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1:-3 resonant centers on 2 with homogeneous cubic nonlinearities. (English) Zbl 1165.34334
Summary: We characterize the collection of systems of differential equations on 2 of the form x ' =x+p(x,y), y ' =-3y+q(x,y), where p and q are homogeneous polynomials of degree three (either of which may be zero), that possess a first integral in a neighborhood of (0,0) of the form x 3 y+, where omitted terms are of order at least five. Such systems are called 1:-3 resonant centers.
MSC:
34C05Location of integral curves, singular points, limit cycles (ODE)
34C20Transformation and reduction of ODE and systems, normal forms
References:
[1]Dulac, H.: Détermination et intégration d’une certaine classe d’équations différentielles ayant pour point singulier un centre, Bull. sci. Math. 32, 230-252 (1908) · Zbl 39.0374.01
[2]Fronville, A.; Sadovski, A. P.; Zoład̨ek, H.: Solution of the 1:-2 resonant center problem in the quadratic case, Fund. math. 157, 191-207 (1998) · Zbl 0943.34018
[3]Christopher, C.; Rousseau, C.: Nondegenerate linearizable centres of complex planar quadratic and symmetric cubic systems in C2, Publ. mat. 45, No. 1, 95-123 (2001) · Zbl 0984.34023 · doi:10.5565/PUBLMAT_45101_04 · doi:http://mat.uab.es/pubmat/v45(1)/45101_04.pdf
[4]Sadovskii, A. P.: Holomorphic integrals of a certain system of differential equations, Differ. uravn. 10, 558-560 (1974) · Zbl 0319.34009
[5]Christopher, C.; Mardešić, P.; Rousseau, C.: Normalizable, integrable and linearizable saddle points for complex quadratic systems in C2, J. dynam. Control systems 9, 311-363 (2003) · Zbl 1022.37035 · doi:10.1023/A:1024643521094
[6]J. Llibre, H. Zołanbsp;dek, The Poincaré’s center problem, preprint, 2007
[7]Zoład̨ek, H.: The problem of center for resonant singular points of polynomial vector fields, J. differential equations 135, 94-118 (1997) · Zbl 0885.34034 · doi:10.1006/jdeq.1997.3260
[8]Romanovski, V.; Robnik, M.: The center and isochronicity problems for some cubic systems, J. phys. A 34, 10267-10292 (2001) · Zbl 1014.34028 · doi:10.1088/0305-4470/34/47/326
[9]V.G. Romanovski, D.S. Shafer, On the center problem for p:-q resonant polynomial vector fields, preprint · Zbl 1163.34020 · doi:euclid:bbms/1228486413
[10]W. Decker, G. Pfister, H.A. Schönemann, SINGULAR 2.0 library for computing the primary decomposition and radical of ideals primdec.lib, 2001
[11]G.-M. Greuel, G. Pfister, H. Schönemann, SINGULAR 3.0. A Computer Algebra System for Polynomial Computations, Centre for Computer Algebra, University of Kaiserslautern, 2005. http://www.singular.uni-kl.de
[12]Llibre, J.: Integrability of polynomial differential systems, Handobook of differential equations (2004)
[13]Faugère, J. -C.:
[14]Buchberger, B.: Ein algorithmisches kriterium fur die lösbarkeit eines algebraischen gleichungssystems, Aequationes math. 4–3, 374-383 (1970) · Zbl 0212.06401 · doi:10.1007/BF01844169
[15]Wang, D.: Elimination methods, (2001)
[16]Cox, D.; Little, J.; O’shea, D.: Ideals, varieties, and algorithms, (1997)