# Existence of invariant set in dynamical system generated by ODE

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

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

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.

## 2 Answers

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

• Thank you. Now I have to solve $x+\sin{(xy+2)}-7=0$, $-y+\arctan{(x^2+y^3-6)}=0$ – MarkT Dec 14 '12 at 9:54
• Don't try for a "closed form" solution, but show there is a solution in, say, the rectangle $6 \le x \le 8$, $-\pi/2 \le y \le \pi/2$. – Robert Israel Dec 14 '12 at 19:56

Hint: Not only that. If you find a bounded trajectory then its closure is also an invariant set. (It can only be made the trajectory and fixed points). I suggest you try to run it through a numerical program so you get the feel of the system.