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We consider the nonlinear damped Klein-Gordon equation \[\partial_{tt}u+2\alpha\partial_{t}u-\Delta u+u-|u|^{p-1}u=0 \quad \text{on} \ \[0,\infty)\times \mathbb{R}^N \] with $\alpha>0$, $2 \le N\le 5$ and energysubcritical exponents $p>2$. We study the behavior of solutions for which it issupposed that only one nonlinear object appears asymptotically for large times,at least for a sequence of times.  We first prove that the nonlinear object is necessarily a bound state. Next,we show that when the nonlinear object is a non-degenerate state or adegenerate excited state satisfying a simplicity condition, the convergenceholds for all positive times, with an exponential or algebraic raterespectively. Last, we provide an example where the solution converges exactlyat the rate $t^{-1}$ to the excited state.

Asymptotics of solutions with a compactness property for the nonlinear damped Klein-Gordon equation