Project Euclid

References

• 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, Mat. Sbor. 30 (72) (1952), 181–196 (in Russian), Amer. Math. Soc. Transl. 100 (1954), 397–413 (in English).
  Zentralblatt MATH: 0059.08201
  
• J. Chavarriga, H. Giacomini, J. Giné and J. Llibre, On the integrability of two-dimensional flows, J. Differ. Equat. 157 (1999), 163–182.
  Zentralblatt MATH: 0940.37005
  Digital Object Identifier: doi:10.1006/jdeq.1998.3621
  
• J. Chavarriga and J. Giné, Integrability of a linear center perturbed by a fourth degree homogeneous polynomial, Publ. Mat. 40 (1996), 21–39.
  Zentralblatt MATH: 0851.34001
  Digital Object Identifier: doi:10.5565/PUBLMAT_40196_03
  
• ––––, Integrability of a linear center perturbed by a fifth degree homogeneous polynomial, Publ. Mat. 41 (1997), 335–356.
  Mathematical Reviews (MathSciNet): MR1485487
  Digital Object Identifier: doi:10.5565/PUBLMAT_41297_02
  Zentralblatt MATH: 0897.34030
  
• L.A. Cherkas, The number of limit cycles of a certain second order autonumous system, Diff. Uravnenija 12 (1976), 944–946.
• C. Christopher, P. Mardešić and C. Rousseau, Normalizable, integrable and linearizable saddle points for complex quadratic systems in $\mathbb{C} \sp 2$, J. Dyn. Contr. Syst. 9 (2003), 311–363.
  Zentralblatt MATH: 1022.37035
  Digital Object Identifier: doi:10.1023/A:1024643521094
  
• C. Christopher and C. Rousseau, Normalizable, integrable and linearizable saddle points in the Lotka-Volterra system, Qual. Th. Dyn. Syst. 5 (2004), 11–61.
  Mathematical Reviews (MathSciNet): MR2197423
  Digital Object Identifier: doi:10.1007/BF02968129
  Zentralblatt MATH: 1100.34022
  
• A. Cima, A. Gasull, V. Mañosa and F. Mañosas, Algebraic properties of the Liapunov and period constants, Rocky Mountain J. Math. 27 (1997), 471–501.
  Zentralblatt MATH: 0911.34025
  Digital Object Identifier: doi:10.1216/rmjm/1181071923
  
• W. Decker, S. Laplagne, G. Pfister and H.A. Schonemann, SINGULAR 3-1 library for computing the prime decomposition and radical of ideals, primdec.lib, 2010.
• P. Gianni, B. Trager and G. Zacharias, Gröbner bases and primary decompositions of polynomials, J. Symb. Comp. 6 (1988), 146–167.
  Mathematical Reviews (MathSciNet): MR988410
  Digital Object Identifier: doi:10.1016/S0747-7171(88)80040-3
  
• J. Giné, The center problem for a linear center perturbed by homogeneous polynomials, Acta Math. Sinica 22 (2006), 1613–1620.
  Zentralblatt MATH: 1124.34326
  Digital Object Identifier: doi:10.1007/s10114-005-0623-4
  
• ––––, On the centers of planar analytic differential systems, Inter. J. Bifurcation Chaos 17 (2007), 3061–3070.
  Zentralblatt MATH: 1158.34018
  Digital Object Identifier: doi:10.1142/S0218127407018865
  
• ––––, On the first integrals in the center problem, Bull. Sci. Math. 137 (2013), 457–465.
  Zentralblatt MATH: 1297.34044
  Digital Object Identifier: doi:10.1016/j.bulsci.2012.11.001
  
• J. Giné and V.G. Romanovski, Integrability conditions for Lotka-Volterra planar complex quintic systems, Nonlin. Anal. 11 (2010), 2100–2105.
  Mathematical Reviews (MathSciNet): MR2646619
  Digital Object Identifier: doi:10.1016/j.nonrwa.2009.06.002
  
• J. Giné and X. Santallusia, On the Poincaré-Lyapunov constants and the Poincaré series, Appl. Math. 28 (2001), 17–30.
• J. Giné and X. Santallusia, Implementation of a new algorithm of computation of the Poincaré-Liapunov constants, J. Comp. Appl. Math. 166 (2004), 465–476.
  Zentralblatt MATH: 1051.65125
  Digital Object Identifier: doi:10.1016/j.cam.2003.08.043
  
