X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/chaos1.git/blobdiff_plain/a1b71af0bb07232355e939a1e1a6a7aedd40b2d2..a1c7d633b750e7417e0e331d9f2df61936c76de7:/main.tex?ds=inline

diff --git a/main.tex b/main.tex
index b72eb6d..8e7f34b 100644
--- a/main.tex
+++ b/main.tex
@@ -58,6 +58,11 @@ IUT de Belfort-Montb\'eliard, BP 527, \\
 \date{\today}
 
 \newcommand{\CG}[1]{\begin{color}{red}\textit{#1}\end{color}}
+\newcommand{\JFC}[1]{\begin{color}{blue}\textit{#1}\end{color}}
+
+
+
+
 
 \begin{abstract}
 %% Text of abstract
@@ -134,7 +139,9 @@ which  is usually  assessed through  the computation  of  the Lyapunov
 exponent.  An alternative approach  is to consider a well-known neural
 network architecture: the  MultiLayer Perceptron (MLP). These networks
 are  suitable to model  nonlinear relationships  between data,  due to
-their universal approximator capacity. Thus, this kind of networks can
+their universal approximator capacity.
+\JFC{Michel, peux-tu donner une ref la dessus}
+Thus, this kind of networks can
 be trained to model a physical  phenomenon known to be chaotic such as
 Chua's circuit  \cite{dalkiran10}.  Sometimes, a  neural network which
 is build by combining  transfer functions and initial conditions that are both
@@ -508,7 +515,9 @@ Fig.~\ref{Fig:perceptron}).
 
 The behavior of the neural network is such that when the initial state
 is  $x^0~\in~\mathds{B}^n$ and  a  sequence $(S^t)^{t  \in \Nats}$  is
-given  as outside  input, then  the sequence  of  successive published
+given  as outside  input,
+\JFC{en dire davantage sur l'outside world}
+ then  the sequence  of  successive published
 output vectors $\left(x^t\right)^{t \in \mathds{N}^{\ast}}$ is exactly
 the  one produced  by  the chaotic  iterations  formally described  in
 Eq.~(\ref{eq:CIs}).  It  means that  mathematically if we  use similar
@@ -539,8 +548,8 @@ without any  convincing mathematical proof. We propose  an approach to
 overcome  this  drawback  for  a  particular  category  of  multilayer
 perceptrons defined below, and for the Devaney's formulation of chaos.
 In  spite of  this restriction,  we think  that this  approach  can be
-extended to  a large variety  of neural networks.  We plan to  study a
-generalization of this approach in a future work.
+extended to  a large variety  of neural networks. 
+
 
 We consider a multilayer perceptron  of the following form: inputs
 are $n$ binary digits and  one integer value, while outputs are  $n$
@@ -556,6 +565,7 @@ connection to an input one.
   compute the new output one $\left(x^{t+1}_1,\dots,x^{t+1}_n\right)$.
   While  the remaining  input receives  a new  integer value  $S^t \in
   \llbracket1;n\rrbracket$, which is provided by the outside world.
+\JFC{en dire davantage sur l'outside world}
 \end{itemize}
 
 The topological  behavior of these  particular neural networks  can be
@@ -563,11 +573,15 @@ proven to be chaotic through the following process. Firstly, we denote
 by  $F:  \llbracket  1;n  \rrbracket \times  \mathds{B}^n  \rightarrow
 \mathds{B}^n$     the     function     that     maps     the     value
 $\left(s,\left(x_1,\dots,x_n\right)\right)    \in    \llbracket    1;n
-\rrbracket      \times      \mathds{B}^n$      into     the      value
+\rrbracket      \times      \mathds{B}^n$   
+\JFC{ici, cela devait etre $S^t$ et pas $s$, nn ?}
+   into     the      value
 $\left(y_1,\dots,y_n\right)       \in       \mathds{B}^n$,       where
 $\left(y_1,\dots,y_n\right)$  is the  response of  the  neural network
 after    the    initialization     of    its    input    layer    with
-$\left(s,\left(x_1,\dots, x_n\right)\right)$.  Secondly, we define $f:
+$\left(s,\left(x_1,\dots, x_n\right)\right)$. 
+\JFC{ici, cela devait etre $S^t$ et pas $s$, nn ?}
+Secondly, we define $f:
 \mathds{B}^n       \rightarrow      \mathds{B}^n$       such      that
 $f\left(x_1,x_2,\dots,x_n\right)$ is equal to
 \begin{equation}
@@ -614,9 +628,12 @@ if  and only  if,  for any  pair  of disjoint  open  sets $U$,$V  \neq
 \emptyset$, we can find some $n_0  \in \mathds{N}$ such that  for any $n$, 
 $n\geq n_0$, we have $f^n(U) \cap V \neq \emptyset$.
 \end{definition}
+\JFC{Donner un sens Ã  ces definitions}
+
 
-As  proven in Ref.~\cite{gfb10:ip},  chaotic iterations  are expansive
-and  topologically mixing when  $f$ is  the vectorial  negation $f_0$.
+It has been proven in Ref.~\cite{gfb10:ip},  that chaotic iterations
+are expansive and  topologically mixing when  $f$ is  the
+vectorial  negation $f_0$.
 Consequently,  these  properties are  inherited  by the  CI-MLP($f_0$)
 recurrent neural network previously presented, which induce a greater
 unpredictability.  Any  difference on the  initial value of  the input
@@ -624,8 +641,10 @@ layer is  in particular  magnified up to  be equal to  the expansivity
 constant.
 
 Let us then focus on the consequences for  a neural  network to  be chaotic
-according  to  Devaney's definition.  First  of  all, the  topological
-transitivity property implies indecomposability.
+according  to  Devaney's definition.  Intuitively, the  topological
+transitivity property implies indecomposability, which is formally defined 
+as follows:
+
 
 \begin{definition} \label{def10}
 A   dynamical   system   $\left(   \mathcal{X},  f\right)$   is   
@@ -784,7 +803,9 @@ such functions into a model amenable to be learned by an ANN.
   
 This  section  presents how  (not)  chaotic  iterations  of $G_f$  are
 translated  into  another  model  more  suited  to  artificial  neural
-networks.  Formally, input and output vectors are pairs~$((S^t)^{t \in
+networks.  
+\JFC{dÃ©tailler le more suited}
+Formally, input and output vectors are pairs~$((S^t)^{t \in
 \Nats},x)$ and $\left(\sigma((S^t)^{t \in \Nats}),F_{f}(S^0,x)\right)$
 as defined in~Eq.~(\ref{eq:Gf}).
 
@@ -965,7 +986,7 @@ configuration is always expressed as  a natural number, whereas in the
 first one  the number  of inputs follows  the increase of  the boolean
 vectors coding configurations. In this latter case, the coding gives a
 finer information on configuration evolution.
-
+\JFC{Je n'ai pas compris le paragraphe precedent. Devrait Ãªtre repris}
 \begin{table}[b]
 \caption{Prediction success rates for configurations expressed with Gray code}
 \label{tab2}
