耦合混沌振子的反向同步与振幅死亡
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摘要
随着人们对非线性系统的深入研究,人们对混沌系统的动力学行为有了更详细的了解,进而掌握了越来越多的混沌同步和混沌控制的方法。1990年物理评论快报上关于混沌同步和混沌控制的最初的两篇论文标志着混沌同步的研究朝着应用的方向发展。在过去的十几年里,人们通过对耦合混沌系统的深入研究发现了许多不同的形式的同步动力学行为(其中包括完全同步,相同步,广义同步,滞后同步,部分同步,间歇滞后同步,哈密顿系统的测度同步等)以及在耦合相互作用下系统间由于存在参数失配或是时延造成的振幅死亡。由于对这些动力学的深入研究可以为混沌控制、混沌加密通信等应用提供非常重要的理论基础,所以对耦合系统动力学的研究具有重要的意义。在本文中,我们研究了耦合混沌振子的几种动力学行为如外信号驱动下的相同步,反向同步,振幅死亡等,并对出现这些动力学行为的内在机制作了相应的分析。主要考查以下几种动力学行为:(1)通过电路实验,研究了几种不同种类的周期外信号驱动下,混沌系统与外信号达到相同步的动力学行为和达到相同步所需的条件;(2)以耦合混沌振子系统为研究对象,分别通过数值计算和电路实验方法来研究耦合混沌系统中的丰富动力学行为如耦合混沌振子的反向同步以及振幅死亡动力学行为,并给出了产生这些动力学的内在机制及产生的条件。(3)研究了耦合映象格子中的扭结和尖峰结构的特征,并用模数分析法近似地分析了扭结和尖峰结构的稳定性及其失稳的机制。
本文的第一章简要地回顾了混沌同步研究的历史与现状,描述了各种混沌同步现象的产生和性质。
在第二章中,我们详细介绍了几种不同周期信号驱动下,混沌系统与周期信号达到相同步的动力学行为和达到相同步所需的条件。以Chua电路为研究对象,我们分别讨论了正弦信号,脉冲信号在各种不同参数(频率、振幅、占空比等)下,使混沌系统达到相同步的参数区间。另外还讨论了噪声驱动下两个混沌振子之间的相同步以及直流信号驱动下两个混沌振子的相同步。
第三章研究了几种耦合混沌系统中的反向同步动力学行为。我们给出了耦合混沌系统出现反向同步的条件,理论上给出了几种耦合混沌振子出现反向同步的参数区间,并观察到了耦合混沌振子走向反向同步时的丰富动力学行为如多态共存、网筛状分形吸引域、开关阵发和迟滞等现象。通过深入分析耦合混沌振子的反向同步动力学,揭示了走向反向同步的道路,以及混沌反向同步与完全同步的异同点,最后通过电子电路实验验证由理论预测的反向同步区间。
第四章从理论和数值上研究了耦合混沌振子在相互作用下出现部分振幅死亡和完全振幅死亡的动力学行为。在特定的耦合方式下,随着耦合强度的增加,耦合Lorenz系统会出现从镜象对称性破缺到平移对称性破缺直到部分振幅死亡(某个振子的部分变量停止振荡,而其它变量保持振荡)的现象。我们从理论上分析了出现这一现象的内在机制并以电子电路实验验证这一现象。讨论了几种不同的耦合混沌系统,在没有参数失配及时间延迟的情形下的振幅死亡动力学。并通过对耦合混沌系统的固定点的稳定性分析,给出了耦合混沌系统出现振幅死亡的参数区间。发现了耦合混沌振子走向振幅死亡的动力学行为(两态共存、对称性破缺,同步混沌的开关阵发)。
第五章以数值计算和理论分析研究了耦合映象格子的斑图动力学行为。发现在最简单的冻结化随机图案模式中除了有扭结(反扭结)结构和平台区外,还有一种新的结构尖峰结构。并采用模数分析方法分析了扭结结构和不同模数的尖峰结构的稳定性。并通过稳定性分析给出了刷子状的分岔图和锯齿状的最大李指数产生的机制,以及随着耦合强度增加,扭结结构、尖峰结构、平台区尺寸变化的关系及在斑图形成中的作用。第六章为全文的总结。
Rich dynamics of the chaotic systems had been explored and revealed by the researchers since great efforts had been stress on the nonlinear dynamics research. The method of chaos control and the synchronization of chaotic systems had been rapidly developed since the two reports on the physics review letter about chaos control and chaos synchronization was released by Peccora. Rich dynamics had been found in the coupled chaotic oscillators such as various of synchronization (including complete synchronization, phase synchronization, generalized synchronization, lagged synchronization, partial synchronization, intermittent lagged synchronization, measure synchronization of the coupled Hamiltonian systems etc) and the amplitude death caused by parameter mismatches or time delay in the coupled systems. It is significant to explore these dynamics for its perspective applications in chaos control and security communications. In this article, several dynamics of the coupled chaotic oscillators and corresponding inner regimes of those dynamics are explored carefully including various synchronization dynamics (phase synchronization, anti-synchronization) and amplitude death dynamics. Detailed dynamics are explored theretically and experimentally such as (1) the dynamics of phase synchronization (PS) and necessary conditions of PS in the signal driving chaotic oscillators. (2) Rich dynamics of the coupled chaotic oscillators with various coupling scheme are explored in detail such as the anti-synchronization and partial or complete amplitude death. (3) The patterns with pulses and kinks are explored in the coupled map lattices and the stability of the pulses structure is analyzed proximately by the modes method and the regimes of the stability losing are explored in the pulses and kinks structure. This paper is arranged as follows:
In the first chapter, the background and recent progresses of the chaos synchronization theory are introduced briefly and various phenomena of chaos synchronization and their characteristics are shown.
