dynamical system has then be realized.
This study is summarized in the next section.
-\subsection{Topological Properties of Chaotic Iterations}
+\subsection{A Topology for Chaotic Iterations}
The topological space on which chaotic iterations are defined has
firstly been investigated, leading to the following result~\cite{gb11:bc,GuyeuxThese10}:
$\forall k \in \mathds{N}, Per_{2k+1}(G_{f_0}) = \varnothing, card\left(Per_{2k+2}(G_{f_0})\right)>0$.
\end{proposition}
-So $\Rightarrow G_{f_0}$ does not present the existence of points of any period referred as chaos in the article of Li and Yorke~\cite{Li75}.
+So $ G_{f_0}$ does not present the existence of points of any period referred as chaos in the article of Li and Yorke~\cite{Li75}.
However~\cite{GuyeuxThese10}:
\begin{itemize}
\item This kind of disorder can be stated on $\mathcal{X}^G = \mathcal{P}\left(\llbracket 1,\mathsf{N}\rrbracket\right)^\mathds{N}\times \mathds{B}^\mathsf{N}$.
Additionally, this existence of points of any period has been rejected
by the community to the benefit of more recent notions of chaos,
like those developed these last decades by Devaney~\cite{Devaney}, Knudsen~\cite{Knudsen94}, etc.
+These approaches are recalled in the next section.
+
+\section{The Mathematical Theory of Chaos}
+
+We will present in this section various understanding of a chaotic behavior for a discrete
+dynamical system.
+
+\subsection{Approaches Similar to Devaney}
In these approaches, three ingredients are required for unpredictability.
Firstly, the system must be intrinsically complicated, undecomposable: it cannot be simplified into two
\end{itemize}
-Concerning the ingredient of sensibility, it can be formulated as follows.
-
-%\frame{
-%\frametitle{Stabilité et expansivité}
-% \begin{block}{Définitions de la sensibilité}
-% \begin{itemize}
-% \item $(\mathcal{X},f)$ est \emph{instable} si tous ses points le sont: $\forall x \in \mathcal{X},$ $\exists \varepsilon >0,$ $\forall \delta > 0,$ $\exists y \in \mathcal{X},$ $\exists n \in \mathbb{N},$ $d(x,y)<\delta$ et $d(f^{(n)}(x),f^{(n)}(y)) \geqslant \varepsilon$
-% \item $(\mathcal{X},f)$ est \emph{expansif} si
-%$\exists \varepsilon >0,$ $\forall x \neq y,$ $\exists n \in \mathbb{N},$ $d(f^{(n)}(x),f^{(n)}(y)) \geqslant \varepsilon$
-% \end{itemize}
-% \end{block}
-%}
+Concerning the ingredient of sensibility, it can be reformulated as follows.
+\begin{itemize}
+ \item $(\mathcal{X},f)$ is \emph{unstable} is all its points are unstable: $\forall x \in \mathcal{X},$ $\exists \varepsilon >0,$ $\forall \delta > 0,$ $\exists y \in \mathcal{X},$ $\exists n \in \mathbb{N},$ $d(x,y)<\delta$ and $d(f^{(n)}(x),f^{(n)}(y)) \geqslant \varepsilon$.
+ \item $(\mathcal{X},f)$ is \emph{expansive} is $\exists \varepsilon >0,$ $\forall x \neq y,$ $\exists n \in \mathbb{N},$ $d(f^{(n)}(x),f^{(n)}(y)) \geqslant \varepsilon$
+\end{itemize}
-%%\frame{
-%% \frametitle{Des systèmes imprévisibles}
-%% \begin{block}{Définitions des systèmes dynamiques désordonnés}
-%% \begin{itemize}
-%% \item \emph{Devaney:} $(\mathcal{X},f)$ est sensible aux conditions initiales, régulier et transitif
-%% \item \emph{Wiggins:} $(\mathcal{X},f)$ est transitif et sensible aux conditions initiales
-%% \item \emph{Knudsen:} $(\mathcal{X},f)$ a une orbite dense et s'il est sensible aux conditions initiales
-%% \item \emph{expansif:} $(\mathcal{X},f)$ est transitif, régulier et expansif
-%% \end{itemize}
-%% \end{block}
-%%}
+These variety of definitions lead to various notions of chaos. For instance,
+a dynamical system is chaotic according to Wiggins if it is transitive and
+sensible to the initial conditions. It is said chaotic according to Knudsen
+if it has a dense orbit while being sensible. Finally, we speak about
+expansive chaos when the properties of transitivity, regularity, and expansivity
+are satisfied.
-%\subsection*{Autres approches}
+\subsection{Other approaches}
%\frame{
