Centers of weight-homogeneous polynomial vector fields on the plane

Giné, Jaume; Llibre, Jaume; Valls, Claudia

doi:10.1090/proc/13393

Centers of weight-homogeneous polynomial vector fields on the plane
HTML articles powered by AMS MathViewer

by Jaume Giné, Jaume Llibre and Claudia Valls PDF

Proc. Amer. Math. Soc. 145 (2017), 2539-2555 Request permission

Abstract:

We characterize all centers of planar weight-homogeneous polynomial vector fields. Moreover we classify all centers of planar weight- homogeneous polynomial vector fields of degrees $6$ and $7$.

References

A. Algaba, N. Fuentes, and C. García, Centers of quasi-homogeneous polynomial planar systems, Nonlinear Anal. Real World Appl. 13 (2012), no. 1, 419–431. MR 2846852, DOI 10.1016/j.nonrwa.2011.07.056
A. F. Andreev, Investigation of the behaviour of the integral curves of a system of two differential equations in the neighbourhood of a singular point, Amer. Math. Soc. Transl. (2) 8 (1958), 183–207. MR 0094499, DOI 10.1090/trans2/008/07
W. Aziz, J. Llibre, and C. Pantazi, Centers of quasi-homogeneous polynomial differential equations of degree three, Adv. Math. 254 (2014), 233–250. MR 3161098, DOI 10.1016/j.aim.2013.12.006
N. N. Bautin, On the number of limit cycles which appear with the variation of coefficients from an equilibrium position of focus or center type, American Math. Soc. Translation 1954 (1954), no. 100, 19. MR 0059426
H. Dulac, Détermination et integration d’une certaine classe d’équations différentielles ayant par point sigulier un centre, Bull. Sci. Math. 32 (1908), 230–252.
Belén García, Jaume Llibre, and Jesús S. Pérez del Río, Planar quasi-homogeneous polynomial differential systems and their integrability, J. Differential Equations 255 (2013), no. 10, 3185–3204. MR 3093361, DOI 10.1016/j.jde.2013.07.032
B. García, A. Lombardero, and J.S. Pérez del Río, Classification and counting of planar quasi-homogeneous differential systems through their weight vectors, preprint, 2015.
Jaume Giné, The center problem for a linear center perturbed by homogeneous polynomials, Acta Math. Sin. (Engl. Ser.) 22 (2006), no. 6, 1613–1620. MR 2262417, DOI 10.1007/s10114-005-0623-4
Jaume Giné, Maite Grau, and Jaume Llibre, Polynomial and rational first integrals for planar quasi-homogeneous polynomial differential systems, Discrete Contin. Dyn. Syst. 33 (2013), no. 10, 4531–4547. MR 3049089, DOI 10.3934/dcds.2013.33.4531
Jaume Giné, Maite Grau, and Jaume Llibre, Limit cycles bifurcating from planar polynomial quasi-homogeneous centers, J. Differential Equations 259 (2015), no. 12, 7135–7160. MR 3401593, DOI 10.1016/j.jde.2015.08.014
Alain Goriely, Integrability, partial integrability, and nonintegrability for systems of ordinary differential equations, J. Math. Phys. 37 (1996), no. 4, 1871–1893. MR 1380879, DOI 10.1063/1.531484
Yanxia Hu, On the integrability of quasihomogeneous systems and quasidegenerate infinity systems, Adv. Difference Equ. , posted on (2007), Art ID. 98427, 10. MR 2322486, DOI 10.1155/2007/98427
Weigu Li, Jaume Llibre, Jiazhong Yang, and Zhifen Zhang, Limit cycles bifurcating from the period annulus of quasi-homogeneous centers, J. Dynam. Differential Equations 21 (2009), no. 1, 133–152. MR 2482012, DOI 10.1007/s10884-008-9126-1
Haihua Liang, Jianfeng Huang, and Yulin Zhao, Classification of global phase portraits of planar quartic quasi-homogeneous polynomial differential systems, Nonlinear Dynam. 78 (2014), no. 3, 1659–1681. MR 3270613, DOI 10.1007/s11071-014-1541-8
Jaume Llibre and Claudio Pessoa, On the centers of the weight-homogeneous polynomial vector fields on the plane, J. Math. Anal. Appl. 359 (2009), no. 2, 722–730. MR 2546789, DOI 10.1016/j.jmaa.2009.06.036
Luisa Mazzi and Marco Sabatini, A characterization of centres via first integrals, J. Differential Equations 76 (1988), no. 2, 222–237. MR 969422, DOI 10.1016/0022-0396(88)90072-1
Yilei Tang, Long Wang, and Xiang Zhang, Center of planar quintic quasi-homogeneous polynomial differential systems, Discrete Contin. Dyn. Syst. 35 (2015), no. 5, 2177–2191. MR 3294245, DOI 10.3934/dcds.2015.35.2177
H. Poincaré, Sur l’intégration des équations différentielles du premier ordre et du premier degré I and II, Rend. Circ. Mat. Palermo 5 (1891), 161–191; 11 (1897), 193–239.
C. Rousseau and D. Schlomiuk, Cubic vector fields symmetric with respect to a center, J. Differential Equations 123 (1995), no. 2, 388–436. MR 1362881, DOI 10.1006/jdeq.1995.1168
Dana Schlomiuk, Algebraic particular integrals, integrability and the problem of the center, Trans. Amer. Math. Soc. 338 (1993), no. 2, 799–841. MR 1106193, DOI 10.1090/S0002-9947-1993-1106193-6
Henryk Żołądek, Quadratic systems with center and their perturbations, J. Differential Equations 109 (1994), no. 2, 223–273. MR 1273302, DOI 10.1006/jdeq.1994.1049
Henryk Żołądek, The classification of reversible cubic systems with center, Topol. Methods Nonlinear Anal. 4 (1994), no. 1, 79–136. MR 1321810, DOI 10.12775/TMNA.1994.024
Henryk Żołądek, Remarks on: “The classification of reversible cubic systems with center” [Topol. Methods Nonlinear Anal. 4 (1994), no. 1, 79–136; MR1321810 (96m:34057)], Topol. Methods Nonlinear Anal. 8 (1996), no. 2, 335–342 (1997). MR 1483632, DOI 10.12775/TMNA.1996.037