Trapping region for $\ddot x + (x^2-2) \dot x + x + \sin(x) =0$

Question

I need to show that the system

$$\ddot x + (x^2-2) \dot x + x + \sin(x) =0 $$

Have a periodic orbit. I always use polar coordinates to find a trapping regio, but with the sine term, I am kinda lost. If someone could help me to find something or give me a hint, I would be glad.

Thanks!

Answer 1

Waiting for a soon-to-come solution by the OP, here is a phase diagram (made using free online tools) which shows the cycle enclosing the origin: