几类高阶时滞系统的动力学与控制
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摘要
对于高维的时滞系统而言,其对应的特征方程是含有指数的超越方程,具有很高的阶数.一般情形下,分析多时滞系统“所有的特征根都具有负实部”这样的稳定性条件几乎是不可能的,甚至给出稳定性的边界条件也是困难的.现有的非线性动力学理论处理低维系统是有效的,但许多的理论难以应用于高维系统,特别是高维时滞系统.事实上,一个切实有效的降维方法将使得复杂系统的稳定性、分岔与混沌的研究变得容易进行,如何将大型、复杂的高维系统进行降维处理已经成为现代非线性动力学研究的热点问题之一.
另外,高维时滞系统的动力学控制是否有新的特点?工程实践中常用的位移反馈控制,速度反馈控制等方法,是否还能有效控制高维时滞系统?这些都是值得探讨的新问题.针对上述几个问题,本文分析了几类高阶时滞系统的动力学与控制,包括高维小世界网络的动力学与控制,具有记忆的小世界网络的动力学与控制,粘弹性振动系统的动力学与控制,参数与时滞相关的振动系统的动力学与控制,以及参数与时滞相关的两个神经元系统的动力学.本文具体的研究内容如下.
(1)详细分析了具有时滞的3-D小世界网络的动力学性质与控制.首先对系统的平衡点进行了研究并指出系统存在复杂的无界动力学行为.然后提出三种动力学控制方法:位移反馈控制,速度反馈控制,速度和加速度反馈控制,文中指出,这三种控制方法中,只有速度和加速度混合反馈控制才能镇定3-D小世界网络.适当的选择控制时滞和控制增益,控制系统可能有稳定的平衡点、由Hopf分岔产生的周期解、或者由倍周期分岔导致的奇怪混沌吸引子.本章的结果可推广至任意的“d”维系统.研究表明,控制“d”维时滞小世界网络,至少需要“d-1”阶微分反馈控制.此工作为高维时滞系统的动力学控制提供了新思路.
(2)提出了一个具有记忆的小世界网络群体动力学模型,分析了分布时滞对小世界网络动力学性质的影响.稳定性分析指出模型的影响总量是无界的,因此导致信息在网络的传播呈指数形式增长.我们还讨论了有限尺寸对网络动力学的影响,给出了所有节点都受到感染的饱和时间.此外,研究了时滞反馈对网络系统动力学的影响,对于适当的反馈增益和时滞,控制模型有稳定的平衡点,周期解,拟周期解以及由倍周期分岔导致的混沌.文中指出,时滞反馈控制可以作为系统管理和动力学控制的重要研究手段,将会促进复杂网络在实际中应用.
(3)详细分析了具有粘弹性项的Duffing系统动力学性质与控制.首先未控系统具有复杂的无界动力学行为,然后引入时滞速度反馈控制来镇定这样的系统.研究结果表明适当的选择控制增益和控制时滞,系统可产生复杂的动力学行为,包括稳定的平衡点、由Hopf分岔产生的周期解、拟周期解及余维2分岔和混沌解.数值模拟证明了结论的正确性.
(4)分析了参数与时滞相关的延迟振子的动力学.应用稳定性切换准则,研究了平衡点的稳定区间,以及由Hopf分岔产生的周期解.结果指出,当时滞从零增加到无穷大,振子可能经历有限次稳定性切换,系统的平衡点最终可能是稳定的或不稳定的,这个结论与参数和时滞无关的系统有很大的不同.另外,数值模拟结果表明,对于非常简单的双曲正切激活函数,系统也会产生复杂的动力学行为,并发现了两条通往混沌的路径:倍周期分岔导致混沌和拟周期分岔导致混沌.研究进一步指出记忆函数显著影响系统的稳定性,因此可以通过选择合适的记忆函数来控制系统的动力学行为.
(5)应用稳定性切换准则分析了参数与时滞相关的具有惯性项的两个神经元系统的稳定性.研究表明,随着时滞的增加,系统会发生有限次的稳定性切换,但系统最终可能是稳定的,不稳定的或者在稳定和不稳定间切换.另外,文中还通过数值模拟分析了系统独特的动力学行为,包括复杂的周期运动与混沌解.
在结束语中,对全文进行了总结,并提出了可能存在的问题和进一步的研究方向.
