Is that possible for a system to have an equilibrium point at infinity?

Question

I know for some system of ODEs, there is no equilibrium at all, but I am wording that is valid to say a system has equilibrium at $\infty$? Consider the following system as an example. $$x' = \frac{y}{x}$$ $$y' = y - 4$$ So for this system, if we want to solve the equilibrium point, we need to solve the system of equation $$\frac{y}{x} = 0$$ $$y - 4 = 0$$ which give us $y = 4$ and $ 4/x = 0$. So the only way for the first equation to be zero is as $x$ approaching $\infty$ $$ \lim_{x \to \infty}\frac{4}{x} = 0$$ but can we say $(\infty, 4)$ is an equilibrium point?

Answer 1

No, you can't, because $\infty$ or $\pm\infty$ is not a point on the real line, it is a symbol to help with convergence or divergence statements.

What you could do is to compactify the real line to an interval with end-points (corona) or to a circle (one-point compactification). Then you could speak of convergence to these newly added points in the modified topology.

But in the traditional sense, your system does not have equilibrium points.