Small perturbations for a Duffing-like evolution equation involving non-commuting operators

We consider an abstract evolution equation with linear damping, a nonlinear term of Duffing type, and a small forcing term. The abstract problem is inspired by some models for damped oscillations of a beam subject to external loads or magnetic fields, and shaken by a transversal force.The main feature is that very natural choices of the boundary conditions lead to equations whose linear part involves two operators that do not commute.We extend to this setting the results that are known in the commutative case, namely that for asymptotically small forcing terms all solutions are eventually close to the three equilibrium points of the unforced equation, two stable and one unstable.