含参系统及共振系统的最简规范形的研究
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摘要
规范形理论是简化常微分方程的一种十分有效的方法,在研究非线性动力系统在平衡点附近的分叉和稳定性等非线性动力学行为方面扮演着重要的角色。它为非线性动力学问题的求解,开辟了一条新的道路。
本文简述了几种计算常微分方程规范形方法的特点及规范形理论的推广应用。对L-S方法、中心流形理论与规范形理论几种化简常微分方程的方法进行比较,对已有的中心流形的Mathematica电算程序进行优化,实现了程序简洁化、通用化和高效化。
将参数视为状态变量,在不截断的情况下,研究了非共振含参双Hopf分叉系统的最简规范形。在采用非线性恒同变换时引入了变参数尺度变换函数,借助于计算机代数语言Mathematica,推导出一般情况下含有参数的非共振双Hopf分叉系统的最简规范形的前5阶系数的表达式,并根据其中的规律推导出该系统高阶最简规范形的通式。
基于李代数方法,对几种常见的共振双Hopf分叉系统的规范形进行了详细分析,得出了共振双Hopf分叉系统的最简规范形系数和非线性恒同变换系数与原动力系统系数之间关系的直接表达式。并利用符号运算语言Mathematica编制出了一套通用的程序,该程序不仅适用于计算各种常见的共振双Hopf分叉系统的最简规范形,同样可以用于计算非共振情况下双Hopf分叉系统的最简规范形。
Normal form theory is one of the most important methods for reducing ordinarydifferential equations (ODE). It plays an important role in the study of the dynamicalbehavior of the original system in the vicinity of the critical equilibrium. This methodcarves out a new way for solving the nonlinear dynamical system.
A brief introduction to the characteristic of the basic methods for calculatingnormal form of ODE is given in this paper. Expansion and application of this theoryare also discussed. The Liapunov-Schmidt reduction、the Central Manifold theoremand normal form theory are compared with each other. The former program forcalculating the Central Manifold of nonlinear systems was improved in this paper.This program becomes more universal、simpler and more efficient than the formerone.
The simplest normal forms of the non-resonant double Hopf bifurcation (DHB)with parameter are obtained without truncating the original differential equation, i.e.the perturbation parameter of the system is treated as one-dimensional state variable.The computation of the simplest normal form is not only based on near-identitynonlinear transformations, but also based on parameter rescaling. The expressions ofthe simplest normal forms from order two to order five for non-resonant DHB for thesystem with parameter are gotten with the help of computer algebra programMathematica. From which, the simplest normal forms of this system up to any highorder is deduced.
The simplest normal form of resonant DHB was studied based on lie operator.The coefficients of the simplest normal forms of resonant DHB and the non-lineartransformations in terms of the original system coefficients are given explicitly. Ageneral program was achieved with the help of the computer algebra languageMathematica. This program can not only compute the simplest normal form ofresonant DHB but also can compute the non-resonant form. This program couldcompute the simplest normal forms of DHB up to seventh order.
本文简述了几种计算常微分方程规范形方法的特点及规范形理论的推广应用。对L-S方法、中心流形理论与规范形理论几种化简常微分方程的方法进行比较,对已有的中心流形的Mathematica电算程序进行优化,实现了程序简洁化、通用化和高效化。
将参数视为状态变量,在不截断的情况下,研究了非共振含参双Hopf分叉系统的最简规范形。在采用非线性恒同变换时引入了变参数尺度变换函数,借助于计算机代数语言Mathematica,推导出一般情况下含有参数的非共振双Hopf分叉系统的最简规范形的前5阶系数的表达式,并根据其中的规律推导出该系统高阶最简规范形的通式。
基于李代数方法,对几种常见的共振双Hopf分叉系统的规范形进行了详细分析,得出了共振双Hopf分叉系统的最简规范形系数和非线性恒同变换系数与原动力系统系数之间关系的直接表达式。并利用符号运算语言Mathematica编制出了一套通用的程序,该程序不仅适用于计算各种常见的共振双Hopf分叉系统的最简规范形,同样可以用于计算非共振情况下双Hopf分叉系统的最简规范形。
Normal form theory is one of the most important methods for reducing ordinarydifferential equations (ODE). It plays an important role in the study of the dynamicalbehavior of the original system in the vicinity of the critical equilibrium. This methodcarves out a new way for solving the nonlinear dynamical system.
