X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/chaos1.git/blobdiff_plain/8f2ca538d4cca28cafc98b78ce0c0e697085f3ee..999fadfcb0b5ed18bd495364a07850278fade07d:/main.tex?ds=sidebyside

diff --git a/main.tex b/main.tex
index 545524f..cee8840 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
@@ -107,6 +112,9 @@ 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
@@ -131,7 +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
@@ -146,7 +156,7 @@ 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   chaotic
+\cite{bgs11:ip},   we  establish   the  equivalence  between asynchronous
 iterations and  a class of globally  recurrent MLP.
 The investigation the  converse problem is the second contribution: 
 we indeed study the  ability for
@@ -193,7 +203,10 @@ iterations and Devaney's chaos is  finally presented at the end of this
 section.
 
 In what follows and for  any function $f$, $f^n$ means the composition
-$f \circ f \circ \hdots \circ f$ ($n$ times).
+$f \circ f \circ \hdots \circ f$ ($n$ times) and an \emph{iteration}
+of a \emph{dynamical system} is the step that consists in
+updating the global state $x^t$ at time $t$ with respect to a function $f$ 
+s.t. $x^{t+1} = f(x^t)$.    
 
 \subsection{Devaney's chaotic dynamical systems}
 
@@ -229,59 +242,45 @@ point dependence on initial conditions.
 
 Topological  transitivity   is  checked  when,  for   any  point,  any
 neighborhood of its future evolution eventually overlap with any other
-given region. More precisely,
-
-\begin{definition} \label{def2}
-A continuous function $f$  on a topological space $(\mathcal{X},\tau)$
-is defined  to be  {\bf topologically transitive}  if for any  pair of
-open  sets $U$,  $V  \in  \mathcal{X}$ there  exists  
-$k \in
-\mathds{N}^{\ast}$
- such  that
-$f^k(U) \cap V \neq \emptyset$.
-\end{definition}
-
-This property  implies that a  dynamical system cannot be  broken into
-simpler  subsystems.  
+given region.  This property  implies that a  dynamical system
+cannot be  broken into simpler  subsystems.  
 Intuitively, its complexity does not allow any simplification.
 On  the contrary,  a dense set  of periodic points  is an
 element of regularity that a chaotic dynamical system has to exhibit.
 
-\begin{definition} \label{def3}
-A point $x$ is called a  {\bf periodic point} for $f$ of period~$n \in
-\mathds{N}^{\ast}$ if $f^{n}(x)=x$.
-\end{definition}
-
-\begin{definition} \label{def4}
-$f$ is said to be {\bf  regular} on $(\mathcal{X},\tau)$ if the set of
-  periodic points  for $f$ is  dense in $\mathcal{X}$ ( for any $x \in
+However, chaos need some regularity to ``counteracts'' 
+the  effects of  transitivity.
+%\begin{definition} \label{def3}
+We recall that a  point $x$ is  {\emph{periodic point}} for $f$ of 
+period~$n \in \mathds{N}^{\ast}$ if $f^{n}(x)=x$.
+%\end{definition}
+Then, the map 
+%\begin{definition} \label{def4}
+$f$ is {\emph{ regular}} on $(\mathcal{X},\tau)$ if the set of
+  periodic points  for $f$ is  dense in $\mathcal{X}$ (for any $x \in
   \mathcal{X}$, we can find at least  one periodic point in any of its
   neighborhood).
-\end{definition}
-
-This  regularity ``counteracts'' the  effects of  transitivity.  Thus,
-due to these two properties, two points close to each other can behave
-in a completely different  manner, leading to unpredictability for the
-whole system. Then,
-
-\begin{definition} \label{sensitivity}
-$f$  has {\bf  sensitive dependence  on initial  conditions}  if there
-  exists $\delta  >0$ such  that, for any  $x\in \mathcal{X}$  and any
-  neighborhood $V$ of $x$, there exist  $y\in V$ and $n > 0$ such that
-  $d\left(f^{n}(x), f^{n}(y)\right) >\delta  $. The value $\delta$ is called the
-  {\bf constant of sensitivity} of $f$.
-\end{definition}
-
-Finally,
-
-\begin{definition} \label{def5}
-The dynamical system that iterates $f$ is {\bf chaotic according to Devaney}  
-on $(\mathcal{X},\tau)$ if $f$  is   regular,  topologically  transitive,   
+%\end{definition}
+  Thus,
+  due to these two properties, two points close to each other can behave
+  in a completely different  manner, leading to unpredictability for the
+  whole system.
+
+Let we recall that $f$  
+has {\emph{  sensitive dependence  on initial  conditions}}  if there
+exists $\delta  >0$ such  that, for any  $x\in \mathcal{X}$  and any
+neighborhood $V$ of $x$, there exist  $y\in V$ and $n > 0$ such that
+$d\left(f^{n}(x), f^{n}(y)\right) >\delta  $. The value $\delta$ is called the
+  {\emph{constant of sensitivity}} of $f$.
+
+Finally, The dynamical system that iterates $f$ is {\emph{ chaotic according to Devaney}} on $(\mathcal{X},\tau)$ if $f$  is   regular,  topologically  transitive,   
 and  has  sensitive dependence to its initial conditions.
-\end{definition}
+In what follows, iterations are said to be \emph{chaotic according Devaney}
+when corresponding dynamical system  is chaotic according Devaney.
+
 
