Stability and Exact Multiplicity of Periodic Solutions of Duffing Equations with Cubic Nonlinearities

We study the stability and exact multiplicity of periodic solutions of the Duffing equation with cubic nonlinearities, x + cx + ax − x = h(t), (∗) where a and c > 0 are positive constants and h(t) is a positive T -periodic function. We obtain sharp bounds for h such that (∗) has exactly three ordered T -periodic solutions. Moreover, when h is within these bounds, one of the three solutions is negative, while the other two are positive. The middle solution is asymptotically stable, and the remaining two are unstable.