X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/chaos1.git/blobdiff_plain/a1c7d633b750e7417e0e331d9f2df61936c76de7..999fadfcb0b5ed18bd495364a07850278fade07d:/main.tex?ds=inline

diff --git a/main.tex b/main.tex
index 8e7f34b..cee8840 100644
--- a/main.tex
+++ b/main.tex
@@ -204,9 +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}
-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}
 
@@ -242,57 +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
-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.
 
-\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).
-\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 $f$.
-\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.
-\end{definition}
-
 In what follows, iterations are said to be \emph{chaotic according Devaney}
 when corresponding dynamical system  is chaotic according Devaney.
 
@@ -317,8 +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*}
@@ -335,23 +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
-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}.
-More prÃ©ciselly, we have here
+is classically referred as \emph{asynchronous iterations}.
+More precisely, we have here for any $i$, $1\le i \le n$,  
 $$
+\left\{ \begin{array}{l}
+x^{0}  \in \Bool^n \\ 
 x^{t+1}_i = \left\{ 
 \begin{array}{l}
   f_i(x^t) \textrm{ if $S^t = i$} \\
   x_i^t \textrm{ otherwise} 
  \end{array}
 \right.
+\end{array} \right.
 $$
 
 Next section shows the link between asynchronous iterations and 
@@ -414,7 +399,7 @@ classical iterations of a dynamical system.
 %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
@@ -443,7 +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 $\mathcal{C}$]
@@ -460,8 +445,8 @@ In that case, iterations of the function $G_f$ as defined in
 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(
@@ -520,12 +505,13 @@ given  as outside  input,
  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 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.
 
@@ -610,6 +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}
 
@@ -676,7 +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
@@ -707,7 +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})
 $$h(f)=\lim_{\varepsilon\to      0}      \left(\limsup_{n\to      \infty}
 \frac{1}{n}\log H(n,\varepsilon)\right) \enspace .$$
@@ -1140,6 +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}