-Let us notice that for a  metric space the last condition follows from
-the two first ones~\cite{Banks92}.
+%Let us notice that for a  metric space the last condition follows from
+%the two first ones~\cite{Banks92}.
 
 \subsection{Asynchronous Iterations}
 
@@ -300,8 +299,8 @@ of integers:  $\{a, a+1,  \hdots, b\}$, where $a~<~b$.
 In that section,  a  system 
 under   consideration    iteratively   modifies   a    collection   of
 $n$~components.   Each component  $i \in  \llbracket 1;  n \rrbracket$
-takes  its  value  $x_i$  among the  domain  $\Bool=\{0,1\}$.   A~{\it
-  configuration} of the system at discrete time $t$  is the vector
+takes  its  value  $x_i$  among the  domain  $\Bool=\{0,1\}$.   
+A \emph{configuration} of the system at discrete time $t$  is the vector
 %\begin{equation*}   
 $x^{t}=(x_1^{t},\ldots,x_{n}^{t}) \in \Bool^n$.
 %\end{equation*}
@@ -318,15 +317,28 @@ $N(i,x)=(x_1,\ldots,\overline{x_i},\ldots,x_n)$  is  the configuration
 obtained by switching the $i-$th component of $x$ ($\overline{x_i}$ is
 indeed  the negation  of $x_i$).   Intuitively, $x$  and  $N(i,x)$ are
 neighbors.   The discrete  iterations of  $f$ are  represented  by the
-oriented  {\it graph  of iterations}  $\Gamma(f)$.  In  such  a graph,
+oriented  \emph{graph  of iterations}  $\Gamma(f)$.  In  such  a graph,
 vertices are configurations  of $\Bool^n$ and there is  an arc labeled
 $i$ from $x$ to $N(i,x)$ if and only if  $f_i(x)$ is $N(i,x)$.
 
-In the  sequel, the  {\it strategy} $S=(S^{t})^{t  \in \Nats}$  is the
+In the  sequel, the  \emph{strategy} $S=(S^{t})^{t  \in \Nats}$  is the
 sequence  defining which  component to  update at  time $t$  and $S^{t}$
 denotes its  $t-$th term. 
 This iteration scheme that only modifies one element at each iteration 
-is clasically refered as \emph{asynchronous iterations}. 
+is classically referred as \emph{asynchronous iterations}.
+More precisely, we have here for any $i$, $1\le i \le n$,  
+$$
+\left\{ \begin{array}{l}
+x^{0}  \in \Bool^n \\ 
+x^{t+1}_i = \left\{ 
+\begin{array}{l}
+  f_i(x^t) \textrm{ if $S^t = i$} \\
+  x_i^t \textrm{ otherwise} 
+ \end{array}
+\right.
+\end{array} \right.
+$$
+
 Next section shows the link between asynchronous iterations and 
 Devaney's Chaos.
 
