基于脉冲控制的分叉与混沌系统研究
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摘要
在非线性科学领域中,分叉和混沌的分析、控制及反控制都是前沿的研究课题,极具挑战性。研究的目的是对给定的非线性动力系统设计一个控制器来抑制或者削弱系统中有害的分叉和混沌行为(控制分叉和混沌);产生、保持或增强健康的、有益的分叉和混沌行为(反控制分叉和混沌),从而产生人们所需要的动力学行为,迄今为止,这方面的研究还很少。我们采用的脉冲控制方法是可以直接或者间接地用来控制一个复杂动力系统的分叉和混沌行为的有效控制方法,可以实现对混沌或分叉系统稳定控制;利用脉冲控制使原本非混沌的系统混沌化(混沌反控制);利用脉冲实现动力系统的分叉反控制等都是我们的研究对象。
在全面分析和总结非线性动力系统分叉和混沌研究现状的基础上,基于非线性动力学、分叉和混沌理论等非线性科学的现代分析方法和脉冲控制理论,对非线性动力系统混沌和分叉行为的分析、控制及反控制问题进行了系统和深入的研究,并对相应的控制器的设计进行了理论分析和仿真验证。主要研究内容如下:
讨论了离散脉冲控制系统的稳定性问题。用李雅普诺夫定理和一些引理研究离散脉冲系统的解的稳定性。用李雅普诺夫函数对离散脉冲系统的渐近稳定性进行了分析且得到系统渐近稳定的基本条件,并且举例验证了理论结果。
讨论了脉冲控制非线性动力系统分叉的问题。提出了一种脉冲控制非线性动力系统分叉的策略,用李雅普诺夫函数证明了脉冲控制系统的稳定性并且推导出了该系统渐近稳定的一些充分条件,并且用数值仿真验证了该方法的有效性。
研究了小世界网络分叉和混沌的分析及控制问题。分析了小世界网络的分叉和混沌行为并且推导出分叉和混沌产生的参数条件。提出一种脉冲混杂控制方法来控制倍周期分叉和稳定嵌入混沌吸引子的不稳定的周期轨道。仿真结果表明,此方法可以延迟或完全消除小世界网络中的倍周期分叉,使整个系统能够在较大的参数值变化范围内保持稳定的动态特性。可以稳定嵌入混沌吸引子中的不稳定周期轨道;也可以将任意的p周期轨道有效地控制到2n ,3m及2n×3m周期以及预期的周期轨道。
研究了Internet网络拥塞控制系统分叉和混沌的分析及控制问题。首先分析了TCP-RED网络拥塞控制系统(具有RED(随机早期检测)排队管理的TCP(传输控制协议)网络)的分叉和混沌行为并且推导出分叉和混沌产生的参数条件;由于存在分叉和混沌造成TCP-RED系统的失稳现象,提出了一种脉冲混杂控制方法来稳定TCP-RED系统,实现改进RED路由器队列管理机制的性能。仿真表明这种控制方法可以较大增加系统的稳定工作区域并且增强工作点的稳定性。
研究了基于脉冲控制方法的分叉反控制问题。分析了在离散系统中分叉产生的条件和脉冲控制系统中分叉的存在性。提出了一种在离散非线性系统中用脉冲控制方法实现分叉反控制的方法。仿真结果显示,在受周期性脉冲控制的逻辑斯蒂映射中,产生了新的分叉类型,发现存在一条新的多重倍周期分叉通往混沌的路径,而且很大程度修改了分叉的结构和混沌行为的范围。说明了复杂动力学系统在周期性的脉冲控制作用之下改变系统的分叉和混沌轨道。
研究了基于脉冲控制方法的混沌反控制问题。提出一种反控制混沌的策略,对一个离散系统施加脉冲控制从而改变系统的状态。仿真结果显示可以用脉冲控制方法实现对离散系统混沌反控制,并且用不同的方法验证了,用脉冲控制方法能够把系统从稳定的周期运动状态驱动到随机的混沌运动状态。
最后对全文的研究工作进行了总结,并对将来要做的研究工作进行了展望。
In the nonlinear science field, bifurcation and chaos analysis, control and anti-control have become front research topics, extremely challenging. The research purposes is to de-sign a controller to suppress or reduce some harmful bifurcation and chaotic behavior of a given nonlinear system (control bifurcation and chaos) or to created, maintain, or enhance the healthy and useful behaviors of bifurcation and chaotic for a given nonlinear system (anti-control bifurcation and chaos), thereby achieving some desirable dynamical behav-iors, so far, its concerned work is very little. As one of the effective control methods, im-pulsive control can be directly or indirectly used for eliminating bifurcation or chaos from a complex dynamical system. Utilize the pulse to contr or anti-control the bifurcation or chaotic system are our research objects.
