The system of ODE is
$$\begin{cases} \dot{x}=-y(1-x^{2}) \\ \dot{y}=x+y(1-x^{2}) \end{cases} \tag{$\ast$}$$
Claim: $\forall p\in\left\{ (x,y)\in\mathbb{R}^{2} : |x|<1,\ x^{2}+y^{2}>0\right\} $, $$\omega(p)=\left\{ (-1,y) : y\in\mathbb{R}\right\} \cup\left\{ (1,y) : y\in\mathbb{R}\right\}.$$
I understand that the following are solutions to the system $(\ast)$:
$$x=1 \text{ and } y=t$$
$$x=-1 \text{ and } y=-t$$
When we talk about omega limit set, aren't we supposed to make $t$ go to infinity? Why do we have $(-1,y)$? As $t\rightarrow\infty$, shouldn't $y\rightarrow \infty$?