本文的创新点为:(1)提出了一个具有记忆的小世界网络的群体动力学模型,分析了其动力学与控制;(2)提出了一种有效控制高维小世界网络群体动力学的方法;(3)对于参数与时滞相关的系统,提出了将记忆函数作为系统动力学控制的有效方法.
It is well known that the characteristic equation of higher order delayed systems istranscendental equation which has very high orders. Normally, for the system with multipledelays, it is impossible to get the stability condition that all the eigenvaues have negative realparts; it is also difficult to give the stability boundary conditions for such systems. Existingtheories of nonlinear dynamics are effective for low-dimensional systems, but many of themare difficult to apply to higher order systems, especially higher order delayed systems. In fact,an effective dimension reduction method can make the analysis of stability, bifurcation andchaos of complex system easier. How to reduce the dimension of large, complexhigh-dimensional systems has become one of the hot research of nonlinear dynamics.
In addition, whether the dynamics control of the higher order delayed systems has newproperties? If the displacement feedback control and velocity feedback control methodswhich is used in the engineering practice can effectively control the higher order delayedsystems? These are new problems worthy to be discussed. For the above problems, thispaper analyzes the dynamics and control of several kind of higher order delayed systems,including the dynamics and control of the small-world networks, the small world networkwith memory, dynamics and control of the viscoelastic vibration system, dynamics andcontrol of the vibration system with delay dependent parameters and dynamics of two neuronsystem with delay dependent parameters. This dissertation focuses on the followingproblems.
(1) A detailed analysis on the collective dynamics and delayed state feedback control ofa three-dimensional delayed small-world network is presented. The trivial equilibrium of themodel is first investigated and shows that uncontrolled model exhibits unbounded behavior.Then presente three control strategies, namely a position feedback control, a velocityfeedback control, and a hybrid control combined velocity with acceleration feedback. Itshows in these three control schemes only the hybrid control can easily stabilize the3-Dnetwork system. And with properly chosen delay and gain in the delayed feedback path, thehybrid controlled model may have stable equilibrium, or periodic solutions resulted from theHopf bifurcation, or complex stranger attractor from the period-doubling bifurcation. Theresults of this paper are further extended to any “d” dimensional network. It shows that tostabilize a “d” dimensional delayed small world network, at least a “d-1” order completeddifferential feedback is needed. This work provides a new thought for the high dimensionaldelayed systems.
(2) A nonlinear small-world network model with memory is presented to investigate the effects of the distribution delays on the dynamical properties of small-world networks.Stability analysis shows that the total infected volume of this model are unbounded whichresult in the exponential growth of an infection in networks. We also discuss the effect offinite size on the network dynamics, and give the saturate time which all the nodes areinfected by the initial infection. The dynamical control is also investigated by introducing thedelayed feedback to the small-world networks. For the appropriate feedback gain and timedelay, controlled model may have stable equilibrium, periodic solution, quasi-periodicsolution, or chaos from a sequence of period-doubling bifurcations. It shows delayedfeedback control may find important applications in the management and dynamical control.This shall be the motivations of some further studies of the dynamics of real networks.
(3) The dynamics and delayed state feedback control of oscillator with generalizedviscoelastic item is discussed. It shows that uncontrolled model exhibits complicatedunbounded behavior. It shows that with properly chosen delay and gain in the delayedfeedback path, the velocity controlled model may have stable equilibrium, or periodicsolutions resulted from the Hopf bifurcation, codimension2bifurcation or complex strangerattractor.
(4) The dynamics of a delayed oscillator with delay-dependent parameter is presented.On the basis of stability switch criteria, the equilibrium is studied, and the stabilityconditions and the periodic solutions bifurcating from the equilibrium are determined. Theresults show that as the delay varies from zero to infinity, the oscillator may undergo anumber of stability switches, it could be eventual stable or unstable. This is very differentfrom that in the system without delay-dependent parameters. In addition, the numericalstudies show there are two routes to chaos for this oscillator even with the very simplehyperbolic tangent nonlinear function: period-doubling bifurcations process to Chaos andquasi-periodic solutions bifurcation to Chaos. It shows that appropriate memory function canbe used as a promising scheme to control the system dynamics.
(5) A detailed analysis on the stability switches of an inertial two neurons model withdelay-dependent parameter are presented. It shows that system will undergo a finite numberof stability switches with an increase of time delay. The system finally will be stable orunstable. Moreover, the paper also analyzed the unique dynamics of the system by numericalsimulation, including complex periodic motions and chaotic solutions.
Finally, a summary of this dissertation is given. The further studies on sytems with timedelays are discussed.
