一类四维幂零向量场的超规范形及应用
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摘要
研究一类具有对称性质的四维幂零向量场的超规范形问题,并将其应用于具有实际工程背景的高维非线性动力学模型的简化.发展与完善由Sanders、Baider和KOW提出的规范形进一步简化的理论,利用线性次数函数、多重李括号与分块矩阵的新记号表示相结合的方法,分别获得四维幂零向量场3次、5次截断的超规范形的一般形式,并将超规范形理论应用于研究环型桁架卫星天线模型的化简问题.本文通过引入并完善大尺寸分块矩阵的新记号表示方法,获得一种处理大尺寸分块矩阵运算的新方法,简化繁琐的大尺寸矩阵的运算,为后续的研究带来便利条件.
In this paper,we mainly focus on the hypernormal form for a class of four-dimensional nilpotent vector fields,and the results are applied to simplify the practical engineering models in the high-dimensional nonlinear dynamics. We improve and develop the further reduction theory of the normal form proposed by Sanders,Baider and KOW,and based on the method of linear grading function,multiple Lie bracket and the new notations for block matrices,we obtain the three order and the five order hypernormal form,respectively,and then the results are applied to simplify the circular truss antenna models. The introduction of the new notations for block matrices gives a new method for dealing with the calculation of big size block matrices,which brings conveniences for the later research.
In this paper,we mainly focus on the hypernormal form for a class of four-dimensional nilpotent vector fields,and the results are applied to simplify the practical engineering models in the high-dimensional nonlinear dynamics. We improve and develop the further reduction theory of the normal form proposed by Sanders,Baider and KOW,and based on the method of linear grading function,multiple Lie bracket and the new notations for block matrices,we obtain the three order and the five order hypernormal form,respectively,and then the results are applied to simplify the circular truss antenna models. The introduction of the new notations for block matrices gives a new method for dealing with the calculation of big size block matrices,which brings conveniences for the later research.
引文
1 Arnold V I.Geometrical method in the theory of ordinary differential equation.New York:Spring-Verlag,1988:3～36
2 Ushiki S.Normal forms for singularities of vector fields.Japan Journal of Applied Mathematics,1984,1(1):1～37
3 Kokubu H,Oka H,Wang D.Linear grading function and further reduction of normal forms.Journal of Differential Equations,1996,132(2):293～318
4 Baider A,Sanders J A.Further reduction of the Takens-Bogdanov normal form.Journal of Differential Equations,1992,99(2):205～244
5 Peng J P,Wang D.A sufficient condition for the uniqueness of normal forms and unique normal forms of generalized Hopf singularities.International Journal Bifurcation and Chaos,2004,14(9):3337～3345
6 Wang D,Li J,Huang M H,et al.Unique normal form of Bogdanov-Takens singularities.Journal of Differential Equations,2000,163(1):223～238
7 Li J,Wang D,Zhang W.General forms of the simplest normal form of Bogdanov-Takens singularities.Dynamics of Continuous,Discrete and Impulsive Systems,2001,8(4):519～530
8 Zhang W,Wang F,Zu J.Computation of normal forms for high dimensional non-linear systems and application to non-planar non-linear oscillations of a cantilever beam.Journal of Sound and Vibration,2004,278(4):949～974
9 Chen S P,Zhang W.Further reduction of normal forms for high dimensional nonlinear systems and application to a composite laminated piezoelectric plate.Applied Mechanics and Materials,2013,291:2662～2665
10 陈淑萍,张伟.一类求解非线性动力系统高阶规范形的新方法.动力学与控制学报,2013,11(3):193～198 (Chen S P,Zhang W.A new approach for computing normal forms of high dimensional nonlinear systems.Journal of Dynamics and Control,2013,11(3):193～198 (in Chinese))
11 Li J,Zhang L N,Wang D.Unique normal form of a class of 3 dimensional vector fields with symmetries.Journal of Differential Equations,2014,257(7):2341～359
12 Li J,Kou L Y,Wang D.Unique normal form for a class of three-dimensional nilpotent vector fields.International Journal Bifurcation and Chaos,2017,27(8):1750131
13 Li J,Kou L Y,Wang D,et al.Unique normal form and the associated coefficients for a class of three-dimensional nilpotent vector fields.International Journal Bifurcation and Chaos,2017,27(14):1750224
14 侯飞,张伟.环型桁架卫星天线的非线性动力学分析.动力学与控制学报,2016,14(2):138～142 (Hou F,Zhang W.Analysis on nonlinear dynamics of circular antenna.Journal of Dynamics and Control,2016,14(2):138～142 (in Chinese))
2 Ushiki S.Normal forms for singularities of vector fields.Japan Journal of Applied Mathematics,1984,1(1):1～37
3 Kokubu H,Oka H,Wang D.Linear grading function and further reduction of normal forms.Journal of Differential Equations,1996,132(2):293～318
4 Baider A,Sanders J A.Further reduction of the Takens-Bogdanov normal form.Journal of Differential Equations,1992,99(2):205～244
5 Peng J P,Wang D.A sufficient condition for the uniqueness of normal forms and unique normal forms of generalized Hopf singularities.International Journal Bifurcation and Chaos,2004,14(9):3337～3345
6 Wang D,Li J,Huang M H,et al.Unique normal form of Bogdanov-Takens singularities.Journal of Differential Equations,2000,163(1):223～238
7 Li J,Wang D,Zhang W.General forms of the simplest normal form of Bogdanov-Takens singularities.Dynamics of Continuous,Discrete and Impulsive Systems,2001,8(4):519～530
8 Zhang W,Wang F,Zu J.Computation of normal forms for high dimensional non-linear systems and application to non-planar non-linear oscillations of a cantilever beam.Journal of Sound and Vibration,2004,278(4):949～974
9 Chen S P,Zhang W.Further reduction of normal forms for high dimensional nonlinear systems and application to a composite laminated piezoelectric plate.Applied Mechanics and Materials,2013,291:2662～2665
10 陈淑萍,张伟.一类求解非线性动力系统高阶规范形的新方法.动力学与控制学报,2013,11(3):193～198 (Chen S P,Zhang W.A new approach for computing normal forms of high dimensional nonlinear systems.Journal of Dynamics and Control,2013,11(3):193～198 (in Chinese))
11 Li J,Zhang L N,Wang D.Unique normal form of a class of 3 dimensional vector fields with symmetries.Journal of Differential Equations,2014,257(7):2341～359
12 Li J,Kou L Y,Wang D.Unique normal form for a class of three-dimensional nilpotent vector fields.International Journal Bifurcation and Chaos,2017,27(8):1750131
13 Li J,Kou L Y,Wang D,et al.Unique normal form and the associated coefficients for a class of three-dimensional nilpotent vector fields.International Journal Bifurcation and Chaos,2017,27(14):1750224
14 侯飞,张伟.环型桁架卫星天线的非线性动力学分析.动力学与控制学报,2016,14(2):138～142 (Hou F,Zhang W.Analysis on nonlinear dynamics of circular antenna.Journal of Dynamics and Control,2016,14(2):138～142 (in Chinese))