A brief introduction to the characteristic of the basic methods for calculatingnormal form of ODE is given in this paper. Expansion and application of this theoryare also discussed. The Liapunov-Schmidt reduction、the Central Manifold theoremand normal form theory are compared with each other. The former program forcalculating the Central Manifold of nonlinear systems was improved in this paper.This program becomes more universal、simpler and more efficient than the formerone.
The simplest normal forms of the non-resonant double Hopf bifurcation (DHB)with parameter are obtained without truncating the original differential equation, i.e.the perturbation parameter of the system is treated as one-dimensional state variable.The computation of the simplest normal form is not only based on near-identitynonlinear transformations, but also based on parameter rescaling. The expressions ofthe simplest normal forms from order two to order five for non-resonant DHB for thesystem with parameter are gotten with the help of computer algebra programMathematica. From which, the simplest normal forms of this system up to any highorder is deduced.
The simplest normal form of resonant DHB was studied based on lie operator.The coefficients of the simplest normal forms of resonant DHB and the non-lineartransformations in terms of the original system coefficients are given explicitly. Ageneral program was achieved with the help of the computer algebra languageMathematica. This program can not only compute the simplest normal form ofresonant DHB but also can compute the non-resonant form. This program couldcompute the simplest normal forms of DHB up to seventh order.
引文
[1]陈予恕,非线性振动,天津:天津科学出版社,1983.
[2]陈予恕,唐云等,非线性动力学中的现代分析方法,北京:科学出版社,1992.
[3]陈予恕, 徐健学,非线性动力学 非线性振动和运动稳定性,北京:中国科学技术出版社,1995.
[4]王洪礼,张琪琩,非线性动力学理论及应用,天津:天津科学技术出版社,2002.
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[9]张锦炎,冯贝叶,常微分方程几何理论与分支问题,北京:北京大学出版社,1997.
[10]Duo Wang. An introduction to the normal form theory of ordinary differential equations, Advances in Mathematics. 1990, 19(1): 38-71.
[11]S. Y. Chow, C. Z. Li, Duo Wang. Normal forms and bifurcation of planar vector fields, Cambridge University Press. Cambridge, 1994.
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[13]陈予恕,张琪昌. 一种求解非线性振动系统渐近解的新方法,力学学报. 1990,22(4):413-419.
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[15]L. Jezequel, C. H. Lamarque. Analysis of non-linear dynamical system by the normal form theory, Journal of Sound and Vibration. 1991, 143(9): 425-459.
[16]P. Yu. Computation of normal forms via a perturbation technique, Journal of Sound and Vibration. 1998, 211(1): 19-38.
[17]AYT Leung, Q C Zhang. Higher-order normal form and period averaging, Journal of Sound and Vibration. 1998, 217(5): 795-806.
[18]韩景龙,朱德懋. 非线性振动系统的规范形与平均法,振动工程学报.1996,9(4):371-377.
[19]Weiyi Zhang, K. Huseyin. On the relation between the methods of averaging and normal forms, Applied Mathematical Modeling. 2000, 24(4):279-295.
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[21]张伟. 非线性动力系统的规范形和余维 3 退化分叉,力学学报.1993,25(5):548-559.
[22]张伟. 含有参数激励的非线性动力系统的高余维和复杂动力学,天津大学博士学位论文,1997.
[23]吴志强,陈予恕. 非半简分叉问题的范式,应用数学和力学.1997,18(4):325-330.
[24]陈予恕,吴志强. 多重非内共振 Hopf 分岔的正规形,力学学报.1997,29(6):669-675.
[25]QC Zhang, AYT Leung. Computation of normal forms for higher dimensional semi-simple systems, Dynamics of Continuous, Discrete and Impulsive Systems, Series A: Mathematical Analysis. 2001, 8:559-574.
