This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.
A. Andreev, (1958), Investigation of the behaviour of the integral curves of a system of two differential equations in the neighborhood of a singular point, Translation of Amer. Math. Soc., 8, 187–207.AndreevA.1958Investigation of the behaviour of the integral curves of a system of two differential equations in the neighborhood of a singular point818720710.1090/trans2/008/07Search in Google Scholar
A.A. Andronov, E.A. Leontovich, I.I. Gordon and A.G. Maier, (1973), Theory of Bifurcations of Dynamic Systems on a Plane, John Wiley and Sons, New York.AndronovA.A.LeontovichE.A.GordonI.I.MaierA.G.1973John Wiley and SonsNew YorkSearch in Google Scholar
M. Berthier and R. Moussu, (1994), Réversibilité et classification des centres nilpotents, Ann. Inst. Fourier (Grenoble), 44, 465–494. 10.5802/aif.1406BerthierM.MoussuR.1994Réversibilité et classification des centres nilpotents4446549410.5802/aif.1406Open DOISearch in Google Scholar
J. Chavarriga, H. Giacomini, J. Giné and J. Llibre, (2003), Local analytic integrability for nilpotent centers, Ergodic Theory and Dynamical Systems, 23, 417–428. 10.1017/S014338570200127XChavarrigaJ.GiacominiH.GinéJ.LlibreJ.2003Local analytic integrability for nilpotent centers2341742810.1017/S014338570200127XOpen DOISearch in Google Scholar
A. Cima, A. Gasull, V. Mañosa and F. Mañosas, (1997), Algebraic properties of the Liapunov and period constants, Rocky Mountain J. Math., 27, 471–501. 10.1216/rmjm/1181071923CimaA.GasullA.MañosaV.MañosasF.1997Algebraic properties of the Liapunov and period constants2747150110.1216/rmjm/1181071923Open DOISearch in Google Scholar
H. Dulac, (1908), Détermination et integration d’une certaine classe d’équations différentielle ayant par point singulier un centre, Bull. Sci. Math. Sér. (2), 32, 230–252.DulacH.1908Détermination et integration d’une certaine classe d’équations différentielle ayant par point singulier un centre232230252Search in Google Scholar
F. Dumortier, J. Llibre and J.C. Artés, (2006), Qualitative theory of planar differential systems. Universitext. Springer Verlag, Berlin. 10.1007/978-3-540-32902-2DumortierF.LlibreJ.ArtésJ.C.2006Qualitative theory of planar differential systemsSpringer VerlagBerlin10.1007/978-3-540-32902-2Open DOISearch in Google Scholar
I.A. García, H. Giacomini, J. Giné and J. Llibre, Analytic nilpotent centers as limits of nondegenerated centers revisited. Preprint.GarcíaI.A.GiacominiH.GinéJ.LlibreJ.Analytic nilpotent centers as limits of nondegenerated centers revisitedSearch in Google Scholar
I.A. García, H. Giacomini and M. Grau, (2011), Generalized Hopf bifurcation for planar vector fields via the inverse integrating factor, J. Dynam. Differential Equations, 23, 251–281. 10.1007/s10884-011-9209-2GarcíaI.A.GiacominiH.GrauM.2011Generalized Hopf bifurcation for planar vector fields via the inverse integrating factorDynamJ.25128110.1007/s10884-011-9209-2Open DOISearch in Google Scholar
A. Gasull, J. Llibre, V. Ma¯nosa and F. Ma¯nosas, (2000), The focus–center problem for a type of degenerate systems, Nonlinearity, 13, 699–730. 10.1088/0951-7715/13/3/311GasullA.LlibreJ.Ma¯nosaV.Ma¯nosasF.2000The focus–center problem for a type of degenerate systems1369973010.1088/0951-7715/13/3/311Open DOISearch in Google Scholar
H. Giacomini, J. Giné and J. Llibre, (2006), The problem of distinguishing between a center and a focus for nilpotent and degenerate analytic systems, J. Differential Equations, 227, 406–426. 10.1016/j.jde.2006.03.012GiacominiH.GinéJ.LlibreJ.2006The problem of distinguishing between a center and a focus for nilpotent and degenerate analytic systems22740642610.1016/j.jde.2006.03.012Open DOISearch in Google Scholar
