X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/chaos1.git/blobdiff_plain/ab09441f8e3a9e5926d268f1d06dc420abad97c3..3e9929bb13d395a475ddfd4045cb00b3be5caa7f:/main.tex

diff --git a/main.tex b/main.tex
index 717ae3e..c134123 100644
--- a/main.tex
+++ b/main.tex
@@ -503,7 +503,7 @@ bits.   Moreover, each  binary  output is  connected  with a  feedback
 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
@@ -529,12 +529,15 @@ $f\left(x_1,x_2,\dots,x_n\right)$ is equal to
 \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''.
 
@@ -652,15 +655,32 @@ $\left( \mathcal{X},d\right)$  is compact and  the topological entropy
 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}
 