In the second chapter, phase synchronization of the chaotic oscillators driven by the periodical signals such as sine wave, pulse wave with different parameter (frequency, amplitude, duty circle) are explored extensively in electronics experiment. Moreover, phase synchronization between two chaotic oscillators driven by the common signals such as the noise or the directed voltage is explored.
In the third chapter, the dynamics of anti-synchronization are explored in various coupled chaotic oscillators. The necessary conditions and the parameters of the anti-synchronization are presented theoretically. Rich dynamics are found in the anti-synchronization such as the coexistence of multi attractors, riddled basin and on-off intermittency. Transitions to the anti-synchronizations are revealed and the characters of the anti-synchronization are analyzed compared to the complete synchronization. Electronics circuits are set up to observe the anti-synchronization.
In the fourth chapter the partial amplitude death and amplitude death of the coupled chaotic oscillators are explored numerically and theoretically. In some specific coupling scheme, the coupled Lorenz oscillators will transit from the reflection symmetry breaking to the translational symmetry breaking till partial amplitude death (some of the variables of the oscillators cease oscillating while others not) with increasing coupling constant. The inner schemes of the partial amplitude death are presented theoretically and the electronics circuits are setup to verify this phenomenon. Moreover, the amplitude death in coupled identical chaotic oscillators without time delay are explored numerically and clarified to three types.
In the fifth chapter the random frozened patterns with kink and pulse structure are explored in the coupled map lattices numerically. The stability of these two structures kinks and pulses are analyzed according to the proximate models. Different modes are found important in the formation of the patterns and the inner regimes of the brushlike bifurcation diagrams and the zigzag maximumal lyapunov exponent of the coupled map lattices are explored.
Finally the summary of the whole articles are present.
本文的第一章简要地回顾了混沌同步研究的历史与现状,描述了各种混沌同步现象的产生和性质。
在第二章中,我们详细介绍了几种不同周期信号驱动下,混沌系统与周期信号达到相同步的动力学行为和达到相同步所需的条件。以Chua电路为研究对象,我们分别讨论了正弦信号,脉冲信号在各种不同参数(频率、振幅、占空比等)下,使混沌系统达到相同步的参数区间。另外还讨论了噪声驱动下两个混沌振子之间的相同步以及直流信号驱动下两个混沌振子的相同步。
第三章研究了几种耦合混沌系统中的反向同步动力学行为。我们给出了耦合混沌系统出现反向同步的条件,理论上给出了几种耦合混沌振子出现反向同步的参数区间,并观察到了耦合混沌振子走向反向同步时的丰富动力学行为如多态共存、网筛状分形吸引域、开关阵发和迟滞等现象。通过深入分析耦合混沌振子的反向同步动力学,揭示了走向反向同步的道路,以及混沌反向同步与完全同步的异同点,最后通过电子电路实验验证由理论预测的反向同步区间。
第四章从理论和数值上研究了耦合混沌振子在相互作用下出现部分振幅死亡和完全振幅死亡的动力学行为。在特定的耦合方式下,随着耦合强度的增加,耦合Lorenz系统会出现从镜象对称性破缺到平移对称性破缺直到部分振幅死亡(某个振子的部分变量停止振荡,而其它变量保持振荡)的现象。我们从理论上分析了出现这一现象的内在机制并以电子电路实验验证这一现象。讨论了几种不同的耦合混沌系统,在没有参数失配及时间延迟的情形下的振幅死亡动力学。并通过对耦合混沌系统的固定点的稳定性分析,给出了耦合混沌系统出现振幅死亡的参数区间。发现了耦合混沌振子走向振幅死亡的动力学行为(两态共存、对称性破缺,同步混沌的开关阵发)。
第五章以数值计算和理论分析研究了耦合映象格子的斑图动力学行为。发现在最简单的冻结化随机图案模式中除了有扭结(反扭结)结构和平台区外,还有一种新的结构尖峰结构。并采用模数分析方法分析了扭结结构和不同模数的尖峰结构的稳定性。并通过稳定性分析给出了刷子状的分岔图和锯齿状的最大李指数产生的机制,以及随着耦合强度增加,扭结结构、尖峰结构、平台区尺寸变化的关系及在斑图形成中的作用。第六章为全文的总结。
Rich dynamics of the chaotic systems had been explored and revealed by the researchers since great efforts had been stress on the nonlinear dynamics research. The method of chaos control and the synchronization of chaotic systems had been rapidly developed since the two reports on the physics review letter about chaos control and chaos synchronization was released by Peccora. Rich dynamics had been found in the coupled chaotic oscillators such as various of synchronization (including complete synchronization, phase synchronization, generalized synchronization, lagged synchronization, partial synchronization, intermittent