Editing Lyapunov stability (section)

==Definition for discrete-time systems==
The definition for [[discrete-time]] systems is almost identical to that for continuous-time systems. The definition below provides this, using an alternate language commonly used in more mathematical texts.

Let (''X'', ''d'') be a [[metric space]] and ''f'' : ''X'' → ''X'' a [[continuous function]]. A point ''x'' in ''X'' is said to be '''Lyapunov stable''', if,

:<math>\forall \epsilon>0 \  \exists \delta>0 \  \forall y\in X \ \left [d(x,y)<\delta \Rightarrow \forall n \in \mathbf{N} \  d\left (f^n(x),f^n(y) \right )<\epsilon \right ].</math>

We say that ''x'' is '''asymptotically stable''' if it belongs to the interior of its [[stable manifold|stable set]], ''i.e.'' if,

:<math> \exists \delta>0 \left [ d(x,y)<\delta \Rightarrow \lim_{n\to\infty} d \left(f^n(x),f^n(y) \right)=0\right ].</math>