I would like to know when the solutions of:
$x'' +cx'+\sin(x) = 0$
are bounded in $[t_0,+\infty[$ depending on $c$.
This is a case of second order differential equation that can be interpreted as an oscillator where there is friction, represented by $c$.
The usual techniques fail for this example since $\sin(x)$ is not coercive
$$ \dot x \ddot x + c \dot x ^2+\dot x \sin x = 0 \Rightarrow \frac{1}{2}\frac{d}{dt}(\dot x)^2+c(\dot x)^2 = \frac{d}{dt}\cos(x) $$
Now $(\dot x)^2 + c \int (\dot x)^2 dt = \cos(x) + C_0$
hence if $c > 0 \Rightarrow (\dot x)^2 \le \cos x + C_0$