Abstract
We consider an autonomous system of ordinary differential equations, which is solved with respect to derivatives. To study the local integrability of the system near a degenerate stationary point we use an approach based on the Power Geometry Method and on the computation of resonant normal forms. For a planar system depending on five parameters we give four series of conditions on parameters of the system for which it is integrable near the degenerate stationary point.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Bruno, A.D., Edneral, V.F.: Algorithmic analysis of local integrability. Dokl. Akademii Nauk 424(3), 299–303 (2009) (Russian) = Doklady Mathem. 79(1), 48–52 (2009) (English)
Bruno, A.D.: Power Geometry in Algebraic and Differential Equations. Fizmatlit, Moscow (1998) (Russian) = Elsevier Science, Amsterdam (2000) (English)
Bruno, A.D.: Local Methods in Nonlinear Differential Equations. Nauka, Moscow (1979) (Russian) = Springer-Verlag, Berlin (1989) (English)
Bruno, A.D.: Analytical form of differential equations (I,II). Trudy Moskov. Mat. Obsc. 25, 119–262 (1971); 26, 199–239 (1972) (Russian) = Trans. Moscow Math. Soc. 25, 131–288 (1971); 26, 199–239 (1972) (English)
Bruno, A.D., Edneral, V.F.: On Integrability of a Planar System of ODE’s near Degenerate Stationary Point. Zapiski Nauchnykh Seminarov POMI 373, 34–47 (2009) (Russian)
Bruno, A.D., Edneral, V.F.: On Integrability of a Planar ODE System of near Degenerate Stationary Point. In: Gerdt, V.P., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2009. LNCS, vol. 5743, pp. 45–53. Springer, Heidelberg (2009)
Amelkin, V.V., Lukashevich, N.A., Sadovskii, A.P.: Nonlinear Oscillations in Second Order Systems. BSU, Minsk (1982) (in Russian)
Liu, Y., Li, J.: New study on the center problem and bifurcations of limit cycles for the Lyapunov system (I). Internat. J. Bifur. Chaos Appl. Sci. Engrg. 19(11), 3791–3801 (2009)
Algaba, A., Gamero, E., Garcia, C.: The integrability problem for a class of planar systems. Nonlinearity 22, 395–420 (2009)
Edneral, V.F.: On algorithm of the normal form building. In: Ganzha, V.G., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2007. LNCS, vol. 4770, pp. 134–142. Springer, Heidelberg (2007)
Bateman, H., Erdêlyi, A.: Higer Transcendental Functions, vol. 1. McGraw-Hill Book Company, Inc., New York (1953)
Romanovski, V.G., Shafer, D.S.: The Center and Cyclicity Problems: A Computational Algebra Approach. Birkhüser, Boston (2009)
Decker, W., Pfister, G., Schönemann, H.A.: Singular 2.0 library for computing the primary decomposition and radical of ideals primdec.lib (2001)
Greuel, G.M., Pfister, G., Schönemann, H.: Singular 3.0. A Computer Algebra System for Polynomial Computations. Centre for Computer Algebra, University of Kaiserslautern (2005), http://www.singular.uni-kl.de
Christopher, C., Mardešić, P., Rousseau, C.: Normalizable, integrable, and linearizable saddle points for complex quadratic systems in C 2. J. Dyn. Control Sys. 9, 311–363 (2003)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Edneral, V., Romanovski, V.G. (2010). On Sufficient Conditions for Integrability of a Planar System of ODEs Near a Degenerate Stationary Point. In: Gerdt, V.P., Koepf, W., Mayr, E.W., Vorozhtsov, E.V. (eds) Computer Algebra in Scientific Computing. CASC 2010. Lecture Notes in Computer Science, vol 6244. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15274-0_9
Download citation
DOI: https://doi.org/10.1007/978-3-642-15274-0_9
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-15273-3
Online ISBN: 978-3-642-15274-0
eBook Packages: Computer ScienceComputer Science (R0)