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Uniqueness of rapidly oscillating periodic solutions to a singularly perturbed differential-delay equation. (English) Zbl 0957.34070

The author considers the perturbed differential-delay equation \[ \varepsilon x'(t)= -x(t)+f\bigl(x(t- \tau)\bigr).\tag{1} \] Existence and uniqueness of rapidly oscillating periodic solutions to (1) are investigated. In Section 3, the author gives a brief review of Lin’s results concerning the “slowly oscillating case”. In Section 4, existence of rapidly oscillating periodic solutions to (1) is proved via generalization of Lin’s method. The main result of the paper (Section 5) is the fact that under generic conditions, for a given oscillation rate, there exists exactly one periodic solution to equation (1).
Reviewer: V.Petrov (Plovdiv)

MSC:

34K26 Singular perturbations of functional-differential equations
37G10 Bifurcations of singular points in dynamical systems
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