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

This issue Previous Article Long-time behavior of positive solutions of a differential equation with state-dependent delay Next Article Existence of strictly decreasing positive solutions of linear differential equations of neutral type

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: March 31, 2017

Revised: June 30, 2017

Published: January 2020

Abstract Full Text(HTML) Figure(9) Related Papers Cited by

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.

Keywords:

Mathematics Subject Classification: Primary: 34K13.

Citation:

Full Text(HTML)

Download: Full-size image PowerPoint slide

Download: Full-size image PowerPoint slide

Download: Full-size image PowerPoint slide

Download: Full-size image PowerPoint slide

Download: Full-size image PowerPoint slide

Download: Full-size image PowerPoint slide

Download: Full-size image PowerPoint slide

Download: Full-size image PowerPoint slide

Download: Full-size image PowerPoint slide

Related Papers

Cited by

References

[1]	B. B. Kennedy, The Poincaré-Bendixson theorem for a class of delay equations with state-dependent delay and monotonic feedback, preprint.
[2]	T. Krisztin and O. Arino, The two-dimensional attractor of a differential equation with state-dependent delay, Journal of Dynamics and Differential Equations, 13 (2001), 453-522. doi: 10.1023/A:1016635223074.
[3]	J. Mallet-Paret and R. D. Nussbaum, Boundary layer phenomena for differential-delay equations with state-dependent time lags, I, Arch. Rational Mech. Anal., 120 (1992), 99-146. doi: 10.1007/BF00418497.
[4]	J. Mallet-Paret and G. R. Sell, Systems of differential delay equations: The Poincaré-Bendixson theorem for monotone cyclic feedback systems with delay, Journal of Differential Equations, 125 (1996), 441-489. doi: 10.1006/jdeq.1996.0037.
[5]	H.-O. Walther, Algebraic-delay differential systems, state-dependent delay, and temporal order of reactions, Journal of Dynamics and Differential Equations, 21 (2009), 195-232. doi: 10.1007/s10884-009-9129-6.
[6]	H.-O. Walther, A homoclinic loop generated by variable delay, Journal of Dynamics and Differential Equations, 27 (2015), 1101-1139. doi: 10.1007/s10884-013-9333-2.