@@ -352,7 +364,8 @@ $F_{f}:   \llbracket1;n\rrbracket\times   \mathds{B}^{n}   \rightarrow
     \right. 
 \end{equation}
 
-\noindent With such a notation, configurations are defined for times 
+\noindent With such a notation, configurations 
+asynchronously obtained are defined for times 
 $t=0,1,2,\ldots$ by:
 \begin{equation}\label{eq:sync}   
 \left\{\begin{array}{l}   
@@ -376,16 +389,20 @@ X^{k+1}& = & G_{f}(X^{k})\\
 \label{eq:Gf}
 \end{equation}
 where  $\sigma$ is the  function that  removes the  first term  of the
-strategy  ({\it i.e.},~$S^0$).   This definition  means that  only one
-component  of the  system is  updated  at an  iteration: the  $S^t$-th
-element. But it can be extended by considering subsets for $S^t$.
+strategy  ({\it i.e.},~$S^0$).  
+This definition  allows to links asynchronous iterations with 
+classical iterations of a dynamical system.
 
 
+%means that  only one
+%component  of the  system is  updated  at an  iteration: the  $S^t$-th
+%element. But it can be extended by considering subsets for $S^t$.
 
 
-The   {\bf   distance}   $d$    between   two   points   $(S,x)$   and
-$(\check{S},\check{x})\in  \mathcal{X} = \llbracket1;n\rrbracket^\Nats
-\times \Bool^{n}$ is defined by
+To study topological properties of these iterations, we are then left to
+introduce a  {\emph{  distance}}   $d$    between   two   points   $(S,x)$   and
+$(\check{S},\check{x})\in  \mathcal{X} = \llbracket1;n\rrbracket^\Nats. 
+\times \Bool^{n}$. It is defined by
 \begin{equation}
 d((S,x);(\check{S},\check{x}))=d_{e}(x,\check{x})+d_{s}(S,\check{S})
 \enspace ,
@@ -401,8 +418,7 @@ d_{s}(S,\check{S})=\frac{9}{2n}\sum_{t=0}^{\infty
 }\frac{|S^{t}-\check{S}^{t}|}{10^{t+1}} \in [0 ; 1] \enspace .
 \end{equation}
 
-This    distance    is    defined    to    reflect    the    following
-information. Firstly, the more  two systems have different components,
+Notice that the more  two systems have different components,
 the  larger the  distance  between them is.  Secondly,  two systems  with
 similar components and strategies, which have the same starting terms,
 must  induce only  a small  distance.  The  proposed  distance fulfill
@@ -412,7 +428,7 @@ measures the difference between the strategies.
 
 The relation  between $\Gamma(f)$ and  $G_f$ is clear: there  exists a
 path from  $x$ to $x'$  in $\Gamma(f)$ if  and only if there  exists a
-strategy  $s$ such  that  the  parallel iteration  of  $G_f$ from  the
+strategy  $s$ such  that   iterations  of  $G_f$ from  the
 initial  point $(s,x)$  reaches  the configuration  $x'$.  Using  this
 link, Guyeux~\cite{GuyeuxThese10} has proven that,
 \begin{theorem}%[Characterization of $\mathcal{C}$]
@@ -421,10 +437,23 @@ Let $f:\Bool^n\to\Bool^n$. Iterations of $G_f$ are chaotic  according
 to  Devaney if and only if  $\Gamma(f)$ is strongly connected.
 \end{theorem}
 
