Unique periodic orbits of a delay differential equation with piecewise linear feedback function

Ábel Garab

doi:10.3934/dcds.2013.33.2369

Article Contents

2013, Volume 33, Issue 6: 2369-2387. Doi: 10.3934/dcds.2013.33.2369

This issue Previous Article Thermal runaway for a nonlinear diffusion model in thermal electricity Next Article On the stability of standing waves of Klein-Gordon equations in a semiclassical regime

Unique periodic orbits of a delay differential equation with piecewise linear feedback function

Ábel Garab^1,

1.
Bolyai Institute, University of Szeged, Aradi vértanúk tere 1, Szeged, H-6720

Received: December 31, 2011

Revised: March 31, 2012

Published: June 2013

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Abstract

In this paper we study the scalar delay differential equation \linebreak $\dot{x}(t)=-ax(t) + bf(x(t-\tau))$ with feedback function $f(\xi)=\frac{1}{2}(|\xi+1|-|\xi-1|)$ and with real parameters $a>0,\ \tau>0$ and $b\neq 0$, which can model a single neuron or a group of synchronized neurons. We give necessary and sufficient conditions for existence and uniqueness of periodic orbits with prescribed oscillation frequencies. We also investigate the period of the slowly oscillating periodic solution as a function of the delay. Based on the obtained results we state an analogous theorem concerning existence and uniqueness of periodic orbits of a certain type of system of delay differential equations. The proofs are based among others on theory of monotone systems and discrete Lyapunov functionals.

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Mathematics Subject Classification: Primary: 34K13, 92B20.

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