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