Document Type
Article
Publication Date
6-13-2023
Department 1
Mathematics
Abstract
We study the scalar difference equation x(k + 1) = x(k) + (f(x(k - N))) / N,
where f is nonincreasing with negative feedback. This equation is a discretization of the well-studied differential delay equation x'(t) = f(x(t - 1)).
We examine explicit families of such equations for which we can find, for infinitely many values of N and appropriate parameter values, various dynamical behaviors including periodic solutions with large numbers of sign changes per minimal period, solutions that do not converge to periodic solutions, and chaos. We contrast these behaviors with the dynamics of the limiting differential equation. Our primary tool is the analysis of return maps for the difference equations that are conjugate to continuous self-maps of the circle.
Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 License.
DOI
10.7494/OpMath.2023.43.4.507
Version
Version of Record
Recommended Citation
Benjamin B. Kennedy, Periodic, nonperiodic, and chaotic solutions for a class of difference equations with negative feedback, Opuscula Math. 43, no. 4 (2023), 507-546, https://doi.org/10.7494/OpMath.2023.43.4.507
Required Publisher's Statement
This article is available from the publisher's website.