## Asymptotic phase and invariant foliations near periodic orbits

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- by Freddy Dumortier PDF
- Proc. Amer. Math. Soc.
**134**(2006), 2989-2996 Request permission

## 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

- G.R. Belitskii,
*Functional equations and local conjugacy of mappings of class $C^{\infty }$*, Math. USSR Sb. 20,**4**(1973), 587–602. - Carmen Chicone and Weishi Liu,
*Asymptotic phase revisited*, J. Differential Equations**204**(2004), no. 1, 227–246. MR**2076165**, DOI 10.1016/j.jde.2004.03.011 - F. Dumortier and R. Roussarie,
*Germes de difféomorphismes et de champs de vecteurs en classe de différentiabilité finie*, Ann. Inst. Fourier (Grenoble)**33**(1983), no. 1, 195–267 (French). MR**698855**, DOI 10.5802/aif.910 - Freddy Dumortier, Paulo R. Rodrigues, and Robert Roussarie,
*Germs of diffeomorphisms in the plane*, Lecture Notes in Mathematics, vol. 902, Springer-Verlag, Berlin-New York, 1981. MR**653474**, DOI 10.1007/BFb0095812 - 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**, DOI 10.1007/BFb0092042 - Floris Takens,
*Forced oscillations and bifurcations*, Applications of global analysis, I (Sympos., Utrecht State Univ., Utrecht, 1973) Comm. Math. Inst. Rijksuniv. Utrecht, No. 3-1974, Math. Inst. Rijksuniv. Utrecht, Utrecht, 1974, pp. 1–59. MR**0478235**

## Additional Information

**Freddy Dumortier**- Affiliation: Universiteit Hasselt, Campus Diepenbeek, Agoralaan - Gebouw D, B-3590 Diepenbeek, Belgium
- Email: freddy.dumortier@uhasselt.be
- 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
- © Copyright 2006 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**134**(2006), 2989-2996 - MSC (2000): Primary 34D05, 34C07, 34C20
- DOI: https://doi.org/10.1090/S0002-9939-06-08392-4
- MathSciNet review: 2231624