A state-dependent delay equation with negative feedback and 'mildly unstable' rapidly oscillating periodic solutions

Benjamin B. Kennedy

doi:10.3934/dcdsb.2013.18.1633

Article Contents

2013, Volume 18, Issue 6: 1633-1650. Doi: 10.3934/dcdsb.2013.18.1633

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A state-dependent delay equation with negative feedback and "mildly unstable" rapidly oscillating periodic solutions

Benjamin B. Kennedy^1,

1.
Department of Mathematics, Gettysburg College, Gettysburg, PA 17325-1484

Received: November 30, 2011

Revised: March 31, 2012

Published: July 2013

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Abstract

We consider state-dependent delay equations of the form \[ x'(t) = f(x(t - d(x(t)))) \] where $d$ is smooth and $f$ is smooth, bounded, nonincreasing, and satisfies the negative feedback condition $xf(x) < 0$ for $x \neq 0$. We identify a special family of such equations each of which has a ``rapidly oscillating" periodic solution $p$. The initial segment $p_0$ of $p$ is the fixed point of a return map $R$ that is differentiable in an appropriate setting.
We show that, although all the periodic solutions $p$ we consider are unstable, the stability can be made arbitrarily mild in the sense that, given $\epsilon > 0$, we can choose $f$ and $d$ such that the spectral radius of the derivative of $R$ at $p_0$ is less than $1 + \epsilon$. The spectral radii are computed via a semiconjugacy of $R$ with a finite-dimensional map.

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Mathematics Subject Classification: 34K13.

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