The asymptotic behavior of the solutions of the Duffing-type equation \( \ddot{y}(t)+\delta \dot{y}(t) + \big (\sigma +\varepsilon (t)\big )y(t)+y(t)^3=\mathcal {F}(t) \quad \text { for a.e. }t\ge 0, \) where \(\delta \ge 0\) , \(\sigma > 0\) and \(\varepsilon \) is bounded and nonnegative, is investigated. When \(\mathcal {F} \equiv 0\) , if \(\varepsilon \) is infinitesimal at infinity, it is shown that both vanishing (for \(t \rightarrow +\infty \) ) and unbounded solutions may exist, while this scenario dramatically changes if \(\varepsilon \) is integrable on \((0, +\infty )\) . This brings evidence of a high sensitivity of the response of the considered equation with respect to \(\varepsilon \) . In the periodically forced case, it is then shown that the attractor of the associated Poincaré map can be arcwise disconnected. Applications to models describing the dynamics of suspension bridges are also discussed.
