X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/chaos1.git/blobdiff_plain/4fe362d8ad3a7c5e9198ca3981fa7a3bf01b466f..486652fa99fdbcf0992d32f910b729e648e33644:/main.tex?ds=sidebyside

diff --git a/main.tex b/main.tex
index 82e99bf..016243a 100644
--- a/main.tex
+++ b/main.tex
@@ -154,7 +154,7 @@ iterations  and  a  class  of  globally  recurrent  MLP.   The  second
 contribution is a  study of the converse problem,  indeed we investigate the
 ability  of classical  multiLayer  perceptrons to  learn a  particular
 family of discrete chaotic  dynamical systems.  This family is defined
-by a Boolean vector, an update function, and a sequence defining which
+by a Boolean vector, an update function, and a sequence defining the
 component  to  update  at  each  iteration.  It  has  been  previously
 established that  such dynamical systems are  chaotically iterated (as
 it  is defined by  Devaney) when  the chosen  function has  a strongly
@@ -234,14 +234,15 @@ point dependence on initial conditions.
 neighborhood of its future evolution eventually overlap with any other
 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.
+allow any  simplification.  
 
 However, chaos needs some regularity to ``counteracts'' the effects of
-transitivity.
+transitivity. % Thus, two points close to each other can behave in a completely different manner, the first one visiting the whole space whereas the second one has a regular orbit.
+In the Devaney's formulation, a dense set  of periodic
+points is the element of regularity that a chaotic dynamical system has
+to exhibit.
 %\begin{definition} \label{def3}
-We recall that a  point $x$ is  {\bf periodic point} for $f$ of 
+We recall that a  point $x$ is a {\bf periodic point} for $f$ of 
 period~$n \in \mathds{N}^{\ast}$ if $f^{n}(x)=x$.
 %\end{definition}
 Then, the map 
@@ -261,12 +262,11 @@ Let  us recall  that  $f$  has {\bf  sensitive  dependence on  initial
 $n  > 0$ such  that $d\left(f^{n}(x),  f^{n}(y)\right) >\delta  $. The
 value $\delta$ is called the {\bf constant of sensitivity} of $f$.
 
-Finally,  The  dynamical system  that  iterates  $f$  is {\bf  chaotic
+Finally,  the  dynamical system  that  iterates  $f$  is {\bf  chaotic
   according  to Devaney}  on $(\mathcal{X},\tau)$  if $f$  is regular,
 topologically transitive, and has  sensitive dependence to its initial
 conditions.   In  what follows,  iterations  are  said  to be  chaotic
-according  Devaney  when  corresponding  dynamical system  is  chaotic
-according Devaney.
+(according to Devaney)  when the corresponding  dynamical system  is  chaotic, as it is defined in the Devaney's formulation.
 
 %Let us notice that for a  metric space the last condition follows from
 %the two first ones~\cite{Banks92}.
@@ -311,15 +311,15 @@ $i$ 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
 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 classically referred as {\it asynchronous
+element at each iteration is usually referred as {\it asynchronous
   iterations}.  More precisely, we have for any $i$, $1\le i \le n$,
 \begin{equation}
 \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} 
+  f_i(x^t) \textrm{ if $S^t = i$}\enspace , \\
+  x_i^t \textrm{ otherwise}\enspace . 
  \end{array}
 \right.
 \end{array} \right.
@@ -373,15 +373,15 @@ X^{k+1}& = & G_{f}(X^{k})\\
 \right.
 \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  allows  to  links
+where  $\sigma$ is the so-called shift function that  removes the  first term  of the
+strategy  ({\it  i.e.},~$S^0$).    This  definition  allows  to  link
 asynchronous  iterations  with  classical  iterations of  a  dynamical
 system. Note that it can be extended by considering subsets for $S^t$.
 
 To study topological properties of  these iterations, we are then left
 to  introduce a  {\bf 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
+$(\check{S},\check{x})$ in $\mathcal{X} = \llbracket1;n\rrbracket^\Nats
+\times \Bool^{n}$. Let $\Delta(x,y) = 0$ if $x=y$, and $\Delta(x,y) = 1$ else, be a distance on $\mathds{B}$. The distance $d$ is defined by 
 \begin{equation}
 d((S,x);(\check{S},\check{x}))=d_{e}(x,\check{x})+d_{s}(S,\check{S})
 \enspace ,
@@ -406,10 +406,10 @@ these  requirements: on  the one  hand  its floor  value reflects  the
 difference between  the cells, on  the other hand its  fractional part
 measures the difference between the strategies.
 
-The relation  between $\Gamma(f)$ and  $G_f$ is clear: there  exists a
+The relation  between $\Gamma(f)$ and  $G_f$ is obvious: there  exists a
 path from  $x$ to $x'$  in $\Gamma(f)$ if  and only if there  exists a
 strategy  $s$ such  that iterations  of $G_f$  from the  initial point
-$(s,x)$   reaches   the   configuration   $x'$.   Using   this   link,
+$(s,x)$   reach   the   configuration   $x'$.   Using   this   link,
 Guyeux~\cite{GuyeuxThese10} has proven that,
 \begin{theorem}%[Characterization of $\mathcal{C}$]
 \label{Th:Caracterisation   des   IC   chaotiques}  
@@ -424,7 +424,7 @@ 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
+Let   us  then   define  two   functions  $f_0$   and  $f_1$   both  in
 $\Bool^n\to\Bool^n$ that are used all along this paper.  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
@@ -460,7 +460,7 @@ we obtain  a global recurrent  neural network that behaves  as follows
 \item When  the network  is activated at  the $t^{th}$  iteration, the
   state of the system $x^t  \in \mathds{B}^n$ received from the output
   layer and  the initial  term of the  sequence $(S^t)^{t  \in \Nats}$
-  ($S^0  \in \llbracket 1;n\rrbracket$)  are used  to compute  the new
+  (\textit{i.e.}, $S^0  \in \llbracket 1;n\rrbracket$)  are used  to compute  the new
   output vector.  This  new vector, which represents the  new state of
   the dynamical system, satisfies:
   \begin{equation}
@@ -478,7 +478,6 @@ we obtain  a global recurrent  neural network that behaves  as follows
 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,
-\JFC{en dire davantage sur l'outside world} %% TO BE UPDATED
 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
@@ -488,7 +487,7 @@ $\left(x^t\right)^{t \in \mathds{N}^{\ast}}$,  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  $G_{f_0}$,  whose iterations are
-  chaotic  according  to
+  chaotic  according  to the
 Devaney's definition  of chaos,  we can conclude  that the  network is
 also chaotic in this sense.
 
@@ -947,7 +946,7 @@ scheme, the strategies cannot be predicted.
 Let us now compare the  two coding schemes. Firstly, the second scheme
 disturbs  the   learning  process.   In   fact  in  this   scheme  the
 configuration is always expressed as  a natural number, whereas in the
-first one  the number  of inputs follows  the increase of  the boolean
+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}
