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Using small deformations of the total energy, as introduced in [31], weestablish that damped second order gradient systems$$u^{\prime\prime}(t)+\gamma u^\prime(t)+\nabla G(u(t))=0,$$may be viewed asquasi-gradient systems. In order to study the asymptotic behavior of thesesystems, we prove that any (nontrivial) desingularizing function appearing inKL inequality satisfies $\varphi(s)\ge c\sqrt s$ whenever the original functionis definable and $C^2.$ Variants to this result are given. These facts are usedin turn to prove that a desingularizing function of the potential $G$ alsodesingularizes the total energy and its deformed versions. Our approach bringsforward several results interesting for their own sake: we provide anasymptotic alternative for quasi-gradient systems, either a trajectoryconverges, or its norm tends to infinity. The convergence rates are alsoanalyzed by an original method based on a one-dimensional worst-case gradientsystem.We conclude by establishing the convergence of solutions of dampedsecond order systems in various cases including the definable case. Thereal-analytic case is recovered and some results concerning convex functionsare also derived.

On damped second-order gradient systems