Sufficient conditions for a center at a completely degenerate critical point

International Journal of Bifurcation and Chaos in Applied Sciences and Engineering

Volumn 12, Issue 7, 2002, Pages 1659-1666

  Gine J ^a  

^a UNIVERSITY OF LLEIDA   (Spain)

Author keywords

Center problem; Nonlinear differential equations

Indexed keywords

ALGORITHMS; INTEGRAL EQUATIONS; INTEGRATION; LINEAR EQUATIONS; POLYNOMIALS; PROBLEM SOLVING; THEOREM PROVING; VECTORS;

CENTER PROBLEMS; HOMOGENEOUS POLYNOMIALS;

PARTIAL DIFFERENTIAL EQUATIONS;

EID: 0036657528     PISSN: 02181274     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0218127402005315     Document Type: Article

References (21)

1. Andreev, A. [1958] "Investigations of the behavior of the integral curves of a system of two differential equations in the neighborhood of a singular point," Transl. Amer. Math. Soc. 8, 187-207.
2. Andronov, A.A., Leontovich, E.A., Gordon, I.I. & Maier, A.G. [1973] Theory of Bifurcations of Dynamic Systems on a Plane (John Wiley, NY).
3. Aranda, J.I. [1998] "Métodos simples para el análisis de puntos degenerados de sistemas analíticos planos," PhD, Universidad Complutense de Madrid, Madrid (in Spanish).
4. 2," Ann. Mat. Pure et Appl. IV, 35-70.
5. Berthier, M. & Moussu, R. [1994] "Réversibilité et classification des centres nilpotents," Ann. Inst. Fourier (Grenoble) 44, 465-494.
6. Berthier, M., Cerveau, D. & Lins Neto, A. [1996] "Sur les feuilletages analytiques rels et le problme du centre," J. Diff. Eqs. 131(2), 244-266.
7. Chavarriga, J., Giacomini, H. & Giné, J. [1997] "The null divergence factor," Publ. Mat. 41(1), 41-56.
8. Chavarriga, J., Giacomini, H., Giné, J. & Llibre, J. [2001] "Local analytic integrability for nilpotent centers," Ergod. Th. Dyn. Syst., to appear.
9. Farr, W.W., Li, C., Labouriau, I.S. & Langford, W.F. [1989] "Degenerate Hopf-bifurcation formulas and Hilbert's 16th problem," SIAM J. Math. Anal. 20, 13-29.
10. Foster, H. [1937] "Über das verhalten der integralkurven einer gewöhnlinchen differentialgleichung ester ordnung in der umgebung eines singulären punktes," Math. Zeits. 43, 271-320.
11. Gasull, A., Llibre, J., Man̄osa, V. & Man̄osas, F. [2000a] "The focus-center problem for a type of degenerate systems," Nonlinearity 13, 699-730.
12. Gasull, A., Man̄osa, V. & Man̄osas, F. [2000b] "Monodromy and stability of a generic class of degenerate planar critical points," J. Diff. Eqs., to appear.
13. Giacomini, H., Llibre, J. & Viano, M. [1996] "On the nonexistence, existence and uniqueness of limit cycles," Nonlinearity 9, 501-516.
14. Giné, J. & Santallusia, X. [2001] "On the Poincaré-Liapunov constants and the Poincaré series," Appl. Math. (Warsaw) 28(1), 17-30.
15. Man̄osa, V. [2002] "On the center problem for degenerate singular points of planar vector fields," Int. J. Bifurcation and Chaos 12(4), 687-707.
16. Mazzi, L. & Sabatini, M. [1988] "A characterization of centres via first integrals," J. Diff. Eqs. 76, 222-237.
17. Medvedeva, N.B. [1997] "The principal term of the asymptotic expansion of the monodromy transformation: Calculation in blowing-up geometry," Sib. Math. J. 38, 114-126.
18. Moussu, R. [1982a] "Symétrie et forme normale des centres et foyers dégénérés," Ergod. Th. Dyn. Syst. 2, 241-251.
19. Moussu, R. [1982b] "Une démonstration d'un théorème de Lyapunov-Poincaré," Astérisque 98 & 99, 216-223.
20. Némytskii, V.V. & Stepanov, V.V. [1989] Qualitative Theory of Differential Equations (Dover Publ., NY).
21. Schlomiuk, S. [1993] "Algebraic and geometric aspects of the theory of polynomials vector fields," Bifurcations and Periodics Orbits of Vector Fields (Kluwer Academic Publishers), pp. 429-467.