混沌动力学系统延拓与分析
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摘要
混沌生成系统是混沌理论和应用研究的重要组成部分,而系统中的非线性组成项又是混沌产生的必要条件。非线性项的非线性特征及在系统中的组成形式直接决定了系统的混沌行为和混沌产生机理。本论文在前人工作基础上,对混沌系统的非线性组成特征进行了深入的研究,建立和延拓了一些典型的混沌系统,进而揭示了一些新的混沌行为和产生机理。这些现象和机理进一步丰富了混沌理论,为混沌的工程应用打下基础。
本论文工作主要分为离散时间系统和连续时间系统两部分。
在离散时间系统的延拓和分析方面,重点研究了具有指数二次项的一维和二维离散映射及含斜坡补偿的电流模式控制开关DC-DC变换器的动力学特性。
为了使得离散映射具有较为复杂的动力学特性,提出了一类具有指数二次项的一维和二维离散映射,利用不动点演变轨迹分析和仿真手段构造相应次数的迭代曲线,对它们的动力学特性进行了全面深入的研究,结果表明：一维广义平方映射有着与单峰平方映射相类似的物理现象,其动力学行为分布在一个单位区域内；一维DOG小波映射随着参数变化不动点会发生数量上的增加或减少,存在倍周期分岔、切分岔、边界危机分岔、周期窗以及不完全费根鲍姆树等非线性物理现象；而二维广义平方映射则存在Hopf分岔和锁频等现象,有着复杂多变、形状奇异的极限环和混沌吸引子。
开关DC-DC变换器是一种特殊的时变离散混沌系统,在较宽的电路参数下具有两个边界,存在一般混沌系统所不具有的边界碰撞分岔现象。随着电路参数变化,开关DC-DC变换器存在倍周期和边界碰撞两种分岔路由,并由边界碰撞分岔导致其运行轨道在稳定的周期1态、连续传导模式(CCM)的鲁棒混沌态和不连续传导模式(DCM)的强阵发性的弱混沌态之间发生模式转移现象。通过引入合适的斜坡补偿电流,开关DC-DC变换器的工作模式可以从DCM转移到CCM,也可以被镇定在周期1态。此外,通过导出运行轨道状态发生转移时的两个分界线方程,给出了所对应的工作状态域的电路参数估计,实验结果验证了其正确性。
在连续时间系统的延拓和分析方面,重点研究了三维混沌系统、四维超混沌系统,以及多涡卷混沌系统和光滑忆阻混沌电路。
提出了具有指数项的鲁棒混沌系统和规范形式的改进型广义Lorenz系统。具有指数项的鲁棒混沌系统有着简单的代数方程,能产生双涡卷混沌吸引子,在较宽的参数范围内存在鲁棒混沌现象；而改进型广义Lorenz系统因折叠因子的引入,可生成一类复杂的双涡卷和单涡卷折叠吸引子,有着丰富而复杂的非线性动力学现象。此外,通过加载线性的或者非线性的状态控制器获得了各种超混沌系统或其相应的电路,并由数值仿真和实验仿真得到了这类系统的超混沌吸引子。
进一步地,根据吸引子的形成机理,首先以两个饱和函数序列驱动线性系统为例,提出了生成多涡卷吸引子的一般思路,由此实现了一个(2N+1)×(2M+1)网格涡卷混沌系统；其次基于Colpitts振荡器模型,通过对模型状态方程中非线性项的改造,构造了网格涡卷和多涡卷的混沌和超混沌系统,生成了(2N+1)涡卷混沌吸引子和(2M+1)×(2N+1)网格涡卷超混沌吸引子；最后通过对线性系统中引入的两个锯齿波函数的设计,获得了不同涡卷位置分布的多涡卷吸引子,完成了吸引子涡卷数量、涡卷位置的设计。
在光滑忆阻混沌电路研究方面,提出了三次非线性函数描述的光滑磁控忆阻器,以此替换蔡氏混沌电路中的蔡氏二极管,导出了一个光滑忆阻混沌电路。采用常规的动力学分析手段研究了在电路参数和初始条件发生变化时的动力学特性。结果表明光滑忆阻振荡器与一般的混沌系统不同,它的动力学行为除了与电路参数有关外,还极端依赖于电路的初始状态,其运行轨道因初始状态不同将会在混沌振荡、周期振荡和稳定的汇之间发生转换。另外,分析了光滑忆阻振荡器中的瞬态混沌和状态转移等奇异的非线性现象。
The chaos-generating system is an important part of chaos theory and application study. The nonlinearity in dynamical systems is compulsory to generate chaotic phenomena and its characteristics and composition forms directly determine the chaotic behaviors and chaos-generating mechanisms. With advances in these areas, this dissertation conducts a thorough investigation on the nonlinear composition characteristics of the chaotic systems and extends some typical chaotic systems. Some new chaotic phenomena are found and new chaos-generating mechanisms are revealed.
