Subscribe Now Subscribe Today
Research Article
 

An Improved Recursive Formula for Computing Normal Forms of Multi-dimensional Dynamical Systems



Lei Guo, Qunhong Li and Zigen Song
 
ABSTRACT

In this study, an improved explicit recursive formula of normal forms under nonlinear near-identity 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.

Services
Related Articles in ASCI
Similar Articles in this Journal
Search in Google Scholar
View Citation
Report Citation

 
  How to cite this article:

Lei Guo, Qunhong Li and Zigen Song , 2013. An Improved Recursive Formula for Computing Normal Forms of Multi-dimensional Dynamical Systems. Journal of Applied Sciences, 13: 2924-2928.

DOI: 10.3923/jas.2013.2924.2928

URL: https://scialert.net/abstract/?doi=jas.2013.2924.2928

REFERENCES
Arnold, V.I., 1983. Geometrical Methods in the Theory of Ordinary Differential Equations. Springer-Verlag, Berlin, New York.

Baider, A. and J.A. Sanders, 1992. Further reduction of the takens-bogdanov normal forms. J. Different. Equat., 99: 205-244.
Direct Link  |  

Carr, J., 1981. Applications of Center Manifold Theory. Springer-Verlag, 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: 863-880.
CrossRef  |  Direct Link  |  

Chua, L.O. and H. Kokubu, 1989. Normal forms for nonlinear vector fields. II. Applications. IEEE. Trans. Circuits Syst., 36: 51-70.
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: 95-117.
CrossRef  |  Direct Link  |  

Guckenheimer, J. and P. Holmes, 1993. Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. Springer-Verlag, New York.

Kokubu, H., H. Oka and D. Wang, 1996. Linear grading function and further reduction of normal forms. J. Different. Equat., 132: 293-318.
CrossRef  |  Direct Link  |  

Nayfeh, A.H., 1993. Methods of Normal Forms. Springer-Verlag, 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: 1-37.
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: 123-151.
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: 1105-1116.
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: 219-249.

Yu, P., 1998. Computation of normal forms via a perturbation technique. J. Sound Vibrat., 211: 19-38.
CrossRef  |  Direct Link  |  

Zhang, W., F. Wang and J.W. Zu, 2004. Computation of normal forms for high dimensional non-linear systems and application to non-planar non-linear oscillations of a cantilever beam. J. Sound Vibrat., 278: 949-974.
CrossRef  |  Direct Link  |  

©  2019 Science Alert. All Rights Reserved