omega limit set

Let $\Phi(t,x)$ be the flow of the differential equation $x^{\prime}=f(x)$, where $f\in C^{k}(M,\mathbb{R}^{n})$, with $k\geq 1$ and $M$ an open subset of $\mathbb{R}^{n}$. Consider $x\in M$.

The omega limit set of $x$, denoted $\omega(x)$, is the set of points $y\in M$ such that there exists a sequence $t_{n}\to\infty$ with $\Phi(t_{n},x)=y$.

Similarly, the alpha limit set of $x$, denoted $\alpha(x)$, is the set of points $y\in M$ such that there exists a sequence $t_{n}\to-\infty$ with $\Phi(t_{n},x)=y$.

Note that the definition is the same for more general dynamical systems.

Title omega limit set OmegaLimitSet 2013-03-22 13:18:42 2013-03-22 13:18:42 mathcam (2727) mathcam (2727) 5 mathcam (2727) Definition msc 37B99 msc 34C05 $\omega$-limit set $\alpha$-limit set LimitCycle alpha limit set