This dissertation is divided into two parts of the discrete time systems and the continuous time systems.
In the study on discrete time systems, we pay attention on a class of one-dimensional and two-dimensional discrete maps with exponential quadratic terms and the current mode controlled switching DC-DC converters with ramp compensation.
Generalized square map and DOG wavelet map are constructed and their complex dynamical behaviors are revealed. With the evolving orbits of the fixed points and the iterative curves with corresponding degree, it is found that (1) one-dimensional generalized square map has physical phenomenon similar to single-peak square map and its dynamical behavior distributes in the unit region. (2) DOG wavelet map. whose the number of the fixed points varies with the parameters, exhibits rich nonlinear phenomena, such as period-doubling bifurcation, tangent bifurcation, boundary crisis bifurcation, period-window, and imperfect Feigenbaum-tree. and so on. (3) two-dimensional generalized square map has the phenomena of Hopf bifurcation and locked-frequency with complex, flexible, strange shaped limit cycles and chaotic attractors.
Switching DC-DC converter is a kind of special time-varying discrete chaotic systems and has two boundaries in the wide ranges of circuit parameters. It exhibits the phenomenon of border collision bifurcation. With the variations of circuit parameters, the switching DC-DC converter has two routes to chaos through period-doubling bifurcation and border collision bifurcation. The border collision bifurcation will lead to state shifts between the stable period-one state, robust chaos state in continuous conduction mode (CCM) and weak chaos state with strong intermittency in discontinuous conduction mode (DCM). By introducing suitable ramp compensation, the operation modes of the switching DC-DC converter can shift from DCM to CCM, and can also be controlled at the stable period-1 region. Furthermore, two borderline equations of the orbit state shifts are derived and the circuit parameters corresponding to the operation state regions are presented. The results are confirmed by circuit experiments.
In the study on continuous time systems, we pay attention on three-dimensional chaotic systems, four-dimensional hyperchaotic systems, as well as multi-scroll chaotic systems and smooth memristor chaotic circuits.
A robust chaotic system with an exponential quadratic term and a modified generalized Lorenz system in a canonical form are presented. The robust chaotic system has simple algebraic structure, can generate 2-scroll chaotic attractors and has chaotic phenomena in a very wide parameter ranges. The modified generalized Lorenz system with a folded factor can display complex 2-scroll and 1-scroll folded attractors and exhibits complicated nonlinear dynamical phenomena. In addition, different hyperchaotic systems or their corresponding circuits are obtained by adding linear or nonlinear state controllers to the different three-dimensional chaotic systems or circuits. Their hyperchaotic dynamics are comfirmed by simulations and experiments.
A general guide to design multi-scroll chaotic systems is proposed. With Colpitts oscillator model, by modifying the nonlinear term in model state equations, grid-scroll and multi-scroll chaotic and hyperchaotic systems are constructed, which can generate a (2N+1)-scroll chaotic attractor and a (2M+1)x(2N+1)-grid scroll hyperchaotic attractor. By designing two different saw-tooth functions in the linear system, some multi-scroll chaotic attractors with different scroll-location distributions are obtained and the designs of the scroll number and scroll-location distributions of the chaotic attractors are realized.
In the study on smooth memristor chaotic circuits, a smooth flux-controlled memristor having a cubic nonlinearity is presented and a smooth memristor chaotic circuit is derived by replacing Chua's diode in Chua's chaotic circuit with the memristor. The dynamical behaviors are analysed under different circuit parameters and initial states. The research results demonstrate that the dynamics of the smooth memristor oscillator, different from general chaotic systems, depend not only on circuit parameters, but also closely on the initial states. With different initial states, the circuit orbits will transfer among chaotic oscillation or periodic oscillation or stable sink. In addition, some transient chaos and state transitions in the smooth memristor oscillator are revealed and analyzed.