The innovations of this dissertation are as follows:(1) A collective model of asmall-world network with memory is proposed, and a dynamical control scheme is given forthe effective control.(2) An effective delayed control scheme is proposed for the higher order system control.(3) For the sysems with delay-dependent parameters, memory functioncould be one of the promising schemes for system control.
另外,高维时滞系统的动力学控制是否有新的特点?工程实践中常用的位移反馈控制,速度反馈控制等方法,是否还能有效控制高维时滞系统?这些都是值得探讨的新问题.针对上述几个问题,本文分析了几类高阶时滞系统的动力学与控制,包括高维小世界网络的动力学与控制,具有记忆的小世界网络的动力学与控制,粘弹性振动系统的动力学与控制,参数与时滞相关的振动系统的动力学与控制,以及参数与时滞相关的两个神经元系统的动力学.本文具体的研究内容如下.
(1)详细分析了具有时滞的3-D小世界网络的动力学性质与控制.首先对系统的平衡点进行了研究并指出系统存在复杂的无界动力学行为.然后提出三种动力学控制方法:位移反馈控制,速度反馈控制,速度和加速度反馈控制,文中指出,这三种控制方法中,只有速度和加速度混合反馈控制才能镇定3-D小世界网络.适当的选择控制时滞和控制增益,控制系统可能有稳定的平衡点、由Hopf分岔产生的周期解、或者由倍周期分岔导致的奇怪混沌吸引子.本章的结果可推广至任意的“d”维系统.研究表明,控制“d”维时滞小世界网络,至少需要“d-1”阶微分反馈控制.此工作为高维时滞系统的动力学控制提供了新思路.
(2)提出了一个具有记忆的小世界网络群体动力学模型,分析了分布时滞对小世界网络动力学性质的影响.稳定性分析指出模型的影响总量是无界的,因此导致信息在网络的传播呈指数形式增长.我们还讨论了有限尺寸对网络动力学的影响,给出了所有节点都受到感染的饱和时间.此外,研究了时滞反馈对网络系统动力学的影响,对于适当的反馈增益和时滞,控制模型有稳定的平衡点,周期解,拟周期解以及由倍周期分岔导致的混沌.文中指出,时滞反馈控制可以作为系统管理和动力学控制的重要研究手段,将会促进复杂网络在实际中应用.
(3)详细分析了具有粘弹性项的Duffing系统动力学性质与控制.首先未控系统具有复杂的无界动力学行为,然后引入时滞速度反馈控制来镇定这样的系统.研究结果表明适当的选择控制增益和控制时滞,系统可产生复杂的动力学行为,包括稳定的平衡点、由Hopf分岔产生的周期解、拟周期解及余维2分岔和混沌解.数值模拟证明了结论的正确性.
(4)分析了参数与时滞相关的延迟振子的动力学.应用稳定性切换准则,研究了平衡点的稳定区间,以及由Hopf分岔产生的周期解.结果指出,当时滞从零增加到无穷大,振子可能经历有限次稳定性切换,系统的平衡点最终可能是稳定的或不稳定的,这个结论与参数和时滞无关的系统有很大的不同.另外,数值模拟结果表明,对于非常简单的双曲正切激活函数,系统也会产生复杂的动力学行为,并发现了两条通往混沌的路径:倍周期分岔导致混沌和拟周期分岔导致混沌.研究进一步指出记忆函数显著影响系统的稳定性,因此可以通过选择合适的记忆函数来控制系统的动力学行为.
(5)应用稳定性切换准则分析了参数与时滞相关的具有惯性项的两个神经元系统的稳定性.研究表明,随着时滞的增加,系统会发生有限次的稳定性切换,但系统最终可能是稳定的,不稳定的或者在稳定和不稳定间切换.另外,文中还通过数值模拟分析了系统独特的动力学行为,包括复杂的周期运动与混沌解.
在结束语中,对全文进行了总结,并提出了可能存在的问题和进一步的研究方向.
本文的创新点为:(1)提出了一个具有记忆的小世界网络的群体动力学模型,分析了其动力学与控制;(2)提出了一种有效控制高维小世界网络群体动力学的方法;(3)对于参数与时滞相关的系统,提出了将记忆函数作为系统动力学控制的有效方法.
