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Proceedings of the American Mathematical Society

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Asymptotic phase and invariant foliations near periodic orbits

Author: Freddy Dumortier
Journal: Proc. Amer. Math. Soc. 134 (2006), 2989-2996
MSC (2000): Primary 34D05, 34C07, 34C20
Published electronically: May 5, 2006
MathSciNet review: 2231624
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Abstract: The paper deals with asymptotic phase and invariant foliations near periodic orbits, extending for two-dimensional smooth vector fields results that have been obtained by Chicone and Liu (2004). The problem of the existence of asymptotic phase is completely solved for analytic vector fields and is reduced to a problem of infinite codimension for $ C^{\infty }$ systems. Moreover it is proven that whenever asymptotic phase occurs, or in other words, when the periodic orbit is isochronous, then there also exists a $ C^{\infty }$ foliation, with leaves transversally cutting the periodic orbit and invariant under the flow of the vector field. The paper also provides some results in three dimensions.

References [Enhancements On Off] (What's this?)

  • [B] G.R. Belitskii, Functional equations and local conjugacy of mappings of class $ C^{\infty }$, Math. USSR Sb. 20, 4 (1973), 587-602.
  • [CL] Carmen Chicone and Weishi Liu, Asymptotic phase revisited, J. Differential Equations 204 (2004), no. 1, 227–246. MR 2076165,
  • [DR] F. Dumortier, R. Roussarie, Germes de difféomorphismes et de champs de vecteurs en classe de différentiabilité finie (Germs of diffeomorphisms and of vector fields in a finite differentiability class), Ann. Inst. Fourier, Grenoble 33 no.1, (1983), 195-267. MR 0698855 (84m:58114)
  • [DRR] F. Dumortier, P. Rodrigues, R. Roussarie, Germs of diffeomorphisms in the plane, Lecture Notes in Mathematics, vol. 902 Springer-Verlag, Berlin-New York, 1981. MR 0653474 (83f:58008)
  • [HPS] M. W. Hirsch, C. C. Pugh, and M. Shub, Invariant manifolds, Lecture Notes in Mathematics, Vol. 583, Springer-Verlag, Berlin-New York, 1977. MR 0501173
  • [T] Floris Takens, Forced oscillations and bifurcations, Applications of global analysis, I (Sympos., Utrecht State Univ., Utrecht, 1973) Math. Inst. Rijksuniv. Utrecht, Utrecht, 1974, pp. 1–59. Comm. Math. Inst. Rijksuniv. Utrecht, No. 3-1974. MR 0478235
    Floris Takens, Forced oscillations and bifurcations, Global analysis of dynamical systems, Inst. Phys., Bristol, 2001, pp. 1–61. MR 1858470

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Additional Information

Freddy Dumortier
Affiliation: Universiteit Hasselt, Campus Diepenbeek, Agoralaan - Gebouw D, B-3590 Diepenbeek, Belgium

Keywords: Periodic orbit, asymptotic phase, isochronous, invariant foliations, Poincar\'{e} map, return-time map
Received by editor(s): May 4, 2005
Published electronically: May 5, 2006
Additional Notes: The author thanks the Ministerio de Educacion y Ciencia from Spain for their financial support and the Centre de Recerca Matemàtica at Bellaterra for their hospitality during the preparation of this paper.
Communicated by: Carmen C. Chicone
Article copyright: © Copyright 2006 American Mathematical Society

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