Consider a discrete-time system $$x_{t+1} = f(x_t),$$ where $t \in \mathbb{N}_0$ and $x_t \in \mathbb{R}^n$. This system has only one equilibrium $x^* = 0$.
Is there any example for $f: \mathbb{R}^n \to \mathbb{R}^n$ such that the system is asymptotically stable but not uniformly stable?
Thanks!
PS: We need two definitions to define asymptotic stability and uniform asymptotic stability.
Definition 1 (Lyapunov Stable and Uniformly Stable) The equilibrium $x^* = 0$ is (Lyapunov) stable at $t = t_0$ if for any $\epsilon > 0$, there exists a $\delta(t_0,\epsilon) > 0$ such that $$\|x(t_0)\| \leq \delta(t_0,\epsilon) \rightarrow \|x(t)\| < \epsilon, \forall t \geq t_0.$$ Furthermore, if $\delta(t_0,\epsilon)$ is not dependent on $t_0$, then the equilibrium $x^* = 0$ is uniformly stable.
Definition 2 (Asymptotic Stability and Uniform Asymptotic Stability) The equilibrium $x^* = 0$ is asymptotically stable if:
$x^* = 0$ is stable;
There exists $\delta(t_0)$ such that $$\|x(t_0)\| \leq \delta(t_0) \rightarrow \lim_{t\to\infty}\|x(t)\| = 0.$$ Furthermore, if $\delta(t_0)$ is not dependent on $t_0$, then the equilibrium $x^* = 0$ is uniformly asymptotically stable.