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(3) = 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.