Periodic Solutions of Second-Order Differential Equations with Large Parameters

Abstract

A second-order differential equation containing a large parameter is considered. Such an equation can be interpreted as an equation of constrained oscillations of a mechanical system with one degree of freedom, provided that the fundamental frequency of the system substantially exceeds the external frequency. We provide a new proof of the existence of a periodic solution of that equation such that it is close to the periodic solution of the corresponding degenerate equation. That proof is obtained by means of the Poincaré method.