Stability properties of periodic solutions of a Duffing equation in the presence of lower and upper solutions

摘要

We prove that a periodic solution of the Duffing equationx″+cx′+g(t,x)=h(t),is asymptotically stable if and only if it is bracketed by a lower solution α and an upper solution β satisfying α(t)>β(t) for every t, provided that the derivative of g with respect to x is not too large. We also produce a characterization of the asymptotic stability of the periodic solutions of the above equation in terms of certain stability properties of the corresponding fixed points of a related infinite dimensional order-preserving discrete-time dynamical system. These results have a local flavour and therefore they naturally apply to the study of the stability in cases where g is not defined everywhere as a function of x, or several periodic solutions exist. As an application, we briefly discuss the existence of stable and unstable periodic solutions for some classes of Duffing equations with singular or oscillating nonlinearities.