Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-10T14:45:22.859Z Has data issue: false hasContentIssue false
  • English
  • Français

A note on analytic integrability of planar vector fields

Published online by Cambridge University Press:  23 April 2012

A. ALGABA ,
C. GARCÍA  and
M. REYES
A. ALGABA
Affiliation:
Department of Mathematics, Facultad de Ciencias Experimentales, Campus del Carmen, University of Huelva, Spain e-mail: algaba@uhu.es, cristoba@uhu.es, colume@uhu.es
C. GARCÍA
Affiliation:
Department of Mathematics, Facultad de Ciencias Experimentales, Campus del Carmen, University of Huelva, Spain e-mail: algaba@uhu.es, cristoba@uhu.es, colume@uhu.es
M. REYES
Affiliation:
Department of Mathematics, Facultad de Ciencias Experimentales, Campus del Carmen, University of Huelva, Spain e-mail: algaba@uhu.es, cristoba@uhu.es, colume@uhu.es
Get access Rights & Permissions [Opens in a new window]

Abstract

We give a new characterisation of integrability of a planar vector field at the origin. This allows us to prove that the analytic systems where h, K, Ψ and ξ are analytic functions defined in the neighbourhood of O with K(O) ≠ 0 or Ψ(O) ≠ 0 and n ≥ 1, have a local analytic first integral at the origin. We show new families of analytically integrable systems that are held in the above class. In particular, this class includes all the nilpotent and generalised nilpotent integrable centres that we know.

Keywords

First integralCentreSaddleMonodromic
Type
Papers
Information
European Journal of Applied Mathematics , Volume 23 , Issue 5 , October 2012 , pp. 555 - 562
Copyright
Copyright © Cambridge University Press 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Algaba, A., Fuentes, N. & García, C. (2012) Centres of quasi-homogeneous polynomial planar systems. Nonlinear Anal. 13, 419–431.CrossRefGoogle Scholar
[2]Algaba, A., Gamero, E. & García, C. (2009) The integrability problem for a class of planar systems. Nonlinearity 22, 395–420.CrossRefGoogle Scholar
[3]Algaba, A., García, C. & Reyes, M. (2008) The centre problem for a family of systems of differential equations having a nilpotent singular point. J. Math. Anal. Appl. 340, 32–43.CrossRefGoogle Scholar
[4]Andreev, A., Sadovskii, A. P. & Tskialyuk, V. A. (2003) The centre-focus problem for a system with homogeneous nonlinearities in the case zero eigenvalues of the linear part. Diff. Equ. 39 (2), 155–164.CrossRefGoogle Scholar
[5]Bruno, A. D. (1989) Local Methods in Nonlinear Differential Equations, Springer-Verlag, New York.CrossRefGoogle Scholar
[6]Chavarriga, J., Giacomini, H., Giné, J. & Llibre, J. (2003) Local analytic integrability for nilpotent centres. Ergod. Th. Dynam. Sys. 23, 417–428.CrossRefGoogle Scholar
[7]Chavarriga, J., García, I. & Giné, J. (1997) Integrability of centres perturbed by quasihomogeneous polynomials. J. Math. Anal. Appl. 210 (1), 268–278.CrossRefGoogle Scholar
[8]Chavarriga, J. & Giné, J. (1999) Local integrability for nilpotent critical point. Dyn. Syst. Plasmas Gravit. (Lect. Notes Phys.) 518, 278–287 (Springer, Berlin, Germany).Google Scholar
[9]Chavarriga, J., Giné, J. & Sorolla, J. (2002) Analytic integrability of a class of nilpotent cubic systems. Math. Comput. Simul. 59, 489–495.CrossRefGoogle Scholar
[10]García, I. & Giné, J. (2010) Analytic nilpotent centres with analytic first integral. Nonlinear Anal. 72 (9–10), 3732–3738.CrossRefGoogle Scholar
[11]Gasull, A. & Torregrosa, J. (2001) A new algorithm for the computation of the Lyapunov constants for some degenerate critical points. Nonlinear Anal. 47, 4479–4490.CrossRefGoogle Scholar
[12]Giacomini, H., Giné, J. & Llibre, J. (2006) The problem of distinguishing between a centre and a focus for nilpotent and dgenerate analytic systems. J. Differ. Equ. 227 (2), 406–426.CrossRefGoogle Scholar
[13]Giacomini, H., Giné, J. & Llibre, J. (2007) Corrigendum to: “The problem of distinguishing between a centre and a focus for nilpotent and dgenerate analytic systems” (J. Differential Equations 227(2) (2006), 406–426). J. Differ. Equ. 232 (2), 702.CrossRefGoogle Scholar
[14]Giné, J. (2002) Sufficient conditions for a centre at a completely degenerate critical point. Int. J. Bifurcation Chaos Appl. Sci. Eng. 12 (7), 1659–1666.CrossRefGoogle Scholar
[15]Giné, J. (2004) Analytic integrability and characterization of centre for nilpotent singular point. Z. Angew. Math. Phys. 55, 725–740.CrossRefGoogle Scholar
[16]Giné, J. (2004) Analytic integrability and characterization of centre for generalized nilpotent singular point. Appl. Math. Comput. 148, 849–868.Google Scholar
[17]Giné, J. & Maza, S. (2011) The reversibility and the centre problem. Nonlinear Anal. 74 (2), 695–704.CrossRefGoogle Scholar
[18]Li, W., Llibre, J., Nicolau, M. & Zhang, X. (2002) On the differentiability of first integrals of two-dimensional flows. Proc. Am. Math. Soc. 130, 2079–2088.CrossRefGoogle Scholar
[19]Liapunov, M. A. (1949) Problème général de la stabilité du mouvement. Annals of Mathematics Studies, vol. 17, Princeton University Press, Princeton, NY.Google Scholar
[20]Mattei, J. F. & Moussu, R. (1980) Holonomie et intégrales premières. Ann. Sci. Ecole Normale Superieure 13, 469–523.CrossRefGoogle Scholar
[21]Mazzi, L. & Sabatini, M. (1988) A characterization of centres via first integrals. J. Differ. Equ. 76, 222–237.CrossRefGoogle Scholar
[22]Moussu, R. (1982) Symétrie et forme normale des centres et foyers dégénérés. Ergodic Theory Dynam. Sys. 2, 241–251.CrossRefGoogle Scholar
[23]Nemytskii, V. V. & Stepanov, V. V. (1960) Qualitative Theory of Differential Equations, Princeton University Press, Princeton, NY.Google Scholar
[24]Poincaré, H. (1881) Mémoire sur les courbes définies par les équations différentielles. J. Math. 37, 375–422.Google Scholar
[25]Strozyna, E. & Zoladek, H. (2002) The analytic and formal normal form for the nilpotent singularity. J. Differ. Equ. 179, 479–537.CrossRefGoogle Scholar