Let $\Phi(t,x)$ be the flow of the differential equation $x'=f(x)$ , 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.