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The Saddle-Node of Nearly Homogeneous Wave Trains in Reaction–Diffusion Systems

  • Jens D. M. Rademacher
  • Arnd Scheel
Article

We study the saddle-node bifurcation of a spatially homogeneous oscillation in a reaction-diffusion system posed on the real line. Beyond the stability of the primary homogeneous oscillations created in the bifurcation, we investigate existence and stability of wave trains with large wavelength that accompany the homogeneous oscillation. We find two different scenarios of possible bifurcation diagrams which we refer to as elliptic and hyperbolic. In both cases, we find all bifurcating wave trains and determine their stability on the unbounded real line. We confirm that the accompanying wave trains undergo a saddle-node bifurcation parallel to the saddle-node of the homogeneous oscillation, and we also show that the wave trains necessarily undergo sideband instabilities prior to the saddle-node.

Keywords

Saddle-node bifurcation wave trains homogeneous oscillation stability reaction diffusion systems 

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References

  1. 1.
    Brusch L., Torcini A., van Hecke M., Zimmermann M.G., Bär M. (2001). Modulated amplitude waves and defect formation in the one-dimensional complex Ginzburg–Landau equation. Phys. D 160: 127–148MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Coullet P., Risler E., Vandenberghe N. (2001). Spatial unfolding of elementary bifurcations. J. Stat. Phys. 101, 521–541CrossRefMathSciNetGoogle Scholar
  3. 3.
    Cross M.C., Hohenberg P.C. (1993). Pattern formation outside of equilibrium. Rev. Mod. Phys. 65, 851–1112CrossRefGoogle Scholar
  4. 4.
    Doelman, A., Sandstede, B., Scheel, A., and Schneider, G. (2006). The dynamics of modulated wave trains. To appear in Mem. AMS.Google Scholar
  5. 5.
    Gardner R.A. (1993). On the structure of the spectra of periodic travelling waves. J. Math. Pures Appl. 72, 415–439MATHMathSciNetGoogle Scholar
  6. 6.
    Bordiougov G., Engel H. (2006). From trigger to phase waves and back again. Physica D 215, 25–37MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Kuramoto Y., Yamada T. (1976). Pattern formation in oscillatory chemical reactions. Prog. Theor. Phys., 56, 724–740CrossRefMathSciNetGoogle Scholar
  8. 8.
    Rademacher, J. D. M., and Scheel, A. (2006). Instabilities of wave trains and turing patterns in large domains. To appear in Int. J. Bif. Chaos.Google Scholar
  9. 9.
    Reed M., Simon B. (1978). Methods of Modern Mathematical Physics. IV. Analysis of Operators. Academic Press, New York-LondonMATHGoogle Scholar
  10. 10.
    Sandstede B., Scheel A. (2000). Absolute and convective instabilities of waves on unbounded and large bounded domains. Phys. D, 145: 233–277MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2006

Authors and Affiliations

  1. 1.Centrum voor Wiskunde en Informatica (CWI)AmsterdamThe Netherlands
  2. 2.School of MathematicsUniversity of MinnesotaMinneapolisUSA

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