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We consider an abstract nonlinear second order evolution equation, inspiredby some models for damped oscillations of a beam subject to external loads ormagnetic fields, and shaken by a transversal force. When there is no externalforce, the system has three stationary positions, two stable and one unstable,and all solutions are asymptotic for $t$ large to one of these stationarysolutions.We show that this pattern extends to the case where the externalforce is bounded and small enough, in the sense that solutions can exhibit onlythree different asymptotic behaviors.

An infinite dimensional Duffing-like evolution equation with linear dissipation and an asymptotically small source term