\begin{xpl}
  Without  loss  of generality,  we  consider  Boolean  domains.  We  have  three
  elements taking their value in the set $\{0,1\}$.  Thus, a configuration is an
  element of  $\{0,1\}^3$, \textit{i.e.},  a binary  number between 0  and 7. In
  \Fig{fig:map}   is    given   the   function   defining    the   dynamic   and
  \Fig{fig:xplgraph}  depicts   its  associated  connection   graph.   Operators
  $\overline{a}$, sum and  product are the classical Boolean  operators.  In the
  incidence graph,  or connection  graph, the vertices  are the elements  of the
  system and an edge from $i$ to $j$ indicates that $f_j$ depends on $i$. Notice
  that  the connection  graph contains  five cycles.

\begin{figure}[ht]
  \centering
  \subfloat[Maps]{
    \begin{minipage}%[h]
{.6\linewidth}
      \begin{center}
        %\vspace{1.6em}
        $ F(X)= \left \{
%         \begin{array}{rcl}
%           f_1(x_1,x_2,x_3) & = & x_1.(x_2 + x_3)  \\
%           f_2(x_1,x_2,x_3) & = & \overline{x_1}.\overline{x_2}.x_3 \\
%           f_3(x_1,x_2,x_3) & = & x_1.x_2.x_3
%         \end{array}
         \begin{array}{rcl}
           f_1(x_1,x_2,x_3) & = & x_1.\overline{x_2} + x_3  \\
           f_2(x_1,x_2,x_3) & = & x_1 + \overline{x_3} \\
           f_3(x_1,x_2,x_3) & = & x_2.x_3
         \end{array}
        \right.
        $        
        %\vspace{1.6em}
      \end{center}
    \end{minipage}
    \label{fig:map}
  }
  \subfloat[Connexion Graph]{
    \begin{minipage}%[h]
{.35\linewidth}
      \begin{center}
        \includegraphics[width=4cm]{xplCnx.pdf}
      \end{center}
    \end{minipage}
    \label{fig:xplgraph}
  }
%  \vspace{-.5em}
  \caption{Dynamic of the running example.}
  %\vspace{-2em}
\end{figure}

% \begin{figure}
% \begin{center}
% %\includegraphics[width=3cm]{xplgraph.ps}
% \includegraphics[width=3cm]{xplgraph.pdf}
% \end{center}
% \caption{Incidence graph of running example\label{fig:xplgraph}}
% \end{figure}


% \begin{figure}
% \begin{center}
% \includegraphics[width=3cm]{xplgraph.ps}
% %\includegraphics[width=3cm]{xplgraph.pdf}
% \end{center}
% \caption{Incidence graph of running example\label{fig:xplgraph}}
% \end{figure}


% The prototype developed for~\cite{BCVC10:ir} has shown 
% that pure parallel iterations are convergent. 
% It has helped us to find pseudo-periodic strategies making chaotic 
% iterations (and then asynchronous iterations) diverging.
% This example shows then a case when convergence is established  
% under some conditions whereas universal convergence is not.

% \begin{figure}[ht]
% \begin{center}
% \includegraphics[width=12cm]{implementation/para_iterate_dec.ps}
% \end{center}
% \caption{Parallel iterations of running example\label{fig:xplparaFig}}
% \end{figure}

% In  what follows  and for  brevity  reasons, configurations  are represented  as
% decimal numbers instead of binary  numbers.  The graph of parallel iterations is
% given in Fig.~\ref{fig:xplparaFig}. 
% Starting  from any configuration, the network
% converges to the fixed point corresponding to the decimal number 19.

% \begin{figure}[ht]
% \begin{center}
% \includegraphics[width=4cm]{chao_iterate_excerpt.ps}
% \end{center}
% \caption{Excerpt of chaotic iterations of running example 
% \label{fig:xplchaoFig}}
% \end{figure}

% An extract of the  graph of  chaotic iterations  is given  in  Fig.~\ref{fig:xplchaoFig}.  
% Arc  label is  the  characteristic function  of  activated elements 
% (i.e. the matrix $J^t$)  expressed as  a
% decimal number.  It  allows us to shortly represent  the strategy.  
% For instance,
%  in this graph we only consider the pseudo-periodic strategy that  alternately activates  both the
% first two elements and the last four elements.  This is formalized as 
% $ J^{2p} =
% \left(
% \begin{array}{lllll}
% 1&0 &0 &0 &0  \\
% 0&1 &0 &0 &0  \\
% 0&0 &0 &0 &0  \\
% 0&0 &0 &0 &0  \\
% 0&0 &0 &0 &0  
% \end{array}
% \right)
% $
% and 
% $J^{2p+1} = 
% \left(
% \begin{array}{lllll}
% 0&0 &0 &0 &0  \\
% 0&1 &0 &0 &0  \\
% 0&0 &1 &0 &0  \\
% 0&0 &0 &1 &0  \\
% 0&0 &0 &0 &1  
% \end{array}
% \right)
% $, $p=0,1,2,\ldots$.
% The former matrix is represented as 24 and the later as 15.
 
% Iterations  do not  converge  for this strategy: for  instance,
% starting from the configuration 3 (resp. configuration 7), the  network goes to the configuration 11 (resp. configuration 15) and
% goes back to the configuration 3 (resp. configuration 7) by applying the strategy depicted above.


% Since chaotic iterations do not converge for some strategies, it is obvious that
% convergence  cannot  be  observed  for  some asynchronous  iterations  with  the
% corresponding  strategies.  However, even  if all  the components  
% are activated, that is the asynchronous counterpart of pