×

Algebraic decoupling of variables for systems of ODEs of quasipolynomial form. (English) Zbl 0929.34030

Summary: A generalization of the reduction technique for ODEs recently introduced by Gao and Liu is given. It is shown that the use of algebraic methods allows an extension of the procedure to much more general flows, and the derivation of simple criteria for the identification of reducible systems.

MSC:

34C20 Transformation and reduction of ordinary differential equations and systems, normal forms
34A25 Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc.
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Olver, P. J., (Applications of Lie Groups to Differential Equations (1993), Springer: Springer New York) · Zbl 0785.58003
[2] Kowalski, K.; Steeb, W.-H., (Nonlinear Dynamical Systems and Carleman Linearization (1991), World Scientific: World Scientific Singapore) · Zbl 0753.34003
[3] Prelle, M. J.; Singer, M. F., Trans. Am. Math. Soc., 279, 215 (1983)
[4] Steeb, W.-H.; Euler, N., (Nonlinear Evolution Equations and Painlevé Test (1988), World Scientific: World Scientific Singapore) · Zbl 0723.34001
[5] Peschel, M.; Mende, W., (The Predator-Prey Model (1986), Springer: Springer Vienna, New York) · Zbl 0601.92023
[6] Brenig, L., Phys. Lett. A, 133, 378 (1988)
[7] Brenig, L.; Goriely, A., Phys. Rev. A, 40, 4119 (1989)
[8] Gouzé, J. L., Report INRIA 1308, 1 (1990)
[9] Goriely, A.; Brenig, L., Phys. Lett. A, 145, 245 (1990)
[10] Goriely, A., J. Math. Phys., 33, 2728 (1992)
[11] Figueiredo, A.; Rocha Filho, T. M.; Brenig, L., J. Math. Phys., 39, 2929 (1998)
[12] Hernández-Bermejo, B.; Fairén, V., Phys. Lett. A, 206, 31 (1995)
[13] Hernández-Bermejo, B.; Fairén, V.; Brenig, L., J. Phys. A, 31, 2415 (1998)
[14] Fairén, V.; Hernández-Bermejo, B., J. Phys. Chem., 100, 19023 (1996)
[15] Hernández-Bermejo, B.; Fairén, V., Math. Biosci., 140, 1 (1997)
[16] Louies, S.; Brenig, L., Phys. Lett. A, 233, 184 (1997)
[17] Hernández-Bermejo, B.; Fairén, V., Hamiltonian structure and Darboux theorem for families of generalized Lotka-Volterra systems, J. Math. Phys. (1998), to be published · Zbl 0929.37021
[18] Gao, P.; Liu, Z., Phys. Lett. A, 244, 49 (1998)
[19] Fu, Z.; Heidel, J., Nonlinearity, 10, 1289 (1997)
[20] Hietarinta, J.; Grammaticos, B.; Dorizzi, B.; Ramani, A., Phys. Rev. Lett., 53, 1707 (1984)
[21] Arnold, V. I., (Mathematical Methods of Classical Mechanics (1989), Springer: Springer New York)
[22] Halphen, M., C.R. Acad. Sci. Paris, 92, 1101 (1881)
[23] Atiyah, M.; Hitchin, N., (The Geometry and Dynamics of Magnetic Monopoles (1988), Princeton Univ. Press: Princeton Univ. Press Princeton, NJ) · Zbl 0671.53001
[24] Gümral, H.; Nutku, Y., J. Math. Phys., 34, 5691 (1993)
[25] Arecchi, F. T., Order and Chaos in Nonlinear Physical Systems, (Lundqvist, S.; March, N. H.; Tosi, M. P. (1988), Plenum Press: Plenum Press New York), 193 · Zbl 0794.58028
[26] Shapovalov, A. V.; Evdokimov, E. V., Physica D, 112, 441 (1998)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.