Proving that the potential solution to a differential equation converges (or not)

Question

I'm looking at this equation:

$$x''(t) - \sin(x(t))\cdot x'(t) + a ^{2}x(t) = 0$$

My question is therefore / how to prove that a solution is not convergent. Numerically we can see it oscillates but I'm not sure how to prove it.

So far I've proven it stays at a bounded distance to the solution of:

$$x''(t) + a^2x(t) = 0$$

(Gronwall lemma)

Answer 1

Note that the differential equation is invariant under the transformation $x \to -x$, $t \to c -t$. Thus a solution that starts at $x(0) = 0$, $x'(0) = v_0$ and goes to $x(t_1) = 0$, $x'(t_1) = v_1$ will come back to $x(2 t_1) = 0$, $x'(2 t_1) = v_0$ on the other side of the $x'$ axis, and continue in a closed orbit.