• G.M. Greuel, G. Pfister and H.A. Schönemann, SINGULAR $3.0$, A computer algebra system for polynomial computations, Centre for Computer Algebra, University of Kaiserlautern, 2005, http://www.singular.uni-kl.de.
  URL: Link to item
  
• \vbox J. Llibre and C. Valls, Classification of the centers and their cyclicity for the generalized quadratic polynomial differential systems, J. Math. Anal. Appl. 357 (2009), 427–437.
  Zentralblatt MATH: 1190.34033
  Digital Object Identifier: doi:10.1016/j.jmaa.2009.04.036
  
• –––– Classification of the centers, their cyclicity and isochronicity for a class of polynomial differential systems generalizing the linear systems with cubic homogeneous nonlinearities, J. Differ. Equat. 246 (2009), 2192–2204.
  Zentralblatt MATH: 1176.34035
  Digital Object Identifier: doi:10.1016/j.jde.2008.12.006
  
• ––––, Classification of the centers and of the isochronous centers for a class of quartic–like systems, Nonlin. Anal. 71 (2009), 3119–3128.
  Zentralblatt MATH: 1201.34051
  Digital Object Identifier: doi:10.1016/j.na.2009.01.223
  
• ––––, Classification of the centers, of their cyclicity and isochronicity for two classes of generalized quintic polynomial differential systems, Nonlin. Differ. Equat. Appl. 16 (2009), 657–679.
  Zentralblatt MATH: 1195.34046
  Digital Object Identifier: doi:10.1007/s00030-009-0029-6
  
• ––––, Centers and isochronous centers for two classes of generalized seventh and ninth systems, J. Dynam. Differ. Equat. 22 (2010), 657–675.
  Zentralblatt MATH: 1222.34032
  Digital Object Identifier: doi:10.1007/s10884-010-9175-0
  
• ––––, Classification of the centers and their isochronicity for a class of polynomial differential systems of arbitrary degree, Adv. Math. 227 (2011), 472–493.
  Zentralblatt MATH: 1230.37028
  Digital Object Identifier: doi:10.1016/j.aim.2011.02.003
  
• M.A. Lyapunov, Problème général de la stabilité du mouvement, Ann. Math. Stud. 17, Princeton University Press, Princeton, 1947.
• H. Poincaré, Sur l'intégration des équations différentielles du premier ordre et du premier degré I, Rend. Circ. Matem. Palermo 5 (1891), 161–191.
  Zentralblatt MATH: 23.0319.01
  
• ––––, Sur l'intégration des équations différentielles du premier ordre et du premier degré II, Rend. Circ. Matem. Palermo 11 (1897), 193–239.
• V.G. Romanovski and M. Prešern, An approach to solving systems of polynomials via modular arithmetics with applications, J. Comp. Appl. Math. 236 (2011), 196–208.
  Zentralblatt MATH: 1225.13029
  Digital Object Identifier: doi:10.1016/j.cam.2011.06.018
  
• D. Schlomiuk, Algebraic and geometric aspects of the theory of polynomial vector fields, in Bifurcations and periodic orbits of vector fields, Kluwer Academic Pubishers, Dordrecht, 1993.
  Zentralblatt MATH: 0790.34031
  
• ––––, Algebraic particular integrals, integrability and the problem of the center, Trans. Amer. Math. Soc. 338 (1993), 799–841.
  Zentralblatt MATH: 0777.58028
  Digital Object Identifier: doi:10.1090/S0002-9947-1993-1106193-6
  
• K.S. Sibirskii, On the number of limit cycles in the neighborhood of a singular point, Differ. Uravnenija 1 (1965), 53–66 (in Russian), Differ. Equat. 1 (1965), 36–47 (in English).
  Mathematical Reviews (MathSciNet): MR188542
  
• N.I. Vulpe and K.S. Sibirskii, Centro-affine invariant conditions for the existence of a center of a differential system with cubic nonlinearities, Dokl. Akad. Nauk 301 (1988), 1297–1301 (in Russian), Soviet Math. Dokl. 38 (1989), 198–201 (in English).
• H. Żoładek, The classification of reversible cubic systems with center, Topol. Meth. Nonlin. Anal. 4 (1994), 79–136.
• ––––, Remarks on: The classification of reversible cubic systems with center, Topol. Meth. Nonlin. Anal. 8 (1996), 335–342.