Through a comprehensive analysis and summary of the research history and actuality of bifurcation and chaos, we have conducted a systematic and thorough investigation into the fundamental theory and application of the bifurcation or chaos control by using the nonlinear dynamics theory, and the chaos and bifurcation theory and so on modern nonlinear science analysis methods and the impulse control theory.The main achievements and conclusions in this dissertation are obtained as follows:
The stability of discrete impulsive control system was discussed. Based on the theory of impulsive differential, the stability of discrete impulsive control system was studied by using Lyapunov method and some lemmas. Utilizing Lyapunov function proved asymp-totic stability of discrete impulsive system and the essential conditions to this kind of sta-bility have been researched. The theoretical results were verified by an example.
The method of impulsive control bifurcation in a nonlinear system was discussed. The stability of the system was proved by using Lyapunov function. The stability condi-tions have been derived. The simulation results confirmed this method validity.
The bifurcation and chaos behavior of small-world networks model has been investi-gated. The parameter conditions of bifurcation and chaos producing was deduced. An im-pulsive hybrid control method was proposed to control the bifurcations and stabilize the period orbits embedded in the chaotic attractor of a small-world network. Theoretical cal- culations and simulation results showed that the period doubling bifurcations can be de-layed or eliminated completely. The period doubling route to chaos is therefore at least delayed to greater parameter values.The unstable period orbits embedded can be stabilized and a p-period orbit can be controlled to desired orbits by adjusting control parameters.
The problem of analysis and control of bifurcations and chaos has been studied in a TCP-RED congestion control system model. We study the bifurcation and chaotic behav-ior of the TCP-RED system which may cause heavy oscillation of average queue length and induce network instability. We propose an impulsive control method for controlling bifurcations and chaos in the TCP-RED system, which is a simple approach improving the performance of the RED router queue management mechanism by adjusting the control parameters. The simulation experiments show that this method can obtain the stable aver-age queue length without sacrificing the advantages of RED.
The method of anti-control bifurcation via impulsive control has been researched. The bifurcation producing condition and anti-control of bifurcations method were dis-cussed. The simulation results showed the complex dynamics under the periodically im-pulsive control alters the bifurcations and chaos orbits of the system. The existence of a new multiple periods doubling bifurcation route to chaos and considerably modified bi-furcation structures and ranges of chaotic behavior are demonstrated by periodic impulses control. Anti-control bifurcations can be viewed as to design bifurcations into a system via impulsive control when such dynamical behaviors are desirable.
The problem of anti-control chaos in discrete system has been investigated. An anti-control of chaos strategy was proposed to drive a periodic motion of a discrete system into a chaotic motion by using an impulsive control method. The simulation results showed the method is effective and successfully realized to anti-control of chaos in the discrete system by the method. Some different ways confirmed nonlinear systems can be divered from stable period motion to random chaotic motion by the control method.
Finally, a summary has been done for all discussion in the dissertation. The research works in further study are presented.
在全面分析和总结非线性动力系统分叉和混沌研究现状的基础上,基于非线性动力学、分叉和混沌理论等非线性科学的现代分析方法和脉冲控制理论,对非线性动力系统混沌和分叉行为的分析、控制及反控制问题进行了系统和深入的研究,并对相应的控制器的设计进行了理论分析和仿真验证。主要研究内容如下:
讨论了离散脉冲控制系统的稳定性问题。用李雅普诺夫定理和一些引理研究离散脉冲系统的解的稳定性。用李雅普诺夫函数对离散脉冲系统的渐近稳定性进行了分析且得到系统渐近稳定的基本条件,并且举例验证了理论结果。
讨论了脉冲控制非线性动力系统分叉的问题。提出了一种脉冲控制非线性动力系统分叉的策略,用李雅普诺夫函数证明了脉冲控制系统的稳定性并且推导出了该系统渐近稳定的一些充分条件,并且用数值仿真验证了该方法的有效性。
研究了小世界网络分叉和混沌的分析及控制问题。分析了小世界网络的分叉和混沌行为并且推导出分叉和混沌产生的参数条件。提出一种脉冲混杂控制方法来控制倍周期分叉和稳定嵌入混沌吸引子的不稳定的周期轨道。仿真结果表明,此方法可以延迟或完全消除小世界网络中的倍周期分叉,使整个系统能够在较大的参数值变化范围内保持稳定的动态特性。可以稳定嵌入混沌吸引子中的不稳定周期轨道;也可以将任意的p周期轨道有效地控制到2n ,3m及2n×3m周期以及预期的周期轨道。
研究了Internet网络拥塞控制系统分叉和混沌的分析及控制问题。首先分析了TCP-RED网络拥塞控制系统(具有RED(随机早期检测)排队管理的TCP(传输控制协议)网络)的分叉和混沌行为并且推导出分叉和混沌产生的参数条件;由于存在分叉和混沌造成TCP-RED系统的失稳现象,提出了一种脉冲混杂控制方法来稳定TCP-RED系统,实现改进RED路由器队列管理机制的性能。仿真表明这种控制方法可以较大增加系统的稳定工作区域并且增强工作点的稳定性。
研究了基于脉冲控制方法的分叉反控制问题。分析了在离散系统中分叉产生的条件和脉冲控制系统中分叉的存在性。提出了一种在离散非线性系统中用脉冲控制方法实现分叉反控制的方法。仿真结果显示,在受周期性脉冲控制的逻辑斯蒂映射中,产生了新的分叉类型,发现存在一条新的多重倍周期分叉通往混沌的路径,而且很大程度修改了分叉的结构和混沌行为的范围。说明了复杂动力学系统在周期性的脉冲控制作用之下改变系统的分叉和混沌轨道。
