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Contracting return maps for monotone delayed feedback

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  • For equations

    $\dot{x}(t)=-\mu x(t)+f(x(t-1))$

    with continuous odd nonlinearities close to a step function $f_a(\xi)=-a$sign$\xi$, $a>0,$ we find sets of initial data to which solutions return. For Lipschitz nonlinearities the associated return maps become Lipschitz continuous. Monotonicity of $f$ close to 0 permits sharp estimates of the Lipschitz constants of the return map. If outside a neighbourhood of 0 the Lipschitz constant of $f$ is sufficiently small then the return map becomes a contraction, whose fixed point defines an attracting periodic orbit of the differential equation. Applications include the cases $f(x)=g(\gamma x)$ with $g'(x)=-(1+x^r)^{-1}$ for $x>0$, $r>\frac{3}{2}$ and $\gamma>0$ sufficiently large.

    Mathematics Subject Classification: 34K15, 58F22.


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