The prototype equation for delayed negative feedback: periodic solutions

  • Odo Diekmann
  • Sjoerd M. Verduyn Lunel
  • Stephan A. van Gils
  • Hanns-Otto Walther
Part of the Applied Mathematical Sciences book series (AMS, volume 110)


In this chapter we study the nonlinear equation
$$ \dot x(t) = f(x(t - \alpha )) $$
where f: ℝ→ℝ is a continuous function. The positive parameter α is called the time lag. Equation (1.1) is the prototype for nonlinear delayed feedback. We shall see that the lag α can cause much more complex behaviour of solutions than in the ODE case α=0 where only monotone solutions converging to equilibria or infinity are possible. In particular, we shall prove the existence of periodic solutions (Section 5).


Periodic Solution Negative Feedback Primary Branch Global Bifurcation Unstable Behaviour 
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Copyright information

© Springer-Verlag New York, Inc. 1995

Authors and Affiliations

  • Odo Diekmann
    • 1
    • 2
  • Sjoerd M. Verduyn Lunel
    • 3
  • Stephan A. van Gils
    • 4
  • Hanns-Otto Walther
    • 5
  1. 1.Centrum voor Wiskunde en InformaticaAmsterdamThe Netherlands
  2. 2.Instituut voor Theoretische BiologieRijkuniversiteit LeidenLeidenThe Netherlands
  3. 3.Faculteit der Wiskunde en InformaticaUniversiteit van AmsterdamAmsterdamThe Netherlands
  4. 4.Faculteit der Toegepaste WiskundeUniversiteit TwenteEnschedeThe Netherlands
  5. 5.Mathematisches InstitutJustus-Leibig-UniversitätGiessenGermany

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