Annali di Matematica Pura ed Applicata

, Volume 145, Issue 1, pp 33–128 | Cite as

Global continuation and asymptotic behaviour for periodic solutions of a differential-delay equation

  • John Mallet-Paret
  • Roger D. Nussbaum


The singularly perturbed differential-delay equation
$$\varepsilon \dot x(t) = - x(t) + f(x(t - 1))$$
is studied. Existence of periodic solutions is shown using a global continuation technique based on degree theory. For small ɛ these solutions are proved to have a square-wave shape, and are related to periodic points of the mappingf:RR.Whenfis not monotone the convergence of x(t) to the square-wave typically is not uniform, and resembles the Gibbs phenomenon of Fourier series.


Asymptotic Behaviour Periodic Solution Fourier Series Periodic Point Degree Theory 
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.


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© Fondazione Annali di Matematica Pura ed Applicata 1985

Authors and Affiliations

  • John Mallet-Paret
    • 1
  • Roger D. Nussbaum
    • 2
  1. 1.Division of Applied Mathematics (Box F)Brown UniversityProvidence
  2. 2.Department of MathematicsRutgers UniversityNew Brunswick

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