-Checking  if a  graph  is  strongly connected  is  not difficult.  For
-example,  consider the  function $f_1\left(x_1,\dots,x_n\right)=\left(
-\overline{x_1},x_1,x_2,\dots,x_{n-1}\right)$. As $\Gamma(f_1)$ is
-obviously strongly connected, then $G_{f_1}$ is a chaotic map.
+Checking  if a  graph  is  strongly connected  is  not difficult 
+(by the Tarjan's algorithm for instance). 
+Let be given  a strategy $S$ and a function $f$ such that 
+$\Gamma(f)$ is strongly connected.
+In that case, iterations of the function $G_f$ as defined in 
+Eq.~(\ref{eq:Gf}) are chaotic according to Devaney.
+
+
+Let us then define two function $f_0$ and $f_1$ both in 
+$\Bool^n\to\Bool^n$ that are used all along this article. 
+The former is the  vectorial negation, \textit{i.e.},
+$f_{0}(x_{1},\dots,x_{n}) =(\overline{x_{1}},\dots,\overline{x_{n}})$.
+The latter is  $f_1\left(x_1,\dots,x_n\right)=\left(
+\overline{x_1},x_1,x_2,\dots,x_{n-1}\right)$.
+It is not hard to see that  $\Gamma(f_0)$ and $\Gamma(f_1)$ are 
+both strongly connected, then iterations of $G_{f_0}$ and of 
+$G_{f_1}$ are chaotic according to Devaney.
 
 With this  material, we are now  able to build a  first chaotic neural
 network, as defined in the Devaney's formulation.
@@ -432,14 +461,13 @@ network, as defined in the Devaney's formulation.
 \section{A chaotic neural network in the sense of Devaney}
 \label{S2}
 
-Let     us     firstly     introduce    the     vectorial     negation
-$f_{0}(x_{1},\dots,x_{n}) =(\overline{x_{1}},\dots,\overline{x_{n}})$,
-which is  such that $\Gamma(f_0)$ is  strongly connected.  Considering
-the map  $F_{f_0}:\llbracket 1;  n \rrbracket \times  \mathds{B}^n \to
-\mathds{B}^n$  associated to the  vectorial negation,  we can  build a
-multilayer perceptron  neural network modeling  $F_{f_0}$.  Hence, for
-all inputs  $(s,x) \in \llbracket  1;n\rrbracket \times \mathds{B}^n$,
-the output layer will produce  $F_{f_0}(s,x)$.  It is then possible to
+Firstly, let us build a
+multilayer perceptron  neural network modeling 
+$F_{f_0}:\llbracket 1;  n \rrbracket \times  \mathds{B}^n \to
+\mathds{B}^n$  associated to the  vectorial negation.
+More precisely, for all inputs  
+$(s,x) \in \llbracket  1;n\rrbracket \times \mathds{B}^n$,
+the output layer  produces  $F_{f_0}(s,x)$.  It is then possible to
 link  the output  layer  and the  input  one, in  order  to model  the
 dependence between two successive iterations.  As a result we obtain a
 global  recurrent   neural  network  that  behaves   as  follows  (see
@@ -472,15 +500,18 @@ 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
+Eq.~(\ref{eq:Gf}).  It  means that  mathematically if we  use similar
 input  vectors   they  both  generate  the   same  successive  outputs
 $\left(x^t\right)^{t \in \mathds{N}^{\ast}}$,  and therefore that they
 are  equivalent  reformulations  of  the iterations  of  $G_{f_0}$  in
-$\mathcal{X}$. Finally, since the  proposed neural network is build to
-model  the  behavior  of  $G_{f_0}$,  which is  chaotic  according  to
+$\mathcal{X}$. Finally, since the  proposed neural network is built to
+model  the  behavior  of  $G_{f_0}$,  whose iterations are
+  chaotic  according  to
 Devaney's definition  of chaos,  we can conclude  that the  network is
 also chaotic in this sense.
 
@@ -503,8 +534,8 @@ 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  $n$
@@ -520,6 +551,7 @@ 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
@@ -527,11 +559,15 @@ 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}
@@ -560,6 +596,10 @@ like  topology  to  establish,  for  example,  their  convergence  or,
 contrarily, their unpredictable behavior.   An example of such a study
 is given in the next section.
 
