# American Institute of Mathematical Sciences

June  2013, 33(6): 2369-2387. doi: 10.3934/dcds.2013.33.2369

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

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

Received  January 2012 Revised  April 2012 Published  December 2012

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.
Citation: Ábel Garab. Unique periodic orbits of a delay differential equation with piecewise linear feedback function. Discrete & Continuous Dynamical Systems, 2013, 33 (6) : 2369-2387. doi: 10.3934/dcds.2013.33.2369
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