A new method is proposed for constructing the asymptotics of periodic solutions of nonlinear delay differential equations with a small parameter in the derivative expansion. The method is based on a uniform (with respect to parameters) normalization of the differential equation, which allows reducing the problem of finding periodic solutions to the construction of nontrivial solutions of some nonlinear infinite system of algebraic equations. The conditions for the stability of periodic solutions have been studied. A comparative analysis with the results of direct numerical integration of the differential equation has been carried out.
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