Stability properties of periodic solutions of a Duffing equation in the presence of lower and upper solutions. (Q1856003)

The paper is concerned with the perturbed Duffing equation \[ x''+cx'+g(t,x)=h(t). \tag{1} \] It is proved that a periodic solution to equation (1) is asymptotically stable if and only if it is bracketed by a lower solution \(\alpha\) and an upper solution \(\beta\) satisfying \(\alpha(t)>\beta(t)\) for every \(t\), provided that the derivative of \(g\) with respect to \(x\) is not too large. Furthermore, the asymptotic stability of the \(T\)-periodic solutions to equation (1) is characterized in terms of the order stability: under certain assumptions, a \(T\)-periodic solution is asymptotically stable if and only if it is isolated and order stable. As an application, the existence of stable and unstable periodic solutions for Duffing equations with singular and oscillating nonlinearities is discussed.