear function of chaos system, a novel chaotic attractor is found by using simple absolute term to substitute for exponent term in normalized chaotic Colpitts system. By differentiating the parameters of constant term and coefficient term, it is verified that the Lyapunov exponent spectrum remains constant when the amplitude of the system variables are modulated by the constant term and the phase of a certain variable is inversed by the coefficient term. The system is implemented in physics after the analog circuit is designed.
2. Followed with the extension of CCSCLE, a chaotic family with invariable Lyapunov exponent spectrum is proposed and which is also implemented successfully by an analog switchable circuit.
By introducing linear and constant terms in system equations of CCSCLE, and based on that, after further adjusting the absolute term and introducing a new absolute term in the dynamical equation, the extension system of CCSCLE is obtained. A class of subsystems with the same properties but different phase trajectories is achieved through different combinations of linear terms, and another novel chaotic attractor is found, all state variables can also be modified linearly by constant term while the Lyapunov exponent spectrum remains stable. An analog switchable circuit is proposed, by the choosing of a jump line or switch and all kinds of novel chaotic attractors generated in various systems are shown on an oscillograph.
3. Synchronization study on CCSCLE is conducted, the adjustable flexibility on state variables of synchronization system is pointed out, the synchronization system is constructed and based on modularization theory a synchronization circuit is designed.
Synchronization study using feedback control, generalized synchronization and generalized projective synchronization methods for CCSCLE are conducted. The proper controller gain region to realize synchronization is obtained, the proper driving and response systems are constructed and the appropriate nonlinear feedback controllers are designed, so then the synchronization systems are constructed, the synchronization circuits are also designed. There exists adjustable flexibility on amplitude and phase of state variables owing to the special parameters, i.e. constant term and coefficient of CCSCLE. Two generalized projective synchronization methods of CCSCLE with the same and different structure are studied, inner and outer amplitude adjusters are obtained, i.e., any scale signal of the driving system state variable can be obtained by adjusting scaling factor, and when the global linear amplitude adjuster is changed, the state variables of the synchronization system and the response system are modulated to increase and decrease in their own evolvement region synchronously in linearity.
引文
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