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:
\begin{equation} \dot{x}=f(x) \tag{4.8} \end{equation} 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 \begin{equation} ||φ(t, x_0 )|| ≤ \epsilon e^{-\alpha t} ~for ~all~ t ≥ 0 \end{equation} 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 \begin{equation} ||φ(t, x_0 )|| ≤ k(β)|| x_0|| e^{−\alpha t} ~for ~all ~t ≥ 0 \end{equation} 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.
Thus the first one says essentially "if you start close enough to zero, you will decay to zero exponentially fast". By contrast, the second one says "no matter where you start, you will decay to zero exponentially fast; at worst, the speed may slow down if you start further away from zero".