This article analyzes the asymptotic behavior of three classes of second-order inertial systems with vanishing damping when applied to non-potential operators. We consider the dynamics \(\text {(V-AVD)}_\alpha \) , \(\text {(V-DIN-AVD)}_{\alpha ,\beta }\) , and \(\text {(V-ISIHD)}_{\alpha ,\beta }\) , where \(V: \mathcal {H} \rightarrow \mathcal {H}\) is a \(\lambda \) -cocoercive operator. By using a time-scale change \(t = \sqrt{2(\alpha +1)s}\) , we show that as the damping parameter \(\alpha \rightarrow +\infty \) , the rescaled trajectories converge uniformly on compact time intervals to solutions of first-order differential equations. Specifically, we establish convergence to the monotone flow \(\dot{y}(s) + V(y(s)) = 0\) for (V-AVD), to a Levenberg-Marquardt type equation \(\dot{y}(s) + V(y(s)) + \beta _0 \frac{d}{ds}V(y(s)) = 0\) for (V-DIN-AVD) in finite dimensions under the condition \(\lambda > 2\beta _0\) , and to an implicit dynamics \(\dot{y}(s) + V(y(s) + \beta _0\dot{y}(s)) = 0\) for (V-ISIHD) when \(x_1 = 0\) and \(2\beta _0 < \lambda \) . The proofs rely on Lyapunov-type energy estimates and provide a unified framework for understanding the high-damping limit of inertial systems in a non-potential setting. These results show that the non-potential setting cannot be treated as a mere extension of the gradient case \(V=\nabla f\) , but requires fundamentally new Lyapunov constructions and stability arguments beyond those available in the potential framework.
