LettersNo Access

SUFFICIENT CONDITIONS FOR A CENTER AT A COMPLETELY DEGENERATE CRITICAL POINT

J. GINÉ

Department de Matemàtica, Universitat de Lleida, Avda. Jaume II, 69. 25001 Lleida, Spain

https://doi.org/10.1142/S0218127402005315Cited by:32 (Source: Crossref)

Abstract

Consider the two-dimensional autonomous systems of differential equations of the form

where P₃(x, y) and Q₃(x, y) are homogeneous polynomials of degree 3, and P₄(x, y) and Q₄(x, y) are homogeneous polynomials of degree 4. The origin is a completely degenerate critical point of this system. In this work we give sufficient conditions in order to have a center at the origin.

Keywords:

References

A. Andreev, Transl. Amer. Math. Soc. 8, 187 (1958). Web of Science, Google Scholar
A. A. Andronov et al. , Theory of Bifurcations of Dynamic Systems on a Plane ( John Wiley , NY , 1973 ) . Google Scholar
Aranda, J. I. [1998] "Métodos simples para el análisis de puntos degenerados de sistemas analíticos planos," PhD, Universidad Complutense de Madrid, Madrid (in Spanish) . Google Scholar
J. Argemí, Ann. Mat. Pure et Appl. IV, 35 (1968). Google Scholar
M. Berthier and R. Moussu, Ann. Inst. Fourier (Grenoble) 44, 465 (1994). Web of Science, Google Scholar
M. Berthier, D. Cerveau and A. Lins Neto, J. Diff. Eqs. 131(2), 244 (1996). Web of Science, Google Scholar
J. Chavarriga, H. Giacomini and J. Giné, Publ. Mat. 41(1), 41 (1997). Web of Science, Google Scholar
J. Chavarrigaet al., Ergod. Th. Dyn. Syst. (2001). Google Scholar
W. W. Farret al., SIAM J. Math. Anal. 20, 13 (1989). Web of Science, Google Scholar
H. Foster, Math. Zeits. 43, 271 (1937). Web of Science, Google Scholar
A. Gasullet al., Nonlinearity 13, 699 (2000). Web of Science, Google Scholar
A. Gasull, V. Manosa and F. Manosas, J. Diff. Eqs. (2000). Google Scholar
H. Giacomini, J. Llibre and M. Viano, Nonlinearity 9, 501 (1996). Web of Science, Google Scholar
J. Giné and X. Santallusia, Appl. Math. (Warsaw) 28(1), 17 (2001). Google Scholar
V. Manosa, Int. J. Bifurcation and Chaos 12(4), 687 (2002), DOI: 10.1142/S0218127402004693. Link, Web of Science, Google Scholar
L. Mazzi and M. Sabatini, J. Diff. Eqs. 76, 222 (1988). Web of Science, Google Scholar
N. B. Medvedeva, Sib. Math. J. 38, 114 (1997). Web of Science, Google Scholar
R. Moussu, Ergod. Th. Dyn. Syst. 2, 241 (1982). Google Scholar
R. Moussu, Astérisque 98 & 99, 216 (1982). Google Scholar
V. V. Nemytskii and V. V. Stepanov , Qualitative Theory of Differential Equations ( Dover Publ. , NY , 1989 ) . Google Scholar
S. Schlomiuk, Bifurcations and Periodics Orbits of Vector Fields (Kluwer Academic Publishers, 1993) pp. 429–467. Google Scholar