Instability: The equilibrium point x = 0 of (2) is unstable if it is not stable.

The above condition is illustrated in Figure 3.

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

Figure 3: Illustration of unstable equilibrium in two dimension

Uniform Stability: The equilibrium point x = 0 of (2) is uniformly stable if, for each ∈ > 0, there exists a $\delta = \delta(\epsilon) >0$ , independent of k₀, such that $\displaystyle \vert\vert\boldsymbol{x}(k_0)\vert\vert < \delta \;\; \Rightarrow... ...\vert\vert\boldsymbol{x}(k)\vert\vert < \epsilon, \; \forall \; k\ge k_0 \ge 0$
Uniform Asymptotic Stability: The equilibrium point x = 0 of (2) is uniformly asymptotically stable if it is uniformly stable and there is a positive constant c, independent of k₀, such that for all $\vert\vert\boldsymbol{x}(k_0)\vert\vert < c$ , x(k) → 0 as k → ∞ uniformly in k₀.

Global Uniform Asymptotic Stability: The equilibrium point x = 0 of (2) is globally uniformly asymptotically stable if it is uniformly asymptotically stable for such a δ when δ(∈) can be chosen to satisfy the following condition $\displaystyle \lim_{\epsilon \rightarrow \infty} \delta(\epsilon) = \infty$
Exponential Stability: The equilibrium point x = 0 of (2) is exponentially stable if there exist positive constants c, γ and λ such that $\displaystyle \vert\vert\boldsymbol{x}(k)\vert\vert \le \gamma \vert\vert\bolds... ...{-\lambda(k-k_0)}, \;\; \forall \; \vert\vert\boldsymbol{x}(k_0)\vert\vert < c$
Global Exponential Stability: The equilibrium point x = 0 of (2) is globally exponentially stable if it is exponentially stable for any initial state x(k₀) .