PapersNo Access

ON THE DEGENERATE CENTER PROBLEM

JAUME GINÉ

Departament de Matemàtica, Universitat de Lleida, Av. Jaume II, 69 25001 Lleida, Spain

https://doi.org/10.1142/S0218127411029082Cited by:23 (Source: Crossref)

Abstract

In this work, it is proved that any degenerate center is limit of a linear type center and when the degenerate center has an analytic first integral then it is limit of an analytic linear type center. A new method to detect integrability developed in [Giné & Santallusia, 2011] is applied to the degenerate center problem. Moreover, a review of the most important recent contributions to the degenerate center problem is given.

Keywords:

References

A. Algaba, E. Gamero and C. García, Nonlinearity 22, 395 (2009), DOI: 10.1088/0951-7715/22/2/009. Crossref, Web of Science, Google Scholar
A. Algaba, C. García and M. Reyes, Bull. Sci. Math. 133, 806 (2009), DOI: 10.1016/j.bulsci.2009.09.006. Crossref, Web of Science, Google Scholar
A. Algabaet al., Nonlin. Anal. 72, 1726 (2010). Crossref, Web of Science, Google Scholar
A. A. Andronov et al. , Qualitative Theory of Second-Order Dynamic Systems ( Halsted Press, A division of John Wiley & Sons , New York-Toronto, Ont. , 1973 ) . Google Scholar
D. V. Anosov et al. , Ordinary Differential Equations and Smooth Dynamical Systems ( Springer-Verlag , Berlin , 1997 ) . Google Scholar
J. Argemí, Ann. Mat. Pure et Appl. IV, 35 (1968). Google Scholar
Bautin, N. N. [1952] "On the number of limit cycles which appear with the variation of coefficients from an equilibrium position of focus or center type," Mat. Sb. 30, 181–196; Amer. Math. Soc. Transl. 100, 397–413 . Google Scholar
M. Berthier and R. Moussu, Ann. Inst. Fourier (Grenoble) 44, 465 (1994). Crossref, Web of Science, Google Scholar
A. D. Bruno , Local Methods in Nonlinear Differential Equations , Springer Series in Soviet Mathematics ( Springer-Verlag , Berlin , 1989 ) . Crossref, Google Scholar
J. Chavarrigaet al., J. Diff. Eqs. 157, 163 (1999), DOI: 10.1006/jdeq.1998.3621. Crossref, Web of Science, Google Scholar
J. Chavarriga, I. A. García and J. Giné, Nonlinearity 14, 863 (2001), DOI: 10.1088/0951-7715/14/4/313. Crossref, Web of Science, Google Scholar
J. Chavarrigaet al., Erg. Th. Dyn. Syst. 23, 417 (2003), DOI: 10.1017/S014338570200127X. Web of Science, Google Scholar
J. Écalle , Introduction aux Fonctions Analysables et Preuve Constructive de la Conjecture de Dulac , Actualités Mathématiques ( Hermann , Paris , 1992 ) . Google Scholar
H. Foster, Math. Zeits. 43, 271 (1937). Crossref, Web of Science, Google Scholar
I. A. García, J. Giné and M. Grau, J. Symb. Comput. 41, 943 (2006), DOI: 10.1016/j.jsc.2006.04.007. Crossref, Web of Science, Google Scholar
H. Giacomini, J. Giné and J. Llibre, J. Diff. Eqs. 227, 406 (2006), DOI: 10.1016/j.jde.2006.03.012. Crossref, Web of Science, Google Scholar
H. Giacomini, J. Giné and J. Llibre, J. Diff. Eqs. 232, 702 (2007), DOI: 10.1016/j.jde.2006.10.004. Crossref, Web of Science, Google Scholar
J. Giné, Int. J. Bifurcation and Chaos 12, 1659 (2002). Link, Web of Science, Google Scholar
