Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Dynamical system is generated by:


$f,g \in C^1$ 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...

share|improve this question

2 Answers 2

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.

share|improve this answer

Thank you. But I'm still thinking about boundedness of the solutions (writing general form of solution gives me nothing).

share|improve this answer
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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.