In his paper, the author determines the stability of periodic solutions of (1) the general forced Duffing equation without damping,x¨+x+&egr;f(x)=&egr;E cos&ohgr;t,f(x) a polynomial, &egr;≪1; (2) the forced Duffing equation with damping,x¨+&egr;rx˙+&agr;x+&egr;x3=&egr;E cos(&ohgr;t+&psgr;), &egr;≪1; and (3) the general forced Van der Pol equation,x¨−&egr;f(x˙)+x=&egr;E cos(&ohgr;t+&psgr;),f(x˙) an odd polynomial, &egr;≪1, by seeking conditions necessary to insure periodic or almost‐periodic solutions of the corresponding variational equation. The former are obtained rigorously by means of an existence proof and the latter formally by perturbation series. A vertical tangent theorem is derived which states that the locus of the points of contact of the vertical tangents to the response curves is a stability boundary, since it coincides, in the first approximation, with the locus of periodic solutions of the variational equation. These techniques are illustrated by considering the forced vibrations of a triode oscillator with a fifth‐order tube characteristic, this being a particular case of (3), where we setf(x˙)=x˙−13x˙3+&lgr;x˙5.