# American Institute of Mathematical Sciences

2009, 2009(Special): 385-393. doi: 10.3934/proc.2009.2009.385

## Periodic solutions and their stability of a differential-difference equation

 1 Department of Mathematics, Pennsylvania State University, P.O. Box PSU, Lehman, PA 18627, United States 2 Departamento de Matemáticas, Facultad de Ciencias, Universidad de Chile, Casilla 653, Santiago

Received  July 2008 Revised  April 2009 Published  September 2009

Existence, stability, and shape of periodic solutions are derived for the differential-difference equation $\varepsilon\dot x(t)+x(t)=f(x([t-1])), 0<\varepsilon\<\<1,$ where $[\cdot]$ is the integer part function. The equation can be viewed as a special discretization (discrete version) of the singularly perturbed differential delay equation $\varepsilon\dot x(t)+x(t)=f(x(t-1))$. The principal analysis is based on reduction to the two-dimensional map $F: (u,v)\to (v, f(u)+ [v-f(u)]e^{-1/\varepsilon}),$ many relevant properties of which follow from those of the one-dimensional map $f$.
Citation: Anatoli F. Ivanov, Sergei Trofimchuk. Periodic solutions and their stability of a differential-difference equation. Conference Publications, 2009, 2009 (Special) : 385-393. doi: 10.3934/proc.2009.2009.385
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