omega limit set

Let $\mathrm{\Phi }(t,x)$ be the flow of the differential equation $x^{\prime }=f(x)$, where $f\in C^k(M,{ℝ}^n)$, with $k\ge 1$ and $M$ an open subset of ${ℝ}^n$.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 $t_n\to \mathrm{\infty }$ with $\mathrm{\Phi }(t_n,x)=y$.

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

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