本论文工作主要分为离散时间系统和连续时间系统两部分。
在离散时间系统的延拓和分析方面,重点研究了具有指数二次项的一维和二维离散映射及含斜坡补偿的电流模式控制开关DC-DC变换器的动力学特性。
为了使得离散映射具有较为复杂的动力学特性,提出了一类具有指数二次项的一维和二维离散映射,利用不动点演变轨迹分析和仿真手段构造相应次数的迭代曲线,对它们的动力学特性进行了全面深入的研究,结果表明：一维广义平方映射有着与单峰平方映射相类似的物理现象,其动力学行为分布在一个单位区域内；一维DOG小波映射随着参数变化不动点会发生数量上的增加或减少,存在倍周期分岔、切分岔、边界危机分岔、周期窗以及不完全费根鲍姆树等非线性物理现象；而二维广义平方映射则存在Hopf分岔和锁频等现象,有着复杂多变、形状奇异的极限环和混沌吸引子。
开关DC-DC变换器是一种特殊的时变离散混沌系统,在较宽的电路参数下具有两个边界,存在一般混沌系统所不具有的边界碰撞分岔现象。随着电路参数变化,开关DC-DC变换器存在倍周期和边界碰撞两种分岔路由,并由边界碰撞分岔导致其运行轨道在稳定的周期1态、连续传导模式(CCM)的鲁棒混沌态和不连续传导模式(DCM)的强阵发性的弱混沌态之间发生模式转移现象。通过引入合适的斜坡补偿电流,开关DC-DC变换器的工作模式可以从DCM转移到CCM,也可以被镇定在周期1态。此外,通过导出运行轨道状态发生转移时的两个分界线方程,给出了所对应的工作状态域的电路参数估计,实验结果验证了其正确性。
在连续时间系统的延拓和分析方面,重点研究了三维混沌系统、四维超混沌系统,以及多涡卷混沌系统和光滑忆阻混沌电路。
提出了具有指数项的鲁棒混沌系统和规范形式的改进型广义Lorenz系统。具有指数项的鲁棒混沌系统有着简单的代数方程,能产生双涡卷混沌吸引子,在较宽的参数范围内存在鲁棒混沌现象；而改进型广义Lorenz系统因折叠因子的引入,可生成一类复杂的双涡卷和单涡卷折叠吸引子,有着丰富而复杂的非线性动力学现象。此外,通过加载线性的或者非线性的状态控制器获得了各种超混沌系统或其相应的电路,并由数值仿真和实验仿真得到了这类系统的超混沌吸引子。
进一步地,根据吸引子的形成机理,首先以两个饱和函数序列驱动线性系统为例,提出了生成多涡卷吸引子的一般思路,由此实现了一个(2N+1)×(2M+1)网格涡卷混沌系统；其次基于Colpitts振荡器模型,通过对模型状态方程中非线性项的改造,构造了网格涡卷和多涡卷的混沌和超混沌系统,生成了(2N+1)涡卷混沌吸引子和(2M+1)×(2N+1)网格涡卷超混沌吸引子；最后通过对线性系统中引入的两个锯齿波函数的设计,获得了不同涡卷位置分布的多涡卷吸引子,完成了吸引子涡卷数量、涡卷位置的设计。
在光滑忆阻混沌电路研究方面,提出了三次非线性函数描述的光滑磁控忆阻器,以此替换蔡氏混沌电路中的蔡氏二极管,导出了一个光滑忆阻混沌电路。采用常规的动力学分析手段研究了在电路参数和初始条件发生变化时的动力学特性。结果表明光滑忆阻振荡器与一般的混沌系统不同,它的动力学行为除了与电路参数有关外,还极端依赖于电路的初始状态,其运行轨道因初始状态不同将会在混沌振荡、周期振荡和稳定的汇之间发生转换。另外,分析了光滑忆阻振荡器中的瞬态混沌和状态转移等奇异的非线性现象。
The chaos-generating system is an important part of chaos theory and application study. The nonlinearity in dynamical systems is compulsory to generate chaotic phenomena and its characteristics and composition forms directly determine the chaotic behaviors and chaos-generating mechanisms. With advances in these areas, this dissertation conducts a thorough investigation on the nonlinear composition characteristics of the chaotic systems and extends some typical chaotic systems. Some new chaotic phenomena are found and new chaos-generating mechanisms are revealed.
