Integrability of 3-dim polynomial systems with three invariant planes
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O R I G I N A L PA P E R
Integrability of 3-dim polynomial systems with three invariant planes Zhaoping Hu · Maira Aldazharova · Tamasha M. Aldibekov · Valery G. Romanovski
Received: 20 April 2013 / Accepted: 30 July 2013 / Published online: 29 August 2013 © Springer Science+Business Media Dordrecht 2013
Abstract We investigate the problem of integrability for a family of three-dimensional autonomous polynomial systems of ODEs. Necessary and sufficient conditions for the existence of two independent analytic first integrals for systems of the family are given. The linearizability of the systems is studied as well. Keywords Analytic first integrals · Linearizability 3-dim systems of ODE’s · Reversibility · Polynomial systems
Z. Hu Department of Mathematics, Shanghai University, Shanghai 200444, P.R. China e-mail: [email protected] M. Aldazharova · T.M. Aldibekov Al-Farabi Kazakh National University, Al-Farabi Av. 71, Almaty, Kazakhstan M. Aldazharova e-mail: [email protected] T.M. Aldibekov e-mail: [email protected] V.G. Romanovski (B) Center for Applied Mathematics and Theoretical Physics and Faculty of Natural Science and Mathematics, University of Maribor, 2000 Maribor, Slovenia e-mail: [email protected]
1 Introduction The integrability of systems of differential equations is one of central topics in the theory of ordinary differential equations. Although a generic system of ODEs is not integrable, integrable systems are important and frequently arise in studying various mathematical models of the real-world phenomena. Local integrability of polynomial systems of differential equations has been studied by many authors. Most studies have been devoted to two-dimensional systems (see, e.g., [9, 12, 14, 17, 18, 31] and references therein); however, the problem of integrability for 3-dim systems has also attracted much attention (see, for instance, [1, 2, 6, 7, 10, 11, 15, 23, 25, 26, 28, 30, 33] and references given there). In this paper we study local first integrals of threedimensional polynomial autonomous systems with three invariant lines passing through the origin and the matrix of the linear approximation at the origin having the eigenvalues 0, −1, and 1. Such systems can be written in the form m j x˙1 = ai,j,k x1i x2 x3k = P (x), i+j +k=2 i>0,j,k≥0
x˙2 = −x2 + x2
m
j −1 k x3
bi,j −1,k x1i x2
= Q(x), (1)
i+j +k=2 i,k≥0,j >0
x˙3 = x3 + x3
m i+j +k=2 i,j ≥0,k>0
j
ci,j,k−1 x1i x2 x3k−1 = R(x),
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where x = (x1 , x2 , x3 ) ∈ C3 . This system can be considered as a complex or real system with a saddle point at the invariant plane x1 = 0, but (similarly to systems treated in [15, 26]) it also can be considered as a complexification of a real system, in which case we have the so-called case of one zero and two purely imaginary roots of the characteristic equation (see, e.g., [3, 26]). Note, that x1 = 0, x2 = 0, and x3 = 0 are invariant planes of system (1). Consider the set (2) R = α ∈ N3+ | α2 = α3 , where N+ = N ∪ 0. The terms of the form aα x α with α ∈ R are
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