#### Document Type

Article

#### Publication Date

2013

#### Department 1

Mathematics

#### 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)x≠0. We identify a special family of such equations each of which has a ``rapidly oscillating" periodic solution p. The initial segment p0 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 ϵ>0, we can choose f and d such that the spectral radius of the derivative of R at p0 is less than 1+ϵ. The spectral radii are computed via a semiconjugacy of R with a finite-dimensional map.

#### Copyright Note

This is the publisher's version of the work. This publication appears in Gettysburg College's institutional repository by permission of the copyright owner for personal use, not for redistribution.

#### DOI

10.3934/dcdsb.2013.18.1633

#### Recommended Citation

Kennedy, Benjamin B. “A State-Dependent Delay Equation with Negative Feedback and ‘Mildly Unstable’ Rapidly Oscillating Periodic Solutions.” Discrete and Continuous Dynamical System Series B 18.6 (2013): 1633-1650.

#### Required Publisher's Statement

Original version is available from the publisher at: https://www.aimsciences.org/journals/displayArticlesnew.jsp?paperID=8407