derniere modifs

[slides_and.git] / devaney.tex

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-\begin{block}{Definition: Chaotic function [4]$^4$}
-Let $(\mathcal{X}; d)$ be a metric space.
-A function $f: \mathcal{X} \rightarrow \mathcal{X}$ is chaotic on $\mathcal{X}$ if:
+\begin{block}{Définition d'une fonction chaotique selon Devaney}
+Soit $(\mathcal{X}; d)$ un espace métrique.
+Une fonction $f: \mathcal{X} \rightarrow \mathcal{X}$ est chaotique sur $\mathcal{X}$ si:

 \begin{enumerate}
 \begin{enumerate}

-\item $f$: topologically transitive (\textit{i.e.}, indecomposability of the system)\\
-(for any pair of open sets $U,V \subset \mathcal{X}$, $\exists k > 0 . 
+\item $f$: topologiquement transitive (\textit{i.e.}, système indécomposable)\\
+(pour chaque paire d'ensembles ouverts $U,V \subset \mathcal{X}$, $\exists k > 0 . 

 f^k (U) \cap V \neq \emptyset$)
 \\
 f^k (U) \cap V \neq \emptyset$)
 \\

-\onslide<2>{\alert<2>{Addressed property: preimage resistance}}.
-\item  $f$ is regular (\textit{i.e.}, fundamentally different points coexist)\\
-(the set of periodic points is dense in $\mathcal{X}$).
-\item  $f$: sensitive dependent on initial conditions (SDIC)\\
+\item  $f$ est régulière (l'ensemble des points périodiques est dense dans $\mathcal{X}$).
+\item  $f$: sensible aux conditions initiales \\

 ($
 \exists \delta  > 0 .  \forall x \in  \mathcal{X}
 \textrm{ and } 
 ($
 \exists \delta  > 0 .  \forall x \in  \mathcal{X}
 \textrm{ and } 

-\forall V \textrm{ neighborhood of $x$}.
-\exists y \in V \textrm{ and } 
+\forall V \textrm{ voisin de $x$}.
+\exists y \in V \textrm{ et } 

 \exists n \ge 0 . 
 d(f^n(x); f^n(y))> \delta
 $)\\
 \exists n \ge 0 . 
 d(f^n(x); f^n(y))> \delta
 $)\\

-\onslide<2>{\alert<2>{Addressed properties: avalanche effect}}.

 \end{enumerate}
 \end{block}
 \end{enumerate}
 \end{block}

-\footnote{\bibentry{devaney}}
\ No newline at end of file