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Une classe d’hamiltoniens polynomiaux isochrones

  • Bertrand Schuman (a1)
Résumé

Soit un hamiltonien isochrone du plan . On met en évidence une classe d’hamiltoniens isochrones qui sont des perturbations polynomiales de H 0. On obtient alors une condition nécessaire d’isochronisme, et un critère de choix pour les hamiltoniens isochrones. On voit ce résultat comme étant une généralisation du caractère isochrone des perturbations hamiltoniennes homogènes considérées dans [L], [P], [S].

Abstract

Let be an isochronous Hamiltonian of the plane . We obtain a necessary condition for a system to be isochronous. We can think of this result as a generalization of the isochronous behaviour of the homogeneous polynomial perturbation of the Hamiltonian H 0 considered in [L], [P], [S].

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Copyright
References
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[A1] Arnold, V. I., Dynamical systems III. Encyclopaedia Math. Sci. Vol. 3, Springer Verlag, 1988.
[A2] Arnold, V. I., Geometrical methods in the theory of ordinary differential equations. Grundlehren Math. Wiss. 250, Springer-Verlag, 1988; Chapitres supplémentaires de la théorie des équations différentielles ordinaires. Mir, Moscou, 1980.
[B] Birkhoff, G. D., Dynamical systems. Amer. Math. Soc. Colloq. Publ. IX, 1927.
[Ch.D] Christopher, C.-J. and Devlin, J., Isochronous centers in planar polynomial systems. SIAM J. Math. Anal. 28 (1997), 162177.
[F1] Franc¸oise, J.-P., Birkhoff normal forms and analytic geometry. Dans: Symplectic singularities and geometry of gauge fields, Banach Center Publ. 39, Polish Acad. Sci., Warszawa, 1997, 49–56.
[F2] Franc¸oise, J.-P., The successive derivatives of the period function of a plane vector field. J. Differential Equations 146 (1998), 320335.
[G] Gavrilov, L., Isochronicity of plane polynomial Hamiltonian systems. Nonlinearity 10 (1997), 433448.
[L] Loud, W. S., Behaviour of the period of solutions of certain plane autonomous systems near centers. Contributions to Differential Equations 3 (1964), 2136.
[P] Pleshkan, I. I., A new method of investigating the isochronicity of a system of two differential equations. Differential Equations 5 (1969), 796802.
[R.T1] Rousseau, C. and Toni, B., Local bifurcation of critical periods in vector fields with homogeneous nonlinearities of the third degree. Canad. Math. Bull. (4) 36 (1993), 473484.
[R.T2] Rousseau, C. and Toni, B., Local bifurcation of critical periods in the reduced Kukles system. Canad. J. Math. (2) 49 (1997), 338358.
[S] Schuman, B., Sur la forme normale de Birkhoff et les centres isochrones. C. R. Acad. Sci. Paris Sér. I 322 (1996), 2124.
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Canadian Mathematical Bulletin
  • ISSN: 0008-4395
  • EISSN: 1496-4287
  • URL: /core/journals/canadian-mathematical-bulletin
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