Periodic solutions and their stability of a differential-difference equation
Anatoli F. Ivanov - Department of Mathematics, Pennsylvania State University, P.O. Box PSU, Lehman, PA 18627, United States (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
Received: July 2008; Revised: April 2009; Published: September 2009. |