J. Giné and J. Llibre, (2014), A method for characterizing nilpotent centers. J. Math. Anal. Appl., 413, 537–545. 10.1016/j.jmaa.2013.12.013GinéJ.LlibreJ.2014A method for characterizing nilpotent centers41353754510.1016/j.jmaa.2013.12.013Open DOISearch in Google Scholar
M. Grau and J. Llibre, (2015), Divergence and Poincaré–Liapunov constants for analytic differential systems, J. Differential Equations, 258, 4348–4367. 10.1016/j.jde.2015.01.035GrauM.LlibreJ.2015Divergence and Poincaré–Liapunov constants for analytic differential systems2584348436710.1016/j.jde.2015.01.035Open DOISearch in Google Scholar
Yu. S. Il’yashenko, (1972), Algebraic unsolvability and almost algebraic solvability of the problem for the center–focus, Funkcion. Anal. Priloz., 6, No 3, 30–37.Il’yashenkoYu. S.1972Algebraic unsolvability and almost algebraic solvability of the problem for the center–focus63303710.1007/BF01077875Search in Google Scholar
Weigu Li, J. Llibre, M. Nicolau and Xiang Zhang, (2002), On the differentiability of first integrals of two dimensional flows, Proc. Amer. Math. Soc., 130, 2079–2088. 10.1090/S0002-9939-02-06310-4LiWeiguLlibreJ.NicolauM.ZhangXiang2002On the differentiability of first integrals of two dimensional flows1302079208810.1090/S0002-9939-02-06310-4Open DOISearch in Google Scholar
M.A. Liapunov, (1947) Problème général de la stabilité du mouvement, Ann. of Math. Stud. 17, Princeton University Press.LiapunovM.A.1947Problème général de la stabilité du mouvement17Princeton University Press10.5802/afst.246Search in Google Scholar
J. Llibre and H. Zoladek, (2008), The Poincaré center problem, J. Dynamical and Control Systems, 14, 505–535. 10.1007/s10883-008-9049-5LlibreJ.ZoladekH.2008The Poincaré center problem1450553510.1007/s10883-008-9049-5Open DOISearch in Google Scholar
L. Mazzi and M. Sabatini, (1988), A characterization of centres via first integrals, J. Differential Equations, 76, 222– 237. 10.1016/0022-0396(88)90072-1MazziL.SabatiniM.1988A characterization of centres via first integrals76222 23710.1016/0022-0396(88)90072-1Open DOISearch in Google Scholar
R. Moussu, (1982), Symétrie et forme normale des centres et foyers dégénérés, Ergodic Theory and Dynamical Systems, 2, 241–251. 10.1017/S0143385700001553MoussuR.1982Symétrie et forme normale des centres et foyers dégénérés224125110.1017/S0143385700001553Open DOISearch in Google Scholar
R. MOUSSU, (1982), Une démonstration d’un théorème de Lyapunov–Poincaré, Astérisque, 98-99, 216–223.MOUSSUR.1982Une démonstration d’un théorème de Lyapunov–Poincaré98-99216223Search in Google Scholar
V.V. Nemytskii and V.V. Stepanov, (1989) Qualitative theory of differential equations, Dover Publ., New York.NemytskiiV.V.StepanovV.V.1989Qualitative theory of differential equationsNew York10.1515/9781400875955Search in Google Scholar
H. Poincaré, (1881), Mémoire sur les courbes définies par les équations différentielles, Journal de Mathématiques, 37, 375–422; Oeuvres de Henri Poincaré, vol. I, Gauthier-Villars, Paris, 1951, pp 3–84.PoincaréH.1881Mémoire sur les courbes définies par les équations différentielles Journal de Mathématiques37375422IGauthier-VillarsParis1951384Search in Google Scholar
E. Strózyna and H. Zoladek, (2002) The analytic and formal normal form for the nilpotent singularity, J. Differential Equations, 179, 479–537. 10.1006/jdeq.2001.4043StrózynaE.ZoladekH.2002The analytic and formal normal form for the nilpotent singularity17947953710.1006/jdeq.2001.4043Open DOISearch in Google Scholar
F. Takens, (1947) Singularities of vector fields, Inst. Hautes Etudes Sci. Publ. Math., 43, 47–100. 10.1007/BF02684366TakensF.1947Singularities of vector fields434710010.1007/BF02684366Open DOISearch in Google Scholar
M.A. Teixeira and J. Yang, (2001), The Center-focus Problem and Reversibility, J. Differential Equations, 174, 237– 251. 10.1006/jdeq.2000.3931TeixeiraM.A.YangJ.2001The Center-focus Problem and Reversibility17423725110.1006/jdeq.2000.3931Open DOISearch in Google Scholar