Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Homoclinic and heteroclinic orbits for the $ 0^2$ or $ 0^2i \omega$ singularity in the presence of two reversibility symmetries

Author: Ling-Jun Wang
Journal: Quart. Appl. Math. 67 (2009), 1-38
MSC (2000): Primary 34C37
DOI: https://doi.org/10.1090/S0033-569X-08-01077-5
Published electronically: December 22, 2008
MathSciNet review: 2495069
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Abstract: This paper is devoted to the study of the dynamics of reversible vector fields close to an equilibrium when a $ 0^2$ or a $ 0^2i\omega$ resonance occurs in the presence of two symmetries of reversibility. In the presence of a unique reversibility symmetry the existence of a homoclinic connection to 0 is known for the $ 0^2$ resonance whereas for the $ 0^2i\omega$ resonance there is generically no homoclinic connection to 0 but there is always a homoclinic connection to an exponentially small periodic orbit.

In the presence of a second symmetry of reversibility, the situation is more degenerate. Indeed, because of the second symmetry the quadratic part of the normal forms vanishes, and so the dynamics of the normal forms is governed by the cubic part. For the $ 0^2$ resonance we prove the existence of homoclinic connections to 0 and of heteroclinic orbits. For the $ 0^2i\omega$ resonance we prove that in most of the cases the second symmetry induces the existence of homoclinic connections to 0 and of heteroclinic orbits whereas with a unique symmetry there is generically no homoclinic connection to 0. Such a reversible vector field with two reversibility symmetries occurs for instance after center manifold reduction when studying 2-dimensional waves in NLS type systems with one-dimensional potential or when studying localized waves in nonlinear chains of coupled oscillators. It also occurs when studying localized buckling in rods with noncircular cross section.

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

  • 1. C. J. Amick and K. Kirchgässner. Solitary water-waves in the presence of surface tension. In Dynamical Problems in Continuum Physics, I.M.A. Volumes in Mathematics and its Applications 4, Springer, New York, 1987. MR 893395 (88f:76011)
  • 2. C. J. Amick and K. Kirchgässner. A theory of solitary water-waves in the presence of surface tension. Arch. Rational Mech. Anal. 105 (1989), 1-49. MR 963906 (89k:76020)
  • 3. M. Haragus and T. Kapitula. Localized waves in NLS-type equations with nearly one-dimensional potentials, in preparation.
  • 4. G. Iooss and M. Adelmeyer. Topics in bifurcation theory and applications.  Second edition. Adv. Ser. World Sci. (1998). MR 1695170 (2000c:37065)
  • 5. G. Iooss and K. Kirchgässner. Water waves for small surface tension: An approch via normal form. Proc. Roy. Soc. Edinburgh 122A (1992), 267-299. MR 1200201 (94b:76012)
  • 6. G. Iooss and M. C. Pérouème. Perturbed homoclinic solutions in reversible $ 1:1$ resonance vector fields. J. Differential Equations 102 (1993), 62-102. MR 1209977 (94c:58134)
  • 7. G. Iooss and G. James. Localized waves in nonlinear oscillator chains. Chaos 15, 1,015113. (2005). MR 2133464 (2005m:37179)
  • 8. E. Lombardi. Oscillatory integrals and phenomena beyond all algebraic orders.  Lecture Notes in Mathematics, 1741 Springer-Verlag, New York, 2000. MR 1770093 (2002f:34081)
  • 9. S. M. Sun. Non-existence of truly solitary waves in water with small surface tension. Proc. Roy. London A. Math. Phys. Eng. Sci. 455 (1999), 2191-2228. MR 1702734 (2000g:76017)
  • 10. G. H. M. Van der Heijden, A. R. Champneys and J. M. T. Thompson. The spatial complexity of Localized Buckling in Rods with Noncircular Cross Section. SIAM J. Appl. Math Vol. 59, No. 1 (1999), 198-221. MR 1647813 (99g:73066)

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Additional Information

Ling-Jun Wang
Affiliation: School of Mathematics and Statistics, Wuhan University, Wuhan 430072, P.R. China and Laboratoire MIP, Université Paul Sabatier Toulouse 3, 31062 Toulouse Cedex 9, France
Email: lingjun.wang@yahoo.com

DOI: https://doi.org/10.1090/S0033-569X-08-01077-5
Keywords: $0^2$ resonance, $0^2i\omega $ resonance, homoclinic, heteroclinic connection
Received by editor(s): April 9, 2007
Published electronically: December 22, 2008
Article copyright: © Copyright 2008 Brown University
The copyright for this article reverts to public domain 28 years after publication.

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