lagged synchronization, measure synchronization of the coupled Hamiltonian systems etc) and the amplitude death caused by parameter mismatches or time delay in the coupled systems. It is significant to explore these dynamics for its perspective applications in chaos control and security communications. In this article, several dynamics of the coupled chaotic oscillators and corresponding inner regimes of those dynamics are explored carefully including various synchronization dynamics (phase synchronization, anti-synchronization) and amplitude death dynamics. Detailed dynamics are explored theretically and experimentally such as (1) the dynamics of phase synchronization (PS) and necessary conditions of PS in the signal driving chaotic oscillators. (2) Rich dynamics of the coupled chaotic oscillators with various coupling scheme are explored in detail such as the anti-synchronization and partial or complete amplitude death. (3) The patterns with pulses and kinks are explored in the coupled map lattices and the stability of the pulses structure is analyzed proximately by the modes method and the regimes of the stability losing are explored in the pulses and kinks structure. This paper is arranged as follows:
In the first chapter, the background and recent progresses of the chaos synchronization theory are introduced briefly and various phenomena of chaos synchronization and their characteristics are shown.
In the second chapter, phase synchronization of the chaotic oscillators driven by the periodical signals such as sine wave, pulse wave with different parameter (frequency, amplitude, duty circle) are explored extensively in electronics experiment. Moreover, phase synchronization between two chaotic oscillators driven by the common signals such as the noise or the directed voltage is explored.
In the third chapter, the dynamics of anti-synchronization are explored in various coupled chaotic oscillators. The necessary conditions and the parameters of the anti-synchronization are presented theoretically. Rich dynamics are found in the anti-synchronization such as the coexistence of multi attractors, riddled basin and on-off intermittency. Transitions to the anti-synchronizations are revealed and the characters of the anti-synchronization are analyzed compared to the complete synchronization. Electronics circuits are set up to observe the anti-synchronization.
In the fourth chapter the partial amplitude death and amplitude death of the coupled chaotic oscillators are explored numerically and theoretically. In some specific coupling scheme, the coupled Lorenz oscillators will transit from the reflection symmetry breaking to the translational symmetry breaking till partial amplitude death (some of the variables of the oscillators cease oscillating while others not) with increasing coupling constant. The inner schemes of the partial amplitude death are presented theoretically and the electronics circuits are setup to verify this phenomenon. Moreover, the amplitude death in coupled identical chaotic oscillators without time delay are explored numerically and clarified to three types.
In the fifth chapter the random frozened patterns with kink and pulse structure are explored in the coupled map lattices numerically. The stability of these two structures kinks and pulses are analyzed according to the proximate models. Different modes are found important in the formation of the patterns and the inner regimes of the brushlike bifurcation diagrams and the zigzag maximumal lyapunov exponent of the coupled map lattices are explored.
Finally the summary of the whole articles are present.
引文
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