Normal Form for Systems with Linear Part N_3(n)

Grace Gachigua, David Malonza, Johana Sigey

Department of Mathematics, Kenyatta University, Nairobi, Kenya.
Department of Mathematics, Kimathi University College of Technology, Nyeri, Kenya.
Pure and Applied Mathematics (PAM) Department, Jomo Kenyatta University of Agriculture and Technology, Nairobi, Kenya.

DOI: 10.4236/am.2012.311227                

Abstract

The concept of normal form is used to study the dynamics of non-linear systems. In this work we describe the normal form for vector fields on 3 × 3 with linear nilpotent part made up of coupled R³ⁿ Jordan blocks. We use an algorithm based on the notion of transvectants from classical invariant theory known as boosting to equivariants in determining the normal form when the Stanley decomposition for the ring of invariants is known.

Keywords

Transvectant; Equivariants; Box Product; Stanley Decomposition

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	R. Cushman, J. A. Sanders and N. White, “Normal Form for the (2;n)-Nilpotent Vector Field Using Invariant Theory,” Physica D: Nonlinear Phenomena, Vol. 30, No. 3, 1988, pp. 399-412. doi:10.1016/0167-2789(88)90028-0
[2]	D. M. Malonza “Normal Forms for Coupled Takens-Bogdanov Systems,” Journal of Nonlinear Mathematical Physics, Vol. 11, No. 3, 2004, pp. 376-398. doi:10.2991/jnmp.2004.11.3.8
[3]	W. W. Adams and P. Loustaunau, “An Introduction to Gr?bner Bases,” American Mathematical Society, Providence, 1994.
[4]	J. Murdock and J. A. Sanders, “A New Transvectant Algorithm for Nilpotent Normal Forms,” Journal of Differential Equations, Vol. 238, No. 1, 2007, pp. 234-256. doi:10.1016/j.jde.2007.03.016
[5]	N. Sri NAmachchivaya, M. M. Doyle, W. F. Langford and N. W. Evans, “Normal Form for Generalized Hopf Bifurcation with Non-Semisimple 1:1 Resonance,” Zeitschrift für Angewandte Mathematik und Physik (ZAMP), Vol. 45, No. 2, 1994, pp. 312-335. doi:10.1007/BF00943508
[6]	G. Gachigua and D. Malonza, “Stanley Decomposition of Coupled N333 System,” Proceedings of the 1st Kenyatta University International Mathematics Conference, Nairobi, 6-10 June 2011, pp. 39-52.