It is well known that the characteristic equation of higher order delayed systems istranscendental equation which has very high orders. Normally, for the system with multipledelays, it is impossible to get the stability condition that all the eigenvaues have negative realparts; it is also difficult to give the stability boundary conditions for such systems. Existingtheories of nonlinear dynamics are effective for low-dimensional systems, but many of themare difficult to apply to higher order systems, especially higher order delayed systems. In fact,an effective dimension reduction method can make the analysis of stability, bifurcation andchaos of complex system easier. How to reduce the dimension of large, complexhigh-dimensional systems has become one of the hot research of nonlinear dynamics.
In addition, whether the dynamics control of the higher order delayed systems has newproperties? If the displacement feedback control and velocity feedback control methodswhich is used in the engineering practice can effectively control the higher order delayedsystems? These are new problems worthy to be discussed. For the above problems, thispaper analyzes the dynamics and control of several kind of higher order delayed systems,including the dynamics and control of the small-world networks, the small world networkwith memory, dynamics and control of the viscoelastic vibration system, dynamics andcontrol of the vibration system with delay dependent parameters and dynamics of two neuronsystem with delay dependent parameters. This dissertation focuses on the followingproblems.
(1) A detailed analysis on the collective dynamics and delayed state feedback control ofa three-dimensional delayed small-world network is presented. The trivial equilibrium of themodel is first investigated and shows that uncontrolled model exhibits unbounded behavior.Then presente three control strategies, namely a position feedback control, a velocityfeedback control, and a hybrid control combined velocity with acceleration feedback. Itshows in these three control schemes only the hybrid control can easily stabilize the3-Dnetwork system. And with properly chosen delay and gain in the delayed feedback path, thehybrid controlled model may have stable equilibrium, or periodic solutions resulted from theHopf bifurcation, or complex stranger attractor from the period-doubling bifurcation. Theresults of this paper are further extended to any “d” dimensional network. It shows that tostabilize a “d” dimensional delayed small world network, at least a “d-1” order completeddifferential feedback is needed. This work provides a new thought for the high dimensionaldelayed systems.
(2) A nonlinear small-world network model with memory is presented to investigate the effects of the distribution delays on the dynamical properties of small-world networks.Stability analysis shows that the total infected volume of this model are unbounded whichresult in the exponential growth of an infection in networks. We also discuss the effect offinite size on the network dynamics, and give the saturate time which all the nodes areinfected by the initial infection. The dynamical control is also investigated by introducing thedelayed feedback to the small-world networks. For the appropriate feedback gain and timedelay, controlled model may have stable equilibrium, periodic solution, quasi-periodicsolution, or chaos from a sequence of period-doubling bifurcations. It shows delayedfeedback control may find important applications in the management and dynamical control.This shall be the motivations of some further studies of the dynamics of real networks.
(3) The dynamics and delayed state feedback control of oscillator with generalizedviscoelastic item is discussed. It shows that uncontrolled model exhibits complicatedunbounded behavior. It shows that with properly chosen delay and gain in the delayedfeedback path, the velocity controlled model may have stable equilibrium, or periodicsolutions resulted from the Hopf bifurcation, codimension2bifurcation or complex strangerattractor.
(4) The dynamics of a delayed oscillator with delay-dependent parameter is presented.On the basis of stability switch criteria, the equilibrium is studied, and the stabilityconditions and the periodic solutions bifurcating from the equilibrium are determined. Theresults show that as the delay varies from zero to infinity, the oscillator may undergo anumber of stability switches, it could be eventual stable or unstable. This is very differentfrom that in the system without delay-dependent parameters. In addition, the numericalstudies show there are two routes to chaos for this oscillator even with the very simplehyperbolic tangent nonlinear function: period-doubling bifurcations process to Chaos andquasi-periodic solutions bifurcation to Chaos. It shows that appropriate memory function canbe used as a promising scheme to control the system dynamics.
(5) A detailed analysis on the stability switches of an inertial two neurons model withdelay-dependent parameter are presented. It shows that system will undergo a finite numberof stability switches with an increase of time delay. The system finally will be stable orunstable. Moreover, the paper also analyzed the unique dynamics of the system by numericalsimulation, including complex periodic motions and chaotic solutions.
Finally, a summary of this dissertation is given. The further studies on sytems with timedelays are discussed.
The innovations of this dissertation are as follows:(1) A collective model of asmall-world network with memory is proposed, and a dynamical control scheme is given forthe effective control.(2) An effective delayed control scheme is proposed for the higher order system control.(3) For the sysems with delay-dependent parameters, memory functioncould be one of the promising schemes for system control.
引文
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