Ukrainian Mathematical Journal

, Volume 50, Issue 1, pp 108–115 | Cite as

Forced frequency locking of rotating waves

  • L. Recke


We describe the frequency locking of an asymptotically orbitally stable rotating wave solution of an autonomous S1-equivariant differential equation under the forcing of a rotating wave.


Periodic Solution Wave Solution Invariant Manifold Stable Manifold Force Frequency 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    N. V. Butenin, Yu. I. Neimark, and N. A. Fufaev, Introduction to the Theory of Nonlinear Oscillations [in Russian], Nauka, Moscow (1976).Google Scholar
  2. 2.
    C. Chicone, “Bifurcations of nonlinear oscillations and frequency entrainment near resonance,” SIAM J. Math. Anal, 23, 1577–1608 (1992).zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    S. N. Chow and J. K. Hale, Methods of Bifurcation Theory, Springer-Verlag, New York (1982).zbMATHGoogle Scholar
  4. 4.
    J. Cronin, Differential Equations. Introduction and Qualitative Theory, Marcel Dekker, New York (1994).zbMATHGoogle Scholar
  5. 5.
    J. K. Hale and Z. Táboas, “Interaction of damping and forcing in a second order equation,” Nonlinear Anal. TMA., 2, 77–84 (1978).zbMATHCrossRefGoogle Scholar
  6. 6.
    D. Rand, “Dynamics and symmetry. Predictions of modulated waves in rotating fluids,” Arch. Rat. Mech. Anal., 75, 1–38 (1982).CrossRefMathSciNetGoogle Scholar
  7. 7.
    C. Z. Ning and H. Haken, “The geometric phase in nonlinear dissipative systems,” Mod. Phys. Lett. B, 6, 1541–1568 (1992).CrossRefMathSciNetGoogle Scholar
  8. 8.
    M. Renardy, “Bifurcation from rotating waves,” Arch. Rat. Mech. Anal., 79, 43–84 (1982).MathSciNetGoogle Scholar
  9. 9.
    M. Renardy and H. Haken. “Bifurcation of solutions of the laser equations,” Phvs. D, 8, 57–89 (1983).CrossRefMathSciNetGoogle Scholar
  10. 10.
    A. M. Samoilenko, Mathematical Theory of Multifrequency Oscillations, Kluwer, Dordrecht (1991).Google Scholar
  11. 11.
    J. K. Hale, Ordinary Differential Equations, Wiley, New York (1969).zbMATHGoogle Scholar
  12. 12.
    L. Recke and D. Peterhof, “Abstract forced symmetry breaking and forced frequency locking of modulated waves,” J. Different. Equat. (1998) (to appear).Google Scholar

Copyright information

© Plenum Publishing Corporation 1998

Authors and Affiliations

  • L. Recke
    • 1
  1. 1.Institute of MathematicsHumboldt UniversityBerlinGermany

Personalised recommendations