It is given that a point $(x^*,y^*)$ is said to be $\omega$- limit point ($\alpha$ -limit point) of the trajectory of a system if there exists a sequence of times $_ \to \infty (_ \to -\infty)$ such that $\lim_{\to\infty} ((_), (_)) = (^∗, ^∗) $.
1) As $t_n$ is a sequence of time, how can it tend to $-\infty$? Does the sign refer to the direction? If so, how can time have a direction, since it is a scalar quantity?
Example:
I found a system of equations
$\dfrac{dx}{dt}=-y+x(1-x^2-y^2)\\
\dfrac{dy}{dt}=x+y(1-x^2-y^2)$
Transforming this sytem to polar coordinates , I found $\dfrac{dr}{dt}=r(1-r^2)$ and $\dfrac{d\theta}{dt}=1$. So clearly if $0<r<1$, then $r$ increases spirally to the limit cycle $r=1$ in anticlockwise sense from inside. If $r>1$, $r$ decreases spirally from outside in aniticlockwise sense to the limit cycle $r=1$.
In this scenario, which limit point is the origin?
EDIT: After having a conversation with Hans Lundmark, I thought it might be good to post a portrait about my question as I am surely lacking of some basic conceptions about positive and negative time directions. So in this picture, which trajectories are for $t\to-\infty$ and which are for $t\to\infty$? And what limit point is the origin?
I have understood that $r=1$ is the limit cycle and a stable one. But I am having trouble to understand how the $\alpha$ and $\omega$ limit point works.
Another edit:
Adding two paths along which solutions approach to the origin. The left side as $t\to -\infty$ and the right one as $t\to\infty$.