研究了基于脉冲控制方法的混沌反控制问题。提出一种反控制混沌的策略,对一个离散系统施加脉冲控制从而改变系统的状态。仿真结果显示可以用脉冲控制方法实现对离散系统混沌反控制,并且用不同的方法验证了,用脉冲控制方法能够把系统从稳定的周期运动状态驱动到随机的混沌运动状态。
最后对全文的研究工作进行了总结,并对将来要做的研究工作进行了展望。
In the nonlinear science field, bifurcation and chaos analysis, control and anti-control have become front research topics, extremely challenging. The research purposes is to de-sign a controller to suppress or reduce some harmful bifurcation and chaotic behavior of a given nonlinear system (control bifurcation and chaos) or to created, maintain, or enhance the healthy and useful behaviors of bifurcation and chaotic for a given nonlinear system (anti-control bifurcation and chaos), thereby achieving some desirable dynamical behav-iors, so far, its concerned work is very little. As one of the effective control methods, im-pulsive control can be directly or indirectly used for eliminating bifurcation or chaos from a complex dynamical system. Utilize the pulse to contr or anti-control the bifurcation or chaotic system are our research objects.
Through a comprehensive analysis and summary of the research history and actuality of bifurcation and chaos, we have conducted a systematic and thorough investigation into the fundamental theory and application of the bifurcation or chaos control by using the nonlinear dynamics theory, and the chaos and bifurcation theory and so on modern nonlinear science analysis methods and the impulse control theory.The main achievements and conclusions in this dissertation are obtained as follows:
The stability of discrete impulsive control system was discussed. Based on the theory of impulsive differential, the stability of discrete impulsive control system was studied by using Lyapunov method and some lemmas. Utilizing Lyapunov function proved asymp-totic stability of discrete impulsive system and the essential conditions to this kind of sta-bility have been researched. The theoretical results were verified by an example.
The method of impulsive control bifurcation in a nonlinear system was discussed. The stability of the system was proved by using Lyapunov function. The stability condi-tions have been derived. The simulation results confirmed this method validity.
The bifurcation and chaos behavior of small-world networks model has been investi-gated. The parameter conditions of bifurcation and chaos producing was deduced. An im-pulsive hybrid control method was proposed to control the bifurcations and stabilize the period orbits embedded in the chaotic attractor of a small-world network. Theoretical cal- culations and simulation results showed that the period doubling bifurcations can be de-layed or eliminated completely. The period doubling route to chaos is therefore at least delayed to greater parameter values.The unstable period orbits embedded can be stabilized and a p-period orbit can be controlled to desired orbits by adjusting control parameters.
The problem of analysis and control of bifurcations and chaos has been studied in a TCP-RED congestion control system model. We study the bifurcation and chaotic behav-ior of the TCP-RED system which may cause heavy oscillation of average queue length and induce network instability. We propose an impulsive control method for controlling bifurcations and chaos in the TCP-RED system, which is a simple approach improving the performance of the RED router queue management mechanism by adjusting the control parameters. The simulation experiments show that this method can obtain the stable aver-age queue length without sacrificing the advantages of RED.
The method of anti-control bifurcation via impulsive control has been researched. The bifurcation producing condition and anti-control of bifurcations method were dis-cussed. The simulation results showed the complex dynamics under the periodically im-pulsive control alters the bifurcations and chaos orbits of the system. The existence of a new multiple periods doubling bifurcation route to chaos and considerably modified bi-furcation structures and ranges of chaotic behavior are demonstrated by periodic impulses control. Anti-control bifurcations can be viewed as to design bifurcations into a system via impulsive control when such dynamical behaviors are desirable.
The problem of anti-control chaos in discrete system has been investigated. An anti-control of chaos strategy was proposed to drive a periodic motion of a discrete system into a chaotic motion by using an impulsive control method. The simulation results showed the method is effective and successfully realized to anti-control of chaos in the discrete system by the method. Some different ways confirmed nonlinear systems can be divered from stable period motion to random chaotic motion by the control method.
Finally, a summary has been done for all discussion in the dissertation. The research works in further study are presented.
引文
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