# Not empty omega limit set

Dynamical system is generated by:

$x'=-x+f(x,y)$
$y'=-y+g(x,y)$

$f,g \in C^1$ and $f,g$ are bounded.

Prove that the omega limit set of p: $\omega(p) \neq \emptyset$ for all $p \in \mathbb{R}^2$.

I can say that if $f_{y}g_{x} \in (0,1)$ Jacobian \begin{pmatrix}-1&f_{y}\\g_{x}&-1\\ \end{pmatrix} has negative eigenvalues, and $\omega(p) = (0,0)$ for all $p \in \mathbb{R}^2$. But it's not much...

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If $f$ and $g$ are bounded this means that the solutions to the system are bounded. If the solutions are bounded then $\omega$-limit set is nonempty since in any bounded sequence a convergent subsequence can be found, and the limit of this subsequence belongs to $\omega$-limit set.
Do you think why solutions are bounded? Take $V(x,y)=x^2+y^2$ and find the derivative with respect to your vector field. You will find that for sufficiently large $x$ and $y$ it is strictly negative which implies that infinity repels the orbits, which means that the solutions are bounded. –  Artem Dec 14 '12 at 13:31