非线性动力系统双Hopf分叉及在工程中的应用
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摘要
Hopf分叉是一类比较简单但是很重要的动态分叉问题，Hopf分叉理论已经成为研究微分方程小振幅周期解产生和消失的主要工具，它不仅在动态分叉和极限环的研究中具有重要理论与数值研究意义，而且与自激振动的产生机理有着密切联系，所以在工程问题中有着广泛的应用。在工程系统中，许多问题的动力学模型都可以用非线性动力学方程来描述，研究工程实际问题中高维非线性动力学方程的双Hopf分叉及混沌运动是工程领域中非常重要的课题。
双Hopf分叉是一类重要的余维2分叉，对应的分叉方程存在两对纯虚特征值。双Hopf分叉点位于两条Hopf分叉曲线相交处。双Hopf分叉是发生在高维非线性系统中的一类现象丰富的分叉行为。目前，对Hopf分叉的研究已经取得丰富的成果，但对非线性动力系统的双Hopf分叉的研究还有待发展。
本文主要将用来研究Hopf分叉问题的高维Hopf分叉定理和奇异性理论进行了推广。首先推广了高维Hopf分叉定理，使其可以应用到共振情形下的非线性动力系统中，并利用推广的Hopf分叉定理，研究了一类在面内和横向激励联合作用下蜂窝夹层板的双Hopf分叉问题。随后，利用推广的高维Hopf分叉定理研究了一类复合材料层合板的双Hopf分叉问题，并得到系统在参数空间的分叉图，利用数值模拟，给出了参数空间上不同区域里的系统运动形式；其次推广了用来研究Hopf分叉问题的奇异性理论，引入函数空间上的映射算子和坐标变换，使其可以用来分析共振情形下的非线性动力系统的双Hopf分叉问题。利用推广的奇异性理论与数值研究了一类在面内和横向激励联合作用下的薄板和压电复合材料层合板的双Hopf分叉问题，在系统参数空间的分叉图的基础上，分析了不同区域里系统的平衡解和周期解的分类问题。
本文的研究内容主要有以下几个方面，
(1)在高维Hopf分叉定理的基础上进行了推广研究，并应用推广的Hopf分叉定理研究了蜂窝夹层板系统在主参数共振-1:2内共振情形下的双Hopf分叉的动力学响应。首先利用多尺度法得到了系统在直角坐标和极坐标形式下的平均方程，通过对平均方程的区分讨论。表明随着分叉参数的变化，蜂窝夹层板系统会发生共振双Hopf分叉。数值模拟给出了系统的局部分叉图，并给出了不同区域所对应的蜂窝夹层板系统的动力学响应。
(2)利用第2章推广的共振情形下的高维非线性系统的Hopf分叉定理，研究了在面内激励和横向激励联合作用下复合材料层合板的双Hopf分叉的动力学响应。在主参数共振-1:1内共振情形下，考虑弱阻尼和弱参数激励的情况，利用多尺度法得到了系统在两种坐标形式下的平均方程，然后应用高维Hopf分叉定理研究了平均方程的不同稳态解在分叉参数变化时的不同分叉响应。通过数值模拟给出了系统的参数分叉图，数值结果与理论分析相对应。
(3)将用来研究非共振情形下非线性系统的奇异性理论方法，推广到共振情形下非线性系统。引入线性变换证明了函数空间上芽等价的定理，给出了开折和普适开折的概念，推广了奇异性理论中的分类和开折定理。考虑了系统的不同内共振比关系，引入对应的相移变换，推导了非线性系统发生共振双Hopf分叉的非退化条件，在此基础上分析了在1:3内共振情形下非线性系统的双Hopf分叉，并研究了系统分叉出的周期解的稳定性。
(4)基于第4章的理论结果研究了主参数共振-1:3内共振情形下参数激励与外激励联合作用下的压电复合材料层合板的双Hopf分叉。考虑弱阻尼的情况下，利用多尺度法推导了系统的平均方程，应用4.2节的共振情形下的双Hopf分叉理论结果分析了分叉方程的动力学响应。本文对系统的阻尼、外激励、参数激励以及调谐参数进行不同的取值模拟，得到了系统在平衡点附近的运动形式。系统的平衡解是稳定的。当改变动力系统的参数取值时，平衡解会发生Hopf分叉，并分叉出周期解。当系统的参数满足给定条件时会发生1:3内共振双Hopf分叉。
在结束语中，进行了全文总结，提出了课题可能存在的问题以及进一步的研究方向。
Hopf bifurcation is a relatively simple but very important dynamic bifurcaton. It isnot only used to study the dynamic bifurcation and limit cycle, but also is also closelylinked with the generation mechanism of self-excited vibration. Hopf bifurcation theoryhas become a classic tool to study differential equations small amplitude periodic solutionand demise. Research of Hopf bifurcation has important theoretical significance. Researchof the double Hopf bifurcation is a very important issue, which being on the practicalproblems in the high-dimensional nonlinear dynamic equations in the engineering field.