[26]AYT Leung, QC Zhang. Normal form computation without central manifold reduction, Journal of Sound and Vibration,2003, 266(2):261-279.
[27]吴志强. 多自由度非线性系统的非线性模态及 Normal Form 直接方法,天津大学博士论文,1996.
[28]Ushiki S. Normal form for singularities of vector fields, Journal of Applied Mathematics. 1984, 1(1): 1-34.
[29]Guoting Chen, Jean Della Dora. An algorithm for computing a new normal form for dynamical systems,J. Symbolic Computation. 2000, 29(3):393-418.
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[32]P. Yu and Y. Yuan. The simplest normal forms for the singularity of a pure imaginary pair and a zero eigenvalue, Dynamics of Continuous, Discrete and Impulsive Systems (DCDIS) Series B: Applications & Algorithms. 2001, 8: 219-249.
[33]Y. Yuan, P. Yu. Computation of simplest normal forms of differential equations associated with a double-zero eigenvalue, International Journal of Bifurcation and Chaos, 2001, 11(5): 1307-1330.
[34]P. Yu, Y.Yuan. The simplest normal forms associated with a tripple zero eigenvalue of indices one and two, Nonlinear Analysis. 2001, 47(2): 1105-1116.
[35]P. Yu. Computation of the simplest normal forms with perturbation parameters based on Lie transform and rescaling, J. Comput. and Appl. Math., 2002;144(1-2):359-373.
[36]P. Yu, A.Y.T. Leung. The simplest normal form of Hopf bifurcation . Nonlinearity, 2003, 16(1), 277–300.
[37]王铎. 规范形理论的 sl(2,R)表示方法的一些新结果,中国科学 A 辑,1990,4: 343-352.
[38]R. Cushman and J. Sanders. Nilponent normal forms and representation theory of sl(2, R), Multi-parameter bifurcation theory, ed. Golubitsky, M. and Guckenheimer, J., Contemporary Mathematics . 1986, 56:31-51.
[39]A. H. Nayfey. Method of normal forms, John Wiley & Sons, New York, 1993.
[40]A.Y.T. Leung, QC Zhang, Chen Yushu. Normal form analysis of Hopf bifurcation exemplified by duffing's equation. Shock and Vibration. 1994, 11(3): 233-240.
[41]陈予恕,吴志强. 多重非内共振 Hopf 分岔的正规形. 力学学报.1997,29(6):669–675.
[42]吴志强,胡海岩. 非半单分叉 Normal Form 计算,力学学报.1998,4:423-431.
[43]W. Zhang and K. Huseyin. Computation of the coefficients associated with the normal form of a resonant five-dimensional system. Mathematical and Computer Modeling, 1999, 30(11-12): 213–228.
[44]张琪昌,郝淑英,陈予恕. 用范式理论研究强非线性振动问题振动工程学报.2002, 13(3):481-486.
[45]张琪琩,陈予恕. 计算非线性振动系统高阶渐近解的 Normal Form 方法,非线性动力学报,1998, 5(3):269-275.
[46]Q. C. Zhang and A.Y.T. Leung. Normal form of double Hopf bifurcation in forced oscillators. Journal of Sound and vibration, 2000, 231(4): 1057–1069.
[47]A.Y.T.Leung, T.Ge. An algorithm for higher order Hopf normal forms. Shock and Vibration. 1995, 2(4): 307-319.
[48]Xiaofeng Wang, Guoting Chen, Duo Wang. Unique normal forms for the Takens–Bogdanov singularity in a special case, Comptes Rendus de l'Academie des Sciences Series I Mathematics,2001,332(6): 551–555.
[49]P. Yu. Symbolic computation of normal forms for resonant double hopf bifurcations using a perturbation technique, Journal of Sound and Vibration. 2001,247(4): 615-632.
[50]P. Yu, Y. Yuan. Simplest normal form of a codimension-two system, Proc. of 17th CANCAM, Hamilton, Canada.1999, 3: 367-378.
