WIAS Preprint No. 1018, (2005)

Entrainment of modulation frequency: A case study


  • Recke, Lutz
  • Schneider, Klaus R.

2010 Mathematics Subject Classification

  • 34E10 34C14 34C27 34C29 34C30 34C25


  • frequency entrainment, modulated wave, equivariance, approximate synchronization




We consider a system of autonomous ODE's which is $S^1$-equivariant and has a family of asymptotically stable modulated wave solutions with wave frequency $alpha_0$ and modulation frequency $beta_0$. This system will be perturbed, where the applied nonautonomous force also represents a modulated wave, but with wave frequency $alpha$ and modulations frequency $beta$. The strength of this perturbation is not necessarily small. Our goal is to look for conditions such that the perturbed system exhibits an approximate entrainment of the modulation frequency $beta$ on any given finite time interval, where the approximation error can be controlled by the wave frequency.

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