J. Giné, Appl. Math. Comput. 148, 849 (2004), DOI: 10.1016/S0096-3003(02)00941-4. Web of Science, Google Scholar
J. Giné, Bull. Sci. Math. 130, 152 (2006), DOI: 10.1016/j.bulsci.2005.09.001. Crossref, Web of Science, Google Scholar
J. Giné, Int. J. Bifurcation and Chaos 17, 3061 (2007). Link, Web of Science, Google Scholar
J. Giné and X. Santallusia, Phys. Lett. A 375, 291 (2011), DOI: 10.1016/j.physleta.2010.11.026. Crossref, Web of Science, Google Scholar
M. Han and K. Jiang, Ann. Diff. Eqs. 14, 150 (1998). Google Scholar
Yu. S. Il'yashenko, Funct. Anal. Appl. 6, 30 (1972). Google Scholar
Yu. S. Il'yashenko , Finiteness Theorems for Limit Cycles , Translations of Mathematical Monographs 94 ( American Mathematical Society , Providence, RI , 1991 ) . Crossref, Google Scholar
W. Liet al., Proc. Amer. Math. Soc. 130, 2079 (2002), DOI: 10.1090/S0002-9939-02-06310-4. Crossref, Web of Science, Google Scholar
W. Liet al., J. Dyn. Diff. Eqs. 21, 133 (2009), DOI: 10.1007/s10884-008-9126-1. Crossref, Web of Science, Google Scholar
M. A. Lyapunov , Problème Général de la Stabilité du Mouvement , Ann. of Math. Stud. 17 ( Princeton University Press , 1947 ) . Google Scholar
V. Mañosa, Int. J. Bifurcation and Chaos 12, 687 (2002). Link, Web of Science, Google Scholar
J. F. Mattei and R. Moussu, Ann. Sci. Ecole Normale Superieure 13, 469 (1980). Crossref, Web of Science, Google Scholar
L. Mazzi and M. Sabatini, J. Diff. Eqs. 76, 222 (1988), DOI: 10.1016/0022-0396(88)90072-1. Crossref, Web of Science, Google Scholar
N. B. Medvedeva, Siber. Math. J. 33, 280 (1992), DOI: 10.1007/BF00971099. Crossref, Web of Science, Google Scholar
N. B. Medvedeva and E. Batcheva, Electron. J. Qual. Th. Diff. Eqs. 19, 1 (2000). Web of Science, Google Scholar
N. B. Medvedeva, Dokl. Akad. Nauk 394, 735 (2004). Google Scholar
N. B. Medvedeva, Proc. Steklov Inst. Math. 254, 7 (2006), DOI: 10.1134/S0081543806030023. Crossref, Google Scholar
R. Moussu, Erg. Th. Dyn. Syst. 2, 241 (1982), DOI: 10.1017/S0143385700001553. Crossref, Google Scholar
R. Moussu, Astérisque 99, 216 (1982). Google Scholar
V. V. Nemytskii and V. V. Stepanov , Qualitative Theory of Differential Equations ( Dover Publ. , NY , 1989 ) . Google Scholar
H. Poincaré, Rendiconti del Circolo Matematico di Palermo 5, 161 (1891), DOI: 10.1007/BF03015693. Crossref, Google Scholar
A. P. Sadovskii and D. N. Cherginets, Diff. Eqs. 44, 1373 (2008). Google Scholar
E. Strózyna and H. Żołladek, J. Diff. Eqs. 179, 479 (2002), DOI: 10.1006/jdeq.2001.4043. Crossref, Web of Science, Google Scholar
S. Walcher, Proc. Roy. Soc. Edinburgh Sect. A 130, 633 (2000). Crossref, Web of Science, Google Scholar
Q. Zhang, Y. Liu and G. Weihua, Appl. Math. Comput. 215, 1507 (2009), DOI: 10.1016/j.amc.2009.07.013. Web of Science, Google Scholar
Q. Zhang, G. Weihua and Y. Liu, Bull. Sci. Math. 133, 198 (2009), DOI: 10.1016/j.bulsci.2007.10.003. Crossref, Web of Science, Google Scholar
H. Żoładek and J. Llibre, J. Dyn. Contr. Syst. 14, 505 (2008), DOI: 10.1007/s10883-008-9049-5. Crossref, Web of Science, Google Scholar