动力系统中的混沌行为及其控制
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摘要
在许多工程实际和科学研究问题中混沌现象已经成为一个无法避免的存在，因而对混沌的控制显得越来越重要。虽然自九十年代初以来，混沌控制研究得到蓬勃发展并取得了突破性的成果，但对混沌控制方法的研究还处在起步阶段，尚有大量的问题有待解决。本文将分别在扩大系统混沌窗口和抑制系统混沌这两个混沌控制方面进行研究。
第一部分，本文对梅尔尼科夫理论进行拓展，得到了一种高维梅尔尼科夫方法的随机推广形式，并将这种方法应用于带慢变参数和弱反馈控制的非线性哈密顿系统在谐和和随机激励双重作用下的同宿分叉和混沌运动的分析中，同时利用数值试验进行了验证。数值结果和理论结果吻合得很好，证明本文的推广是有效的。结果表明，适当的随机激励可以使得系统出现混沌运动的参数阈值变化范围更大，从而系统的混沌运动更容易出现。
第二部分，本文进行了混沌抑制的研究。在离散动力系统的混沌抑制方面，本文利用新的平衡点邻域线性化方法对改进OGY方法进行了修正，提出了新的控制启动条件和控制反馈参数。利用此方法对一类在zy平面上受到可变外部磁场作用的Bloch系统的混沌运动进行了控制，结果表明，方法成功地抑制了系统的混沌运动，说明此方法的改进是合理和有效的。
在时间连续动力系统方面，基于Krasovskii理论本文提出了一种的新优化控制方法，并利用这一方法对一类具有双奇怪吸引子的Newton-Leipnik(N-L)系统进行了混沌抑制，结果表明一旦控制启动，系统行为能够迅速渐近稳定于目标平衡点。与一类bang-bang控制方法相比，本文的方法具有更好的抑制效果和更灵活的抑制目标选择。
本文的创新之处体现在三个方面。一是提出了一种高维随机梅尔尼科夫推广方法，这一方法使得我们顺利利用合适的噪声扩大系统混沌窗口成为可能。二是在一新的平衡点邻域线性化方法基础上进行了OGY方法的修正，实例控制表明本文进行的修正不仅能使受控系统具有更理想的输出，同时使得输出有更灵活的可调控性。第三就是在Krasovskii理论的基础上提出了一种新的优化控制方法，这一方法使得我们可以根据实际需要随意改变系统状态，并能够以理想的效果将系统混沌运动控制到目标态上去。
Chaos exists and acts an important role in many engineering application and scientific research. It becomes more and more important to controlling chaos. Although studies on the controlling chaos have made a rapid progress since 1990s, the exploration on the methods of controlling chaos still stay at the early stage and there are many problem to be resolved. In this dissertation, the studies are carried out on two aspects: extending the chaos window and suppressing chaos.
In the first part, an extended form of the stochastic Melnikov method is presented and applied in analysis on the homoclinic bifurcation and chaotic behavior of a nonlinear Hamiltonian system with weakly feed-back control and with both harmonic and Gaussian white noisy excitations. Numerical simulation is used to test the form and the results agree well with the theory, which proves the rationality of the form. The results reveal that the addition of stochastic excitation can make the parameter threshold value for the rising of the chaotic motions vary in a wider region, and so, the chaotic motions will appear easily in the system.
In the second part, the chaos suppression is studied. In discrete dynamical systems, a new OGY strategy with new control start condition and feed-back control parameters is proposed. According to the new strategy, the chaotic motions of a Bloch wall, which is located in the zy plane experienced the action of an alternating external magnetic field, are analyzed and controlled. The results show that the chaotic motions of the Bloch wall are suppressed successfully, that is, the new strategy is successful and effective.
In a time-continuous dynamical system, a new optimal control scheme is proposed base on the Krasovskii theory and is utilized to control the chaos in the Newton-Leipnik system which has double strange attractors. It is found that the N-L system can converge onto a selected fixed point rapidly and perfectly through asymptotic mode. Compared with a bang-bang control method, more effective control and flexible choice on control target are shown with the present scheme.
The innovation of the present dissertation can be shown on three aspects. One is the stochastic extended form of the high-dimension Melnikov method which make it possible to choose proper noise excitation to extend the chaos window of a dynamical system. The following one is a new OGY strategy proposed by a new linearization method used in the neighborhood of fixed points, which provide a easy and effective way to control chaos in a discrete dynamical system. The last one is a new optimal control method based on Krasovskii theory, which makes it possible to drive and change the state of a system easily and control the chaotic motions of a system effectively.
