The stability of the periodic solutions of a particular non-linear differential equation of third order is discussed in terms of the asymptotic stability of the corresponding variational equation and, thereby, in terms of the characteristic exponents of this equation. A method, due to Cesari, is used to evaluate the characteristic exponents and thus the boundaries of asymptotic stability. Taking the solution to have, in the first approximation, an amplitudeFand frequency ω, it is shown that one of the asymptotic stability boundaries corresponds to the locus of vertical tangents of the frequency response curves in the ω,Fplane. This result is similar in character to, and a modest generalization of, the well-known stability criterion associated with Duffing's equation.