Existence of invariant set in dynamical system generated by ODE

Question

Is there any nonempty, compact and invariant set in dynamical system generated by this system of equations?

$x'=x+\sin{(xy+2)}-7$
$y'=-y+\arctan{(x^2+y^3-6)}$

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My idea is to use this fact: Not empty omega limit set - because here we have also bounded functions and omega limit set is invariant. But it's hard to say anything about compactness.

Thanks a lot for your help.

Edit: Of course is not true that omega limit set is always invariant - it is only when it lays in trajectory. That makes problem harder and probably it's not a good way.

Answer 1

Hint: a fixed point is a compact invariant set. If that fixed point has a stable manifold, you can include some of that too.