The results of our researching mainly include the promotion of high.dimensionalHopf bifurcation theorem, so that it can be used to study the nonlinear dynamical systemsin the resonance case. Using the Hopf bifurcation theory, the double Hopf bifurcation of aclass of honeycomb sandwich plates was studied, which is under the combined effects ofplane and transverse incentive. Subsequently, appling the multi.scale methods and theHopf bifurcation theory, a class of composite laminated plates double Hopf bifurcationwas studied, and gave the parameter diagrams on the parameter plane. Furthermore,studied the singularity theory, and used it to giving the explaining of the double Hopfbifurcation of piezoelectric laminated composite plate under the bombined effects of theplane and transverse and gave the different forms of movement in different regions.
The main research contents obtained in this dissertation are as follows:
(1) We gave a further study of the high-dimensional Hopf bifurcation theorem, andstudied the double Hopf bifurcation of the honeycomb sandwich plate system as theprimary parametric-1:2internal resnonace. First, by using the method of multiple scales,we studied the averaged equations of the plate system in the form of Cartesian and polarcoordinates. Based on the averaged equations, we gave the parameter bifurcation biagram.The numerical simulation gave the local bifurcation diagram of the system, and sometypes of periodic motions of the honeycomb sandwich plate system.
(2) Using the Hopf bifurcation theorem aboute the high-dimensional nonlinearsystem, we studied the double Hopf bifurcation of the laminated composite plate subjectto the surface incentives and lateral excitation, which is in the primaryparametric-1:1internal resonace. Consider the case of weak damping and weak parametricexcitation, we gave the two coordinate forms of the average equation of the system, andstudied the bifurcation response. The parameter bifurcation diagram of the system wasobtained.
(3) Extended the singularity theory of nonlinear systems as in non.resonant case tothe resoant case. By introducing the linear transformation, we proved the bud equivalencetheorem. Then we gave the concepts of unfolding and universal unfolding. We studied the double Hopf bifurcation of the nonlinear system in different resonance cases. The systemmay present periodic motion or quasi-periodic motion.
(4) Based on the theoretical results of the fourth chapter, we analysis the double Hopfbifurcation of the piezoelectric composite laminated plate as in the primary parameter-1:3internal resonant. Using the method of multiple scales we derived the averaged equationsof the system. Appling the results in4.2, we studied the different bifurcation zones in theparameter bifurcation diagram. We found the euqlibrium solution of the system is instable,and may bifurcate to different types of periodic solutions with different conditions.
In closing remarks, full text summarizes and the further research directions wereproposed.
双Hopf分叉是一类重要的余维2分叉，对应的分叉方程存在两对纯虚特征值。双Hopf分叉点位于两条Hopf分叉曲线相交处。双Hopf分叉是发生在高维非线性系统中的一类现象丰富的分叉行为。目前，对Hopf分叉的研究已经取得丰富的成果，但对非线性动力系统的双Hopf分叉的研究还有待发展。
本文主要将用来研究Hopf分叉问题的高维Hopf分叉定理和奇异性理论进行了推广。首先推广了高维Hopf分叉定理，使其可以应用到共振情形下的非线性动力系统中，并利用推广的Hopf分叉定理，研究了一类在面内和横向激励联合作用下蜂窝夹层板的双Hopf分叉问题。随后，利用推广的高维Hopf分叉定理研究了一类复合材料层合板的双Hopf分叉问题，并得到系统在参数空间的分叉图，利用数值模拟，给出了参数空间上不同区域里的系统运动形式；其次推广了用来研究Hopf分叉问题的奇异性理论，引入函数空间上的映射算子和坐标变换，使其可以用来分析共振情形下的非线性动力系统的双Hopf分叉问题。利用推广的奇异性理论与数值研究了一类在面内和横向激励联合作用下的薄板和压电复合材料层合板的双Hopf分叉问题，在系统参数空间的分叉图的基础上，分析了不同区域里系统的平衡解和周期解的分类问题。
本文的研究内容主要有以下几个方面，
(1)在高维Hopf分叉定理的基础上进行了推广研究，并应用推广的Hopf分叉定理研究了蜂窝夹层板系统在主参数共振-1:2内共振情形下的双Hopf分叉的动力学响应。首先利用多尺度法得到了系统在直角坐标和极坐标形式下的平均方程，通过对平均方程的区分讨论。表明随着分叉参数的变化，蜂窝夹层板系统会发生共振双Hopf分叉。数值模拟给出了系统的局部分叉图，并给出了不同区域所对应的蜂窝夹层板系统的动力学响应。
(2)利用第2章推广的共振情形下的高维非线性系统的Hopf分叉定理，研究了在面内激励和横向激励联合作用下复合材料层合板的双Hopf分叉的动力学响应。在主参数共振-1:1内共振情形下，考虑弱阻尼和弱参数激励的情况，利用多尺度法得到了系统在两种坐标形式下的平均方程，然后应用高维Hopf分叉定理研究了平均方程的不同稳态解在分叉参数变化时的不同分叉响应。通过数值模拟给出了系统的参数分叉图，数值结果与理论分析相对应。
(3)将用来研究非共振情形下非线性系统的奇异性理论方法，推广到共振情形下非线性系统。引入线性变换证明了函数空间上芽等价的定理，给出了开折和普适开折的概念，推广了奇异性理论中的分类和开折定理。考虑了系统的不同内共振比关系，引入对应的相移变换，推导了非线性系统发生共振双Hopf分叉的非退化条件，在此基础上分析了在1:3内共振情形下非线性系统的双Hopf分叉，并研究了系统分叉出的周期解的稳定性。