This dissertation is divided into two parts of the discrete time systems and the continuous time systems.
In the study on discrete time systems, we pay attention on a class of one-dimensional and two-dimensional discrete maps with exponential quadratic terms and the current mode controlled switching DC-DC converters with ramp compensation.
Generalized square map and DOG wavelet map are constructed and their complex dynamical behaviors are revealed. With the evolving orbits of the fixed points and the iterative curves with corresponding degree, it is found that (1) one-dimensional generalized square map has physical phenomenon similar to single-peak square map and its dynamical behavior distributes in the unit region. (2) DOG wavelet map. whose the number of the fixed points varies with the parameters, exhibits rich nonlinear phenomena, such as period-doubling bifurcation, tangent bifurcation, boundary crisis bifurcation, period-window, and imperfect Feigenbaum-tree. and so on. (3) two-dimensional generalized square map has the phenomena of Hopf bifurcation and locked-frequency with complex, flexible, strange shaped limit cycles and chaotic attractors.
Switching DC-DC converter is a kind of special time-varying discrete chaotic systems and has two boundaries in the wide ranges of circuit parameters. It exhibits the phenomenon of border collision bifurcation. With the variations of circuit parameters, the switching DC-DC converter has two routes to chaos through period-doubling bifurcation and border collision bifurcation. The border collision bifurcation will lead to state shifts between the stable period-one state, robust chaos state in continuous conduction mode (CCM) and weak chaos state with strong intermittency in discontinuous conduction mode (DCM). By introducing suitable ramp compensation, the operation modes of the switching DC-DC converter can shift from DCM to CCM, and can also be controlled at the stable period-1 region. Furthermore, two borderline equations of the orbit state shifts are derived and the circuit parameters corresponding to the operation state regions are presented. The results are confirmed by circuit experiments.
In the study on continuous time systems, we pay attention on three-dimensional chaotic systems, four-dimensional hyperchaotic systems, as well as multi-scroll chaotic systems and smooth memristor chaotic circuits.
A robust chaotic system with an exponential quadratic term and a modified generalized Lorenz system in a canonical form are presented. The robust chaotic system has simple algebraic structure, can generate 2-scroll chaotic attractors and has chaotic phenomena in a very wide parameter ranges. The modified generalized Lorenz system with a folded factor can display complex 2-scroll and 1-scroll folded attractors and exhibits complicated nonlinear dynamical phenomena. In addition, different hyperchaotic systems or their corresponding circuits are obtained by adding linear or nonlinear state controllers to the different three-dimensional chaotic systems or circuits. Their hyperchaotic dynamics are comfirmed by simulations and experiments.
A general guide to design multi-scroll chaotic systems is proposed. With Colpitts oscillator model, by modifying the nonlinear term in model state equations, grid-scroll and multi-scroll chaotic and hyperchaotic systems are constructed, which can generate a (2N+1)-scroll chaotic attractor and a (2M+1)x(2N+1)-grid scroll hyperchaotic attractor. By designing two different saw-tooth functions in the linear system, some multi-scroll chaotic attractors with different scroll-location distributions are obtained and the designs of the scroll number and scroll-location distributions of the chaotic attractors are realized.
In the study on smooth memristor chaotic circuits, a smooth flux-controlled memristor having a cubic nonlinearity is presented and a smooth memristor chaotic circuit is derived by replacing Chua's diode in Chua's chaotic circuit with the memristor. The dynamical behaviors are analysed under different circuit parameters and initial states. The research results demonstrate that the dynamics of the smooth memristor oscillator, different from general chaotic systems, depend not only on circuit parameters, but also closely on the initial states. With different initial states, the circuit orbits will transfer among chaotic oscillation or periodic oscillation or stable sink. In addition, some transient chaos and state transitions in the smooth memristor oscillator are revealed and analyzed.
引文
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