第一部分，本文对梅尔尼科夫理论进行拓展，得到了一种高维梅尔尼科夫方法的随机推广形式，并将这种方法应用于带慢变参数和弱反馈控制的非线性哈密顿系统在谐和和随机激励双重作用下的同宿分叉和混沌运动的分析中，同时利用数值试验进行了验证。数值结果和理论结果吻合得很好，证明本文的推广是有效的。结果表明，适当的随机激励可以使得系统出现混沌运动的参数阈值变化范围更大，从而系统的混沌运动更容易出现。
第二部分，本文进行了混沌抑制的研究。在离散动力系统的混沌抑制方面，本文利用新的平衡点邻域线性化方法对改进OGY方法进行了修正，提出了新的控制启动条件和控制反馈参数。利用此方法对一类在zy平面上受到可变外部磁场作用的Bloch系统的混沌运动进行了控制，结果表明，方法成功地抑制了系统的混沌运动，说明此方法的改进是合理和有效的。
在时间连续动力系统方面，基于Krasovskii理论本文提出了一种的新优化控制方法，并利用这一方法对一类具有双奇怪吸引子的Newton-Leipnik(N-L)系统进行了混沌抑制，结果表明一旦控制启动，系统行为能够迅速渐近稳定于目标平衡点。与一类bang-bang控制方法相比，本文的方法具有更好的抑制效果和更灵活的抑制目标选择。
本文的创新之处体现在三个方面。一是提出了一种高维随机梅尔尼科夫推广方法，这一方法使得我们顺利利用合适的噪声扩大系统混沌窗口成为可能。二是在一新的平衡点邻域线性化方法基础上进行了OGY方法的修正，实例控制表明本文进行的修正不仅能使受控系统具有更理想的输出，同时使得输出有更灵活的可调控性。第三就是在Krasovskii理论的基础上提出了一种新的优化控制方法，这一方法使得我们可以根据实际需要随意改变系统状态，并能够以理想的效果将系统混沌运动控制到目标态上去。
Chaos exists and acts an important role in many engineering application and scientific research. It becomes more and more important to controlling chaos. Although studies on the controlling chaos have made a rapid progress since 1990s, the exploration on the methods of controlling chaos still stay at the early stage and there are many problem to be resolved. In this dissertation, the studies are carried out on two aspects: extending the chaos window and suppressing chaos.
In the first part, an extended form of the stochastic Melnikov method is presented and applied in analysis on the homoclinic bifurcation and chaotic behavior of a nonlinear Hamiltonian system with weakly feed-back control and with both harmonic and Gaussian white noisy excitations. Numerical simulation is used to test the form and the results agree well with the theory, which proves the rationality of the form. The results reveal that the addition of stochastic excitation can make the parameter threshold value for the rising of the chaotic motions vary in a wider region, and so, the chaotic motions will appear easily in the system.
In the second part, the chaos suppression is studied. In discrete dynamical systems, a new OGY strategy with new control start condition and feed-back control parameters is proposed. According to the new strategy, the chaotic motions of a Bloch wall, which is located in the zy plane experienced the action of an alternating external magnetic field, are analyzed and controlled. The results show that the chaotic motions of the Bloch wall are suppressed successfully, that is, the new strategy is successful and effective.
In a time-continuous dynamical system, a new optimal control scheme is proposed base on the Krasovskii theory and is utilized to control the chaos in the Newton-Leipnik system which has double strange attractors. It is found that the N-L system can converge onto a selected fixed point rapidly and perfectly through asymptotic mode. Compared with a bang-bang control method, more effective control and flexible choice on control target are shown with the present scheme.
The innovation of the present dissertation can be shown on three aspects. One is the stochastic extended form of the high-dimension Melnikov method which make it possible to choose proper noise excitation to extend the chaos window of a dynamical system. The following one is a new OGY strategy proposed by a new linearization method used in the neighborhood of fixed points, which provide a easy and effective way to control chaos in a discrete dynamical system. The last one is a new optimal control method based on Krasovskii theory, which makes it possible to drive and change the state of a system easily and control the chaotic motions of a system effectively.
引文
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