[51]P. Yu. Symbolic computation of normal forms for resonant double Hopf bifurcations using multiple time scales, Journal of Sound and vibration, 2001, 247(4): 615–63.
[52]S. N. Chow, C. Z. Li and Duo Wang. Normal forms and bifurcations of planar vector fields, Cambridge University Press, 1994.
[53]Jing Li, Duo Wang, Wei Zhang. General forms of the simplest normal forms of Bogdanov-Takens singularities, Dynamics of Continuous, Discrete and Impulsive Systems Series A: Mathematical Analysis, 2001, 8: 519-530.
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[2]陈予恕,唐云等,非线性动力学中的现代分析方法,北京:科学出版社,1992.
[3]陈予恕, 徐健学,非线性动力学 非线性振动和运动稳定性,北京:中国科学技术出版社,1995.
[4]王洪礼,张琪琩,非线性动力学理论及应用,天津:天津科学技术出版社,2002.
[5]褚亦清,李翠英,非线性振动分析,北京:北京理工大学出版社,1996.
[6]朱因远,周纪卿,非线性振动和运动稳定性,西安:西安交通大学出版社,1992.
[7]唐云,对称性分叉理论基础,北京:科学出版社,1998.
[8]陆启韶,常微分方程的定性方法和分叉,北京:北京航空航天大学出版社,1989.
[9]张锦炎,冯贝叶,常微分方程几何理论与分支问题,北京:北京大学出版社,1997.
[10]Duo Wang. An introduction to the normal form theory of ordinary differential equations, Advances in Mathematics. 1990, 19(1): 38-71.
[11]S. Y. Chow, C. Z. Li, Duo Wang. Normal forms and bifurcation of planar vector fields, Cambridge University Press. Cambridge, 1994.
[12]A. H. Nayfeh. Method of normal forms, John Wiley & Sons, New York, 1993.
[13]陈予恕,张琪昌. 一种求解非线性振动系统渐近解的新方法,力学学报. 1990,22(4):413-419.
[14]张琪昌. Hopf 分叉理论及其在非线性振动系统中的应用,天津大学博士论文,天津,1991.
[15]L. Jezequel, C. H. Lamarque. Analysis of non-linear dynamical system by the normal form theory, Journal of Sound and Vibration. 1991, 143(9): 425-459.
[16]P. Yu. Computation of normal forms via a perturbation technique, Journal of Sound and Vibration. 1998, 211(1): 19-38.
[17]AYT Leung, Q C Zhang. Higher-order normal form and period averaging, Journal of Sound and Vibration. 1998, 217(5): 795-806.
[18]韩景龙,朱德懋. 非线性振动系统的规范形与平均法,振动工程学报.1996,9(4):371-377.
[19]Weiyi Zhang, K. Huseyin. On the relation between the methods of averaging and normal forms, Applied Mathematical Modeling. 2000, 24(4):279-295.
[20]张伟,陈予恕. 共轭算子法和非线性动力系统的高阶规范形,应用数学和力学. 1997,18(5):421-432.
[21]张伟. 非线性动力系统的规范形和余维 3 退化分叉,力学学报.1993,25(5):548-559.
[22]张伟. 含有参数激励的非线性动力系统的高余维和复杂动力学,天津大学博士学位论文,1997.
[23]吴志强,陈予恕. 非半简分叉问题的范式,应用数学和力学.1997,18(4):325-330.
[24]陈予恕,吴志强. 多重非内共振 Hopf 分岔的正规形,力学学报.1997,29(6):669-675.
[25]QC Zhang, AYT Leung. Computation of normal forms for higher dimensional semi-simple systems, Dynamics of Continuous, Discrete and Impulsive Systems, Series A: Mathematical Analysis. 2001, 8:559-574.
[26]AYT Leung, QC Zhang. Normal form computation without central manifold reduction, Journal of Sound and Vibration,2003, 266(2):261-279.
[27]吴志强. 多自由度非线性系统的非线性模态及 Normal Form 直接方法,天津大学博士论文,1996.
