corps fini

[chaos1.git] / main.tex

diff --git a/main.tex b/main.tex
index b72eb6d8ba43fb5945c7d9fd561aeddae914b5af..cee88400c5a74d8f73babf97e75ca1fb7f6a49aa 100644 (file)
--- 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}}
 \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
 
 \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
 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
 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


@@ -197,9 +204,9 @@ section.
 
 In what follows and for  any function $f$, $f^n$ means the composition
 $f \circ f \circ \hdots \circ f$ ($n$ times) and an \emph{iteration}
 
 In what follows and for  any function $f$, $f^n$ means the composition
 $f \circ f \circ \hdots \circ f$ ($n$ times) and an \emph{iteration}

-of a \emph{dynamical system} the step that consists in
-updating the global state $x$ with respect to a function $f$ s.t.
-$x^{t+1} = f(x^t)$.    
+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}
 
 
 \subsection{Devaney's chaotic dynamical systems}
 


@@ -235,57 +242,39 @@ point dependence on initial conditions.
 
 Topological  transitivity   is  checked  when,  for   any  point,  any
 neighborhood of its future evolution eventually overlap with any other
 
 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  {\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}
-
-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.
 
 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  {\emph{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 {\emph{ 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}$, we can find at least  one periodic point in any of its
   neighborhood).
   \mathcal{X}$, 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 {\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
+%\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 $f$.
   {\emph{constant of sensitivity}} of $f$.

-\end{definition}
-
-Finally,

 
 

-\begin{definition} \label{def5}
-The dynamical system that iterates $f$ is {\emph{ chaotic according to Devaney}}  
-on $(\mathcal{X},\tau)$ if $f$  is   regular,  topologically  transitive,   
+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.
 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.
 
 In what follows, iterations are said to be \emph{chaotic according Devaney}
 when corresponding dynamical system  is chaotic according Devaney.
 


@@ -310,8 +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  $i \in  \llbracket 1;  n \rrbracket$
 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*}
 %\begin{equation*}   
 $x^{t}=(x_1^{t},\ldots,x_{n}^{t}) \in \Bool^n$.
 %\end{equation*}


@@ -328,23 +317,26 @@ $N(i,x)=(x_1,\ldots,\overline{x_i},\ldots,x_n)$  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
 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 $N(i,x)$.
 
 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 
 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}.
-More préciselly, we have here
+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.
 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 
 $$
 
 Next section shows the link between asynchronous iterations and 


@@ -407,7 +399,7 @@ classical iterations of a dynamical system.
 %element. But it can be extended by considering subsets for $S^t$.
 
 
 %element. But it can be extended by considering subsets for $S^t$.
 
 

-To study topological properties of these iteations, we are then left to
+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
 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


@@ -436,7 +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
 
 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 $\mathcal{C}$]
 initial  point $(s,x)$  reaches  the configuration  $x'$.  Using  this
 link, Guyeux~\cite{GuyeuxThese10} has proven that,
 \begin{theorem}%[Characterization of $\mathcal{C}$]


@@ -453,8 +445,8 @@ In that case, iterations of the function $G_f$ as defined in
 Eq.~(\ref{eq:Gf}) are chaotic according to Devaney.
 
 
 Eq.~(\ref{eq:Gf}) are chaotic according to Devaney.
 
 

-Let us then consider two function $f_0$ and $f_1$ both in 
-$\Bool^n\to\Bool^n$ defined as follows that are used all along this article. 
+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(
 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(


@@ -508,15 +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
 
 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
 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 built to
 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 built to

-model  the  behavior  of  $G_{f_0}$,  which is  chaotic  according  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.
 
 Devaney's definition  of chaos,  we can conclude  that the  network is
 also chaotic in this sense.
 


@@ -539,8 +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
 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  $n$
 
 We consider a multilayer perceptron  of the following form: inputs
 are $n$ binary digits and  one integer value, while outputs are  $n$


@@ -556,6 +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.
   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
 \end{itemize}
 
 The topological  behavior of these  particular neural networks  can be


@@ -563,11 +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
 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
 $\left(y_1,\dots,y_n\right)       \in       \mathds{B}^n$,       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}
 \mathds{B}^n       \rightarrow      \mathds{B}^n$       such      that
 $f\left(x_1,x_2,\dots,x_n\right)$ is equal to
 \begin{equation}


@@ -596,6 +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.
 
 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}
 
 \section{Topological properties of chaotic neural networks}
 \label{S4}
 


@@ -614,9 +618,12 @@ 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}
 \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 à 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
 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 +631,10 @@ 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
 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   
 
 \begin{definition} \label{def10}
 A   dynamical   system   $\left(   \mathcal{X},  f\right)$   is   


@@ -657,7 +666,8 @@ intrinsically complicated and it cannot be decomposed or simplified.
 Furthermore, those  recurrent neural networks  exhibit the instability
 property:
 \begin{definition}
 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
 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


@@ -688,7 +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}
 \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})
 $$h(f)=\lim_{\varepsilon\to      0}      \left(\limsup_{n\to      \infty}
 \frac{1}{n}\log H(n,\varepsilon)\right) \enspace .$$
 Ref.~\cite{Adler65} or Ref.~\cite{Bowen})
 $$h(f)=\lim_{\varepsilon\to      0}      \left(\limsup_{n\to      \infty}
 \frac{1}{n}\log H(n,\varepsilon)\right) \enspace .$$


@@ -784,7 +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
   
 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~$((S^t)^{t \in
+networks.  
+\JFC{détailler le more suited}
+Formally, input and output vectors are pairs~$((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}).
 
 \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 +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.
 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 être repris}

 \begin{table}[b]
 \caption{Prediction success rates for configurations expressed with Gray code}
 \label{tab2}
 \begin{table}[b]
 \caption{Prediction success rates for configurations expressed with Gray code}
 \label{tab2}


@@ -1119,6 +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.
 
 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}
 \bibliography{chaos-paper}% Produces the bibliography via BibTeX.
 
 \end{document}