X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/chaos1.git/blobdiff_plain/999fadfcb0b5ed18bd495364a07850278fade07d..10496d652add2d6fa789a21237cb3cdeb972d257:/main.tex?ds=sidebyside

diff --git a/main.tex b/main.tex
index cee8840..78454a7 100644
--- a/main.tex
+++ b/main.tex
@@ -75,7 +75,7 @@ Devaney  and  a  particular  class of  neural
 networks. On the one hand we show  how to build such a network, on the
 other  hand we provide  a method  to check  if a  neural network  is a
 chaotic one.  Finally, the ability of classical feedforward multilayer
-perceptrons to  learn sets of  data obtained from a  chaotic dynamical
+perceptrons to  learn sets of  data obtained from a  dynamical
 system is regarded.  Various  Boolean functions are iterated on finite
 states. Iterations of some of them  are proven to  be chaotic 
  as it is  defined by
@@ -97,7 +97,7 @@ appealing   properties  of   deterministic   chaos  (unpredictability,
 sensitivity,  and so  on). However,  such networks  are  often claimed
 chaotic without  any rigorous mathematical proof.   Therefore, in this
 work  a theoretical  framework based  on the  Devaney's  definition of
-chaos  is introduced.   Starting with  a relationship  between chaotic
+chaos  is introduced.   Starting with  a relationship  between discrete
 iterations  and  Devaney's chaos,  we  firstly  show  how to  build  a
 recurrent  neural networks  that is  equivalent to  a chaotic  map and
 secondly  a way  to check  whether  an already  available network,  is
@@ -112,9 +112,6 @@ is far more difficult than non chaotic behaviors.
 \section{Introduction}
 \label{S1}
 
-REVOIR TOUT L'INTRO et l'ABSTRACT en fonction d'asynchrone, chaotic
-
-
 Several research  works have proposed  or run chaotic  neural networks
 these last  years.  The complex dynamics  of such a  networks leads to
 various       potential      application       areas:      associative
@@ -156,13 +153,13 @@ using  a theoretical  framework  based on  the  Devaney's  definition of  chaos
 qualitative and quantitative tools to evaluate the complex behavior of
 a  dynamical  system:  ergodicity,   expansivity,  and  so  on.   More
 precisely, in  this paper,  which is an  extension of a  previous work
-\cite{bgs11:ip},   we  establish   the  equivalence  between asynchronous
+\cite{bgs11:ip},   we  establish   the  equivalence  between chaotic
 iterations and  a class of globally  recurrent MLP.
 The investigation the  converse problem is the second contribution: 
 we indeed study the  ability for
 classical MultiLayer Perceptrons  to learn a particular family of
-discrete  chaotic  dynamical  systems.   This family,  called  chaotic
-iterations, is defined by a  Boolean vector, an update function, and a
+discrete  chaotic  dynamical  systems.   This family  
+is defined by a  Boolean vector, an update function, and a
 sequence giving  which component  to update at  each iteration.   It has
 been  previously established  that such  dynamical systems is 
 chaotically iterated (as it is defined by Devaney) when the chosen function has
@@ -574,7 +571,10 @@ $f\left(x_1,x_2,\dots,x_n\right)$ 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 $F=F_f$. If this recurrent  neural network is seeded with 
+Thus, for any $j$, $1 \le j \le n$, we have 
+$f_j\left(x_1,x_2,\dots,x_n\right) = 
+F\left(j,\left(x_1,x_2,\dots,x_n\right)\right)$.
+If this recurrent  neural network is seeded with 
 $\left(x_1^0,\dots,x_n^0\right)$    and   $S   \in    \llbracket   1;n
 \rrbracket^{\mathds{N}}$, it produces  exactly the
 same      output      vectors  than the 
