In the setting ofa real Hilbert space ${\cal H}$, we investigate the asymptotic behavior, as time t goes to infinity, of trajectories of second-order evolutionequations

          ü(t) + γ$\dot{u}$(t) + ∇ϕ(u(t)) + A(u(t)) = 0,

where ∇ϕ is the gradient operator of a convexdifferentiable potential function ϕ : ${\cal H}\to \R$, A : ${\cal H}\to {\cal H}$ is a maximal monotone operator which is assumed to beλ-cocoercive, and γ > 0 is a damping parameter.Potential and non-potential effects are associated respectively to∇ϕ and A. Under condition λγ2 > 1, it is proved that each trajectory asymptotically weaklyconverges to a zero of ∇ϕ + A. This condition, whichonly involves the non-potential operator and the dampingparameter, is sharp and consistent with time rescaling. Passingfrom weak to strong convergence of the trajectories is obtained byintroducing an asymptotically vanishing Tikhonov-like regularizingterm. As special cases, we recover the asymptotic analysis of theheavy ball with friction dynamic attached to a convex potential, thesecond-order gradient-projection dynamic, and the second-orderdynamic governed by the Yosida approximation of a general maximalmonotone operator. The breadth and flexibility of the proposed framework is illustrated through applications in the areas of constrained optimization,dynamical approach to Nash equilibria for noncooperative games, and asymptotic stabilization in the case of a continuum of equilibria.