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

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## Summary

*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 mapping*

*f*:

**R**→

**R**.

*When*

*f*

*is not monotone the convergence of x(t) to the square-wave typically is not uniform, and resembles the Gibbs phenomenon of Fourier series*.

## Keywords

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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