WIAS Preprint No. 257, (1996)

Forced frequency locking in S1-equivariant differential equations


  • Peterhof, Daniela
  • Recke, Lutz

2010 Mathematics Subject Classification

  • 58F35 34A47 34C25 34C27 34D35


  • Forced symmetry breaking, forced frequency locking, bifurcation from solution orbits, rotating waves, modulated waves


The aim of this paper is to present a simple analytic stategy for predicting, or engineering, two frequency locking phenomena for S1-equivariant ordinary differential equations. First we consider the forced frequency locking of a rotating wave solution of the unforced equation with a forcing of "rotating wave type", and we describe the creation of modulated wave solutions which is connected with this locking phenomenon. And second, we consider the forced frequency locking of a modulated wave solution with a forcing of "modulated wave type". Especially, we describe the sets of all control parameters and of all forcings such that frequency locking occures, the dynamic stability and the asymptotic behavior (for the forcing intensity tending to zero) of the locked solutions and the structural stability of all the phenomena. This paper is essentially founded on results from our previous work [41] concerning abstract forced symmetry breaking. The equations considered in the present paper are finite dimensional prototypes of certain infinite dimensional models describing the behavior of continuous wave operated or self-pulsating multisection DFB lasers under continuous or pulsating light injection, respectively.

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