[28]Ushiki S. Normal form for singularities of vector fields, Journal of Applied Mathematics. 1984, 1(1): 1-34.
[29]Guoting Chen, Jean Della Dora. An algorithm for computing a new normal form for dynamical systems,J. Symbolic Computation. 2000, 29(3):393-418.
[30]Wang Xiaofeng , Guoting Chen, Duo Wang. Unique normal forms for the Takens–Bogdanov singularity in a special case,Comptes Rendus de l'Academie des Sciences Series I Mathematics. 2001, 332(6): 551-555.
[31]P. Yu. Simplest normal forms of Hopf and generalized Hopf bifurcations, Int. J. Bifurcation and Chaos. 1999, 9(10): 1917-1939.
[32]P. Yu and Y. Yuan. The simplest normal forms for the singularity of a pure imaginary pair and a zero eigenvalue, Dynamics of Continuous, Discrete and Impulsive Systems (DCDIS) Series B: Applications & Algorithms. 2001, 8: 219-249.
[33]Y. Yuan, P. Yu. Computation of simplest normal forms of differential equations associated with a double-zero eigenvalue, International Journal of Bifurcation and Chaos, 2001, 11(5): 1307-1330.
[34]P. Yu, Y.Yuan. The simplest normal forms associated with a tripple zero eigenvalue of indices one and two, Nonlinear Analysis. 2001, 47(2): 1105-1116.
[35]P. Yu. Computation of the simplest normal forms with perturbation parameters based on Lie transform and rescaling, J. Comput. and Appl. Math., 2002;144(1-2):359-373.
[36]P. Yu, A.Y.T. Leung. The simplest normal form of Hopf bifurcation . Nonlinearity, 2003, 16(1), 277–300.
[37]王铎. 规范形理论的 sl(2,R)表示方法的一些新结果,中国科学 A 辑,1990,4: 343-352.
[38]R. Cushman and J. Sanders. Nilponent normal forms and representation theory of sl(2, R), Multi-parameter bifurcation theory, ed. Golubitsky, M. and Guckenheimer, J., Contemporary Mathematics . 1986, 56:31-51.
[39]A. H. Nayfey. Method of normal forms, John Wiley & Sons, New York, 1993.
[40]A.Y.T. Leung, QC Zhang, Chen Yushu. Normal form analysis of Hopf bifurcation exemplified by duffing's equation. Shock and Vibration. 1994, 11(3): 233-240.
[41]陈予恕,吴志强. 多重非内共振 Hopf 分岔的正规形. 力学学报.1997,29(6):669–675.
[42]吴志强,胡海岩. 非半单分叉 Normal Form 计算,力学学报.1998,4:423-431.
[43]W. Zhang and K. Huseyin. Computation of the coefficients associated with the normal form of a resonant five-dimensional system. Mathematical and Computer Modeling, 1999, 30(11-12): 213–228.
[44]张琪昌,郝淑英,陈予恕. 用范式理论研究强非线性振动问题振动工程学报.2002, 13(3):481-486.
[45]张琪琩,陈予恕. 计算非线性振动系统高阶渐近解的 Normal Form 方法,非线性动力学报,1998, 5(3):269-275.
[46]Q. C. Zhang and A.Y.T. Leung. Normal form of double Hopf bifurcation in forced oscillators. Journal of Sound and vibration, 2000, 231(4): 1057–1069.
[47]A.Y.T.Leung, T.Ge. An algorithm for higher order Hopf normal forms. Shock and Vibration. 1995, 2(4): 307-319.
[48]Xiaofeng Wang, Guoting Chen, Duo Wang. Unique normal forms for the Takens–Bogdanov singularity in a special case, Comptes Rendus de l'Academie des Sciences Series I Mathematics,2001,332(6): 551–555.
[49]P. Yu. Symbolic computation of normal forms for resonant double hopf bifurcations using a perturbation technique, Journal of Sound and Vibration. 2001,247(4): 615-632.
[50]P. Yu, Y. Yuan. Simplest normal form of a codimension-two system, Proc. of 17th CANCAM, Hamilton, Canada.1999, 3: 367-378.
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