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Center conditions for a polynomial differential system. (English) Zbl 1275.34047
Differ. Equ. 49, No. 2, 151-165 (2013); translation from Differ. Uravn. 49, No. 2, 151-164 (2013).
The authors obtain 16 center conditions for a polynomial differential system with 27 parameters.

##### MSC:
 34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations 34C25 Periodic solutions to ordinary differential equations
##### Keywords:
polynomial differential system; center condition
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##### References:
 [1] Sadovskii, AP; Shcheglova, TV, Solution of the center-focus problem for a cubic system with parameters, Differ. Uravn., 47, 209-224, (2011) [2] Sadovskii, AP; Shcheglova, TV, Centers of a cubic system with eleven parameters, 71-75, (2011) · Zbl 1247.34046 [3] Romanovski, V.G. and Shafer, D.S, The Center and Cyclity Problems: A Computational Algebra Approach, Boston, 2009. [4] Sadovskii, A.P., Polinomial’nye idealy i mnogoobraziya (Polynomial Ideals and Manifolds), Minsk, 2008. [5] Cherkas, LA, Conditions for a center for a certain lienard equation, Differ. Uravn., 12, 292-298, (1976) · Zbl 0328.34023 [6] Cherkas, LA, Conditions for a center for the equation yy′ = ∑_{$$i$$=0}\^{}{3}$$P$$_{i}($$x$$)$$y$$\^{}{i}, Differ. Uravn., 14, 1594-1600, (1978) · Zbl 0414.34045 [7] Sadovskii, AP, On conditions for center and focus for nonlinear oscillation equations, Differ. Uravn., 15, 1716-1719, (1979) [8] Amel’kin, V.V., Lukashevich, N.A., and Sadovskii, A.P., Nelineinye kolebaniya v sistemakh vtorogo poryadka (Nonlinear Oscillations in Second-Order Systems), Minsk: Belarus Gos. Univ., 1982. [9] Sadovskii, AP, Lyurot theorem and cherkas method, Tr. 5-i mezhdunar. konf. “Analiticheskie metody analiza i differentsial’nykh uravnenii”, 2, 120-122, (2010) [10] Van der Waerden, B.L., Algebra (Algebra), Moscow, 1976. · Zbl 0997.00501 [11] Cox, D., Little, J., and O’Shea, D., Ideals, Varieties, and Algorithms. An Introduction to Computational Algebraic Geometry and Commutative Algebra, New York: Springer Verlag, 1997. Translated under the title Idealy, mnogoobraziya i algoritmy. Vvedenie v vychislitel’nye aspekty algebraicheskoi geometrii i kommutativnoi algebry, Moscow, 2000.
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