
Research Article


An Improved Recursive Formula for Computing Normal Forms of Multidimensional Dynamical Systems


Lei Guo,
Qunhong Li
and
Zigen Song



ABSTRACT

In this study, an improved explicit recursive formula of normal
forms under nonlinear nearidentity transformations is firstly introduced and
the associated proof is also given out. Compared with traditional method, the
improved method can get the specific expression of the higher order terms of
dynamical systems after nonlinear transformations. So the normal forms of the
original dynamical systems can be obtained by a series of nonlinear transformations
but without changing its topology structure, also by solving a series of algebra
equations with the aid of computing software Maple, not only the coefficients
of the jth order normal form and the associated nonlinear transformations but
also higher (>j) order terms of the original equations can be obtained. The
application of the improved method will greatly simplify the computation during
the research of dynamical systems.








REFERENCES 
Arnold, V.I., 1983. Geometrical Methods in the Theory of Ordinary Differential Equations. SpringerVerlag, Berlin, New York.
Baider, A. and J.A. Sanders, 1992. Further reduction of the takensbogdanov normal forms. J. Different. Equat., 99: 205244. Direct Link 
Carr, J., 1981. Applications of Center Manifold Theory. SpringerVerlag, New York.
Chow, S.N., C. Li and D. Wang, 1983. Normal Forms and Bifurcation of Planar Vector Fields. Cambridge University Press, Cambridge UK.
Chua, L.O. and H. Kokubu, 1988. Normal forms for nonlinear vector fields. I. Theory and algorithm. IEEE. Trans Circuits Syst., 35: 863880. CrossRef  Direct Link 
Chua, L.O. and H. Kokubu, 1989. Normal forms for nonlinear vector fields. II. Applications. IEEE. Trans. Circuits Syst., 36: 5170. CrossRef  Direct Link 
Elphick, C., E. Tirapegui, M.E. Brachet, P. Coullet and G. Iooss, 1987. A simple global characterization for normal forms of singular vector fields. Phys. D: Nonlin. Phenom., 29: 95117. CrossRef  Direct Link 
Guckenheimer, J. and P. Holmes, 1993. Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. SpringerVerlag, New York.
Kokubu, H., H. Oka and D. Wang, 1996. Linear grading function and further reduction of normal forms. J. Different. Equat., 132: 293318. CrossRef  Direct Link 
Nayfeh, A.H., 1993. Methods of Normal Forms. SpringerVerlag, New York.
Taylor, A.E. and D.C. Lay, 1980. Introduction to Functional Analysis. John Wiley and Sons, New York.
Ushiki, S., 1984. Normal forms for singularities of vector fields. Jpn J. Applied Math., 1: 137. CrossRef  Direct Link 
Yu, P. and A.Y.T. Leung, 2003. A perturbation method for computing the simplest normal forms of dynamical systems. J. Sound Vibrat., 26: 123151. CrossRef  Direct Link 
Yu, P. and Y. Yuan, 2001. The simplest normal forms associated with a triple zero eigenvalue of indices one and two. Nonlin. Anal., 47: 11051116. Direct Link 
Yu, P. and Y. Yuan, 2001. The simplest normal form for the singularity of a pure imaginary pair and a zero eigenvalue. J. Math. Res. Exp., 8: 219249.
Yu, P., 1998. Computation of normal forms via a perturbation technique. J. Sound Vibrat., 211: 1938. CrossRef  Direct Link 
Zhang, W., F. Wang and J.W. Zu, 2004. Computation of normal forms for high dimensional nonlinear systems and application to nonplanar nonlinear oscillations of a cantilever beam. J. Sound Vibrat., 278: 949974. CrossRef  Direct Link 



