Periodic solutions and their stability of a differential-difference equation

Pages: 385 - 393, Issue Special, September 2009

 Abstract        Full Text (163.2K)              

Anatoli F. Ivanov - Department of Mathematics, Pennsylvania State University, P.O. Box PSU, Lehman, PA 18627, United States (email)
Sergei Trofimchuk - Departamento de Matemáticas, Facultad de Ciencias, Universidad de Chile, Casilla 653, Santiago, Chile (email)

Abstract: 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 $f$.

Keywords:  Differential delay and difference equations, Singular perturbations, Periodic solutions and their stability, Reduction to discrete maps, Interval maps
Mathematics Subject Classification:  Primary: 34K13, 34K26; Secondary: 37E05

Received: July 2008;      Revised: April 2009;      Published: September 2009.