(4)基于第4章的理论结果研究了主参数共振-1:3内共振情形下参数激励与外激励联合作用下的压电复合材料层合板的双Hopf分叉。考虑弱阻尼的情况下，利用多尺度法推导了系统的平均方程，应用4.2节的共振情形下的双Hopf分叉理论结果分析了分叉方程的动力学响应。本文对系统的阻尼、外激励、参数激励以及调谐参数进行不同的取值模拟，得到了系统在平衡点附近的运动形式。系统的平衡解是稳定的。当改变动力系统的参数取值时，平衡解会发生Hopf分叉，并分叉出周期解。当系统的参数满足给定条件时会发生1:3内共振双Hopf分叉。
在结束语中，进行了全文总结，提出了课题可能存在的问题以及进一步的研究方向。
Hopf bifurcation is a relatively simple but very important dynamic bifurcaton. It isnot only used to study the dynamic bifurcation and limit cycle, but also is also closelylinked with the generation mechanism of self-excited vibration. Hopf bifurcation theoryhas become a classic tool to study differential equations small amplitude periodic solutionand demise. Research of Hopf bifurcation has important theoretical significance. Researchof the double Hopf bifurcation is a very important issue, which being on the practicalproblems in the high-dimensional nonlinear dynamic equations in the engineering field.
The results of our researching mainly include the promotion of high.dimensionalHopf bifurcation theorem, so that it can be used to study the nonlinear dynamical systemsin the resonance case. Using the Hopf bifurcation theory, the double Hopf bifurcation of aclass of honeycomb sandwich plates was studied, which is under the combined effects ofplane and transverse incentive. Subsequently, appling the multi.scale methods and theHopf bifurcation theory, a class of composite laminated plates double Hopf bifurcationwas studied, and gave the parameter diagrams on the parameter plane. Furthermore,studied the singularity theory, and used it to giving the explaining of the double Hopfbifurcation of piezoelectric laminated composite plate under the bombined effects of theplane and transverse and gave the different forms of movement in different regions.
The main research contents obtained in this dissertation are as follows:
(1) We gave a further study of the high-dimensional Hopf bifurcation theorem, andstudied the double Hopf bifurcation of the honeycomb sandwich plate system as theprimary parametric-1:2internal resnonace. First, by using the method of multiple scales,we studied the averaged equations of the plate system in the form of Cartesian and polarcoordinates. Based on the averaged equations, we gave the parameter bifurcation biagram.The numerical simulation gave the local bifurcation diagram of the system, and sometypes of periodic motions of the honeycomb sandwich plate system.
(2) Using the Hopf bifurcation theorem aboute the high-dimensional nonlinearsystem, we studied the double Hopf bifurcation of the laminated composite plate subjectto the surface incentives and lateral excitation, which is in the primaryparametric-1:1internal resonace. Consider the case of weak damping and weak parametricexcitation, we gave the two coordinate forms of the average equation of the system, andstudied the bifurcation response. The parameter bifurcation diagram of the system wasobtained.
(3) Extended the singularity theory of nonlinear systems as in non.resonant case tothe resoant case. By introducing the linear transformation, we proved the bud equivalencetheorem. Then we gave the concepts of unfolding and universal unfolding. We studied the double Hopf bifurcation of the nonlinear system in different resonance cases. The systemmay present periodic motion or quasi-periodic motion.
(4) Based on the theoretical results of the fourth chapter, we analysis the double Hopfbifurcation of the piezoelectric composite laminated plate as in the primary parameter-1:3internal resonant. Using the method of multiple scales we derived the averaged equationsof the system. Appling the results in4.2, we studied the different bifurcation zones in theparameter bifurcation diagram. We found the euqlibrium solution of the system is instable,and may bifurcate to different types of periodic solutions with different conditions.
In closing remarks, full text summarizes and the further research directions wereproposed.
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