A periodic solution with non-simple oscillation for an equation with state-dependent delay and strictly monotonic negative feedback

Benjamin B. Kennedy

doi:10.3934/dcdss.2020003

Article Contents

2020, Volume 13, Issue 1: 47-66. Doi: 10.3934/dcdss.2020003

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A periodic solution with non-simple oscillation for an equation with state-dependent delay and strictly monotonic negative feedback

Benjamin B. Kennedy

Department of Mathematics, Gettysburg College, 300 N. Washington St., Gettysburg, PA 17325, USA

Received: April 2017

Revised: July 2017

Published: January 2020

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Abstract

We consider a real-valued differential equation

$ \begin{equation*} x'(t) = f(x(t - d(x_t))), \end{equation*} $

with strictly monotonic negative feedback and state-dependent delay, that has a nontrivial periodic solution $ q $ for which the planar map $ q_t \mapsto (q(t),q(t - d(q_t))) $ is not injective on the orbit of $ q $ in phase space. This solution demonstrates that Mallet-Paret and Sell's version of the Poincaré-Bendixson theorem for delay equations with constant delay and monotonic feedback does not carry over entirely to the state-dependent delay case.

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Mathematics Subject Classification: Primary: 34K13.

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References

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