Article Contents
Article Contents

Periodic solutions and their stability of a differential-difference equation

• 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 $f$.
Mathematics Subject Classification: Primary: 34K13, 34K26; Secondary: 37E05.

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