Integrability Conditions for Lotka-Volterra Planar Complex Quartic Systems Having Homogeneous Nonlinearities
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Integrability Conditions for Lotka-Volterra Planar Complex Quartic Systems Having Homogeneous Nonlinearities Brigita Ferˇcec · Jaume Giné · Yirong Liu · Valery G. Romanovski
Received: 12 April 2012 / Accepted: 27 June 2012 / Published online: 18 July 2012 © Springer Science+Business Media B.V. 2012
Abstract In this paper we investigate the integrability problem for the two-dimensional Lotka-Volterra complex quartic systems which are linear systems perturbed by fourth degree homogeneous polynomials, that is, we consider systems of the form x˙ = x(1 − a30 x 3 − a21 x 2 y − a12 xy 2 − a03 y 3 ), y˙ = −y(1 − b30 x 3 − b21 x 2 y − b12 xy 2 − b03 y 3 ). Conditions for the integrability of this system are found. From them the center conditions for corresponding real system can be derived. The study relays on making use of algorithms of computational algebra based on the Groebner basis theory. To simplify laborious manipulations with polynomial modular arithmetics is involved.
The first and the fourth authors are supported by the Slovenian Research Agency. The second author is partially supported by a MICINN/FEDER grant number MTM2011-22877 and by a Generalitat de Catalunya grant number 2009SGR 381. The third author is partially supported by the National Natural Science Foundation of China (11071222). The fourth author also acknowledges the support by the Transnational Access Programme at RISC-Linz of the European Commission Framework 6 Programme for Integrated Infrastructures Initiatives under the project SCIEnce (Contract No. 026133). B. Ferˇcec · V.G. Romanovski Center for Applied Mathematics and Theoretical Physics, University of Maribor, Krekova 2, 2000 Maribor, Slovenia B. Ferˇcec e-mail: [email protected] J. Giné () Departament de Matemàtica, Universitat de Lleida, Av. Jaume II, 69, 25001 Lleida, Spain e-mail: [email protected] Y. Liu School of Mathematics, Central South University, Changsha, Hunan, 410083, P.R. China e-mail: [email protected] V.G. Romanovski Faculty of Natural Science and Mathematics, University of Maribor, Koroška cesta 160, 2000 Maribor, Slovenia e-mail: [email protected]
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Keywords Integrability · Linearizability · Polynomial vector field · Polynomial differential system Mathematics Subject Classification 34C05 · 34A05 · 34C20
1 Introduction and Statement of the Results The integrability problem consists in the determination of local or global first integrals, see for instance [1, 2, 9, 23, 26], and is one of the main open problems in the qualitative theory of differential systems. The integrability problem is directly connected through the PoincaréLyapunov theorem with the center problem as we shall soon see. Consider a planar analytic differential system in the form of a linear center perturbed by higher order terms, that is, u˙ = −v + U (u, v),
v˙ = u + V (u, v),
(1)
where U and V are real analytic functions whose series expansions in a neighborhood of the origin start in at least second order terms. Taking polar coordinates we can see that
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