+\JFC{Ce qui suit est davantage qu'un exemple.Il faudrait 
+motiver davantage, non?}
+
+
 \section{Topological properties of chaotic neural networks}
 \label{S4}
 
@@ -567,20 +607,23 @@ Let us first recall  two fundamental definitions from the mathematical
 theory of chaos.
 
 \begin{definition} \label{def8}
-A  function   $f$  is   said  to  be   {\bf  expansive}   if  $\exists
+A  function   $f$  is   said  to  be   {\emph{  expansive}}   if  $\exists
 \varepsilon>0$, $\forall  x \neq y$,  $\exists n \in  \mathds{N}$ such
 that $d\left(f^n(x),f^n(y)\right) \geq \varepsilon$.
 \end{definition}
 
 \begin{definition} \label{def9}
-A discrete dynamical  system is said to be  {\bf topologically mixing}
+A discrete dynamical  system is said to be  {\emph{ topologically mixing}}
 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
@@ -588,12 +631,14 @@ 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   {\bf
-not decomposable}  if it  is not  the  union of  two closed  sets $A,  B
+A   dynamical   system   $\left(   \mathcal{X},  f\right)$   is   
+{\emph{not decomposable}}  if it  is not  the  union of  two closed  sets $A,  B
 \subset \mathcal{X}$ such that $f(A) \subset A, f(B) \subset B$.
 \end{definition}
 
@@ -603,8 +648,8 @@ strongly connected.  Moreover, under this  hypothesis CI-MLPs($f$) are
 strongly transitive:
 
 \begin{definition} \label{def11}
-A  dynamical system  $\left( \mathcal{X},  f\right)$ is  {\bf strongly
-transitive} if $\forall x,y  \in \mathcal{X}$, $\forall r>0$, $\exists
+A  dynamical system  $\left( \mathcal{X},  f\right)$ is  {\emph{ strongly
+transitive}} if $\forall x,y  \in \mathcal{X}$, $\forall r>0$, $\exists
 z  \in  \mathcal{X}$,  $d(z,x)~\leq~r  \Rightarrow  \exists  n  \in
 \mathds{N}^{\ast}$, $f^n(z)=y$.
 \end{definition}
@@ -621,7 +666,8 @@ intrinsically complicated and it cannot be decomposed or simplified.
 Furthermore, those  recurrent neural networks  exhibit the instability
 property:
 \begin{definition}
-A dynamical  system $\left( \mathcal{X}, f\right)$ is  unstable if for
+A dynamical  system $\left( \mathcal{X}, f\right)$ is  \emph{unstable}
+if for
 all  $x  \in  \mathcal{X}$,   the  orbit  $\gamma_x:n  \in  \mathds{N}
 \longmapsto f^n(x)$  is unstable,  that means: $\exists  \varepsilon >
 0$, $\forall  \delta>0$, $\exists y  \in \mathcal{X}$, $\exists  n \in
@@ -652,7 +698,7 @@ at  least $\varepsilon$  apart in  the metric  $d_n$. Denote  by $H(n,
 \varepsilon)$     the    maximum     cardinality     of    an     $(n,
 \varepsilon)$-separated set,
 \begin{definition}
-The {\it topological entropy} of the  map $f$ is defined by (see e.g.,
+The \emph{topological entropy} of the  map $f$ is defined by (see e.g.,
 Ref.~\cite{Adler65} or Ref.~\cite{Bowen})
 $$h(f)=\lim_{\varepsilon\to      0}      \left(\limsup_{n\to      \infty}
 \frac{1}{n}\log H(n,\varepsilon)\right) \enspace .$$
@@ -748,7 +794,9 @@ 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}).
 
@@ -929,7 +977,7 @@ 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}
@@ -1083,6 +1131,21 @@ consequences in biology, physics, and computer science security fields
 will be  stated.  Lastly,  thresholds separating systems  depending on
 the ability to learn their dynamics will be established.
 
+% \appendix{}
+
+
+
+% \begin{definition} \label{def2}
+% A continuous function $f$  on a topological space $(\mathcal{X},\tau)$
+% is defined  to be  {\emph{topologically transitive}}  if for any  pair of
+% open  sets $U$,  $V  \in  \mathcal{X}$ there  exists  
+% $k \in
+% \mathds{N}^{\ast}$
+%  such  that
+% $f^k(U) \cap V \neq \emptyset$.
+% \end{definition}
+
+
 \bibliography{chaos-paper}% Produces the bibliography via BibTeX.
 
 \end{document}
