From Wiggins' book, "Let $\cal{M}$ be a trapping region. Then $A=\cap_{t>0}\phi(t,\cal{M})$ is called an attracting set".
Then he gives an example:
$\dot{x}=x-x^3$
$\dot{y}=-y$,
and claims that the closed interval $[-1,1]$ is an attracting set with some appropriate trapping region (namely an ellipse $\cal{M}$ that surrounds all three critical points of the system. But I cannot relate this to the definition of an attracting set; I mean, if $t$ is arbitrarily large, then shouldn't all points be either at $(-1,0),(0,0)$ or $(1,0)$? How is the containment $[-1,1] \subset \cap_{t>0}\phi(t,\cal{M})$ seen? I have attached a diagram that he has provided below.
$(0.5,0)$ is in $\phi(t,\mathcal{M})$ for all $t$, because we can start at $(\epsilon_t,0)$ for arbitrarily small $\epsilon_t>0$, which moves us very very slowly toward $(1,0)$, arriving at $(0.5,0)$ exactly at time $t$.