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.

我们考虑一个具有线性阻尼，一个Duffing类型的非线性项和一个小的强迫项的抽象演化方程。抽象问题是受到一些模型的启发，这些模型针对梁在外部载荷或磁场作用下的阻尼振荡，并受到横向力的影响而振动。主要特征是边界条件的自然选择导致方程的线性部分涉及两个算符我们将这种可交换情况下的结果扩展到该设置，即对于渐近小的强迫项，所有解最终都接近于无力方程的三个平衡点，两个稳定点和一个不稳定点。