多自由度非线性系统的霍普分岔与鞍结分岔控制
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摘要
非线性微分动力系统的分岔控制与混沌控制研究近年来引起了科技工作者的浓厚兴趣,混沌控制的研究成果较多,而分岔控制的研究成果相对来说要少些。分岔是非线性系统特有的现象,在力学、物理、化学、医学、生物学、经济学等领域都普遍存在。分岔控制是分岔研究的重要内容。分岔控制指的是设计一个控制器去改变非线性系统的分岔特性,典型的分岔控制包括:延迟分岔的出现,设计合适的参数使之产生新的分岔,改变分岔点的参数值,稳定分岔解,控制极限环的多重性、幅值和频率,优化系统在分岔点附近的动力学行为,缩小不稳定解的区域等等。在工程问题中,分岔控制的目的就是避免系统因分岔现象产生有害的动力学行为,使系统得到监控。在分岔研究中,霍普分岔是一类重要的动态分岔,鞍结分岔是一类基本的静态分岔。目前对多自由度系统的霍普分岔与鞍结分岔的研究内容还不丰富。
论文运用数理理论对非线性动力系统的分岔和混沌的基础理论和控制进行了较为系统和深入的研究,为应用于工程实际奠定了理论基础。论文的主要创新性工作有:
1.将反馈控制方法应用于多自由度耦合非线性范德波系统,对系统的极限环幅值和霍普分岔进行控制,获得了系统的控制策略和方法。
2.将反馈控制方法应用于多自由度复杂非线性系统的鞍结分岔控制,揭示了控制参数与鞍结分岔“跳跃”不稳定区间之间的联系。
3.对多自由度非线性耦合范德波系统的稳定周期运动进行混沌反控制,成功使原系统的稳定周期运动能够产生一个对称的混沌吸引子,实现对高维非线性耦合范德波混沌控制。
4.研究了类Chen系统的混沌现象,获得了该系统在空间存在的混沌吸引子,成功的将系统运动收敛到一个空间平衡点,实现了对该系统的混沌控制。
Control of bifurcation and chaos in nonlinear differential dynamic systems has recently caused great interests in researchers. Compared to chaos control, reports on the former were much less. Bifurcation is a specific phenomenon which generally existed in such fields as mechanics, physics, chemistry, medical, biology and economics. Bifurcation control plays a vital roll in bifurcation research, which means a controller should be recalled to change the characteristic of bifurcation. Typical bifurcation control includes: delay appearance point of bifurcation, generate a new bifurcation, change parameter of a bifurcation point, stabilize bifurcation solution, control the multiplicity/ amplitude/frequency of a limit cycle, optimize system behavior around bifurcation point, reduce instable zone and so on. In engineering, bifurcation control is to avoid harmful kinetics behavior and take supervisory to the system. Hopf bifurcation is an important dynamic bifurcation in bifurcation research and saddle-node bifurcation is one of three rudimentary static bifurcations. Reports on multiple-degree-of-freedom nonlinear systems were still minor at the present time.
The paper did a systematic and profound research in control of bifurcation and chaos based on mathematical theory, thus, set a theoretical foundation for its application in engineering projects. Some innovative conclusions are drawn as follows:
1. Applied feedback control method in coupled multiple-degree-of-freedom nonlinear van der Pol systems to control Hopf bifurcation and its amplitude, control strategy and method were achieved.
2. Applied feedback control method to control the saddle-node bifurcation in complicated multiple-degree-of-freedom nonlinear systems, revealed the relationship of controlling parameter and the instable intervals.
3. Proceed anti-chaos control in coupled multiple-degree-of-freedom nonlinear van der Pol systems, produced a symmetric chaotic attractor, thus, made anti-chaos control in high dimensional nonlinear coupled van der Pol system possible.
4. Studied a three-dimensional system derived from the Chen system, obtained its chaotic attractor, applied feedback controller to perform chaos control of the system, and finally contract the system moving to a equilibrium point.
论文运用数理理论对非线性动力系统的分岔和混沌的基础理论和控制进行了较为系统和深入的研究,为应用于工程实际奠定了理论基础。论文的主要创新性工作有:
1.将反馈控制方法应用于多自由度耦合非线性范德波系统,对系统的极限环幅值和霍普分岔进行控制,获得了系统的控制策略和方法。
2.将反馈控制方法应用于多自由度复杂非线性系统的鞍结分岔控制,揭示了控制参数与鞍结分岔“跳跃”不稳定区间之间的联系。
3.对多自由度非线性耦合范德波系统的稳定周期运动进行混沌反控制,成功使原系统的稳定周期运动能够产生一个对称的混沌吸引子,实现对高维非线性耦合范德波混沌控制。
4.研究了类Chen系统的混沌现象,获得了该系统在空间存在的混沌吸引子,成功的将系统运动收敛到一个空间平衡点,实现了对该系统的混沌控制。
Control of bifurcation and chaos in nonlinear differential dynamic systems has recently caused great interests in researchers. Compared to chaos control, reports on the former were much less. Bifurcation is a specific phenomenon which generally existed in such fields as mechanics, physics, chemistry, medical, biology and economics. Bifurcation control plays a vital roll in bifurcation research, which means a controller should be recalled to change the characteristic of bifurcation. Typical bifurcation control includes: delay appearance point of bifurcation, generate a new bifurcation, change parameter of a bifurcation point, stabilize bifurcation solution, control the multiplicity/ amplitude/frequency of a limit cycle, optimize system behavior around bifurcation point, reduce instable zone and so on. In engineering, bifurcation control is to avoid harmful kinetics behavior and take supervisory to the system. Hopf bifurcation is an important dynamic bifurcation in bifurcation research and saddle-node bifurcation is one of three rudimentary static bifurcations. Reports on multiple-degree-of-freedom nonlinear systems were still minor at the present time.
The paper did a systematic and profound research in control of bifurcation and chaos based on mathematical theory, thus, set a theoretical foundation for its application in engineering projects. Some innovative conclusions are drawn as follows:
1. Applied feedback control method in coupled multiple-degree-of-freedom nonlinear van der Pol systems to control Hopf bifurcation and its amplitude, control strategy and method were achieved.
2. Applied feedback control method to control the saddle-node bifurcation in complicated multiple-degree-of-freedom nonlinear systems, revealed the relationship of controlling parameter and the instable intervals.
3. Proceed anti-chaos control in coupled multiple-degree-of-freedom nonlinear van der Pol systems, produced a symmetric chaotic attractor, thus, made anti-chaos control in high dimensional nonlinear coupled van der Pol system possible.
4. Studied a three-dimensional system derived from the Chen system, obtained its chaotic attractor, applied feedback controller to perform chaos control of the system, and finally contract the system moving to a equilibrium point.
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