# local and global exponential stability

I read the book "A Linear Systems Primer" [1] and confused about the differences between the definitions of local and global exponential stability. They are defined as:

$$\dot{x}=f(x) \tag{4.8}$$ Definition 4.8 (local exponential stability). The equilibrium $x = 0$ of (4.8) is exponentially stable if there exists an $\alpha > 0$, and for every $\epsilon > 0$, there exists a $\delta(\epsilon)>0$, such that $$||φ(t, x_0 )|| ≤ \epsilon e^{-\alpha t} ~for ~all~ t ≥ 0$$ whenever $||x_0||<\delta(\epsilon)$

Definition 4.11 (global exponential stability). The equilibrium $x = 0$ of (4.8) is exponentially stable in the large if there exists $\alpha > 0$ and for any β > 0, there exists $k(β) > 0$ such that $$||φ(t, x_0 )|| ≤ k(β)|| x_0|| e^{−\alpha t} ~for ~all ~t ≥ 0$$ whenever $||x_0|| < β$.

The only major difference I could see is that in the definition of the global exponential stability, there is a $||x_0||$ in the equation. But I just do not understand how definition 4.11 can guarantee that the exponential stability is global since it also restricts the domain of $||x_0||$ by $||x_0|| < β$ just like that in the definition of the local exponential stability?

References:
[1] Antsaklis, Panos J., and Anthony N. Michel. A linear systems primer. Vol. 1. Boston: Birkhäuser, 2007.

The important thing you're missing is the restrictions on $x_0$. In the first case you need $\| x_0 \|<\delta(\epsilon)$, and $\delta(\epsilon)$ might be small. More importantly, it might remain bounded as $\epsilon \to +\infty$, so that some $x_0$ don't satisfy the inequality for any choice of $\epsilon$. By contrast, in the second case you need $\| x_0 \|<\beta$ but $\beta$ is arbitrary.