connection to an input one.
\begin{itemize}
-\item During initialization, the network if we seed it with $n$~bits denoted
+\item During initialization, the network is seeded with $n$~bits denoted
$\left(x^0_1,\dots,x^0_n\right)$ and an integer value $S^0$ that
belongs to $\llbracket1;n\rrbracket$.
\item At iteration~$t$, the last output vector
\left(F\left(1,\left(x_1,x_2,\dots,x_n\right)\right),\dots,
F\left(n,\left(x_1,x_2,\dots,x_n\right)\right)\right) \enspace .
\end{equation}
-Then $F=F_f$ and this recurrent neural network produces exactly the
-same output vectors, when feeding it with
+Then $F=F_f$. If this recurrent neural network is seeded with
$\left(x_1^0,\dots,x_n^0\right)$ and $S \in \llbracket 1;n
-\rrbracket^{\mathds{N}}$, than chaotic iterations $F_f$ with initial
+\rrbracket^{\mathds{N}}$, it produces exactly the
+same output vectors than the
+chaotic iterations of $F_f$ with initial
condition $\left(S,(x_1^0,\dots, x_n^0)\right) \in \llbracket 1;n
-\rrbracket^{\mathds{N}} \times \mathds{B}^n$. In the rest of this
+\rrbracket^{\mathds{N}} \times \mathds{B}^n$.
+Theoretically speakig, such iterations of $F_f$ are thus a formal model of
+these kind of recurrent neural networks. In the rest of this
paper, we will call such multilayer perceptrons CI-MLP($f$), which
stands for ``Chaotic Iterations based MultiLayer Perceptron''.
of $(\mathcal{X},G_{f_0})$ is infinite.
\end{theorem}
-We have explained how to construct truly chaotic neural networks, how
-to check whether a given MLP is chaotic or not, and how to study its
-topological behavior. The last thing to investigate, when comparing
-neural networks and Devaney's chaos, is to determine whether
-artificial neural networks are able to learn or predict some chaotic
-behaviors, as it is defined in the Devaney's formulation (when they
+\begin{figure}
+ \centering
+ \includegraphics[scale=0.625]{scheme}
+ \caption{Summary of addressed membership problems}
+ \label{Fig:scheme}
+\end{figure}
+
+The Figure~\ref{Fig:scheme} is a summary of the addressed problems.
+Section~\ref{S2} has explained how to construct a truly chaotic neural
+networks $A$ for instance.
+Section~\ref{S3} has shown how to check whether a given MLP
+$A$ or $C$ is chaotic or not in the sens of Devaney.
+%, and how to study its topological behavior.
+The last thing to investigate, when comparing
+neural networks and Devaney's chaos, is to determine whether
+an artificial neural network $A$ is able to learn or predict some chaotic
+behaviors of $B$, as it is defined in the Devaney's formulation (when they
are not specifically constructed for this purpose). This statement is
studied in the next section.
+
+
+
+
+
+
\section{Suitability of Artificial Neural Networks
for Predicting Chaotic Behaviors}
