The above condition is illustrated in Figure 1.

$\begin{figure}\centering \input{stab.pstex_t}\end{figure}$

Figure 1: Illustration of stable equilibrium in the sense of Lyapunov in two dimension

Asymptotic Stability: The equilibrium point x = 0 of (2) is asymptotically stable if it is stable and there is a positive constant such that x(k) → 0 as $\displaystyle \; k \rightarrow \infty, \;\;\; \forall \; \vert\vert\boldsymbol{x}(k_0)\vert\vert < c$ The above condition is illustrated in Figure 2.

$\begin{figure}\centering \input{astab.pstex_t}\end{figure}$

Figure 2: Illustration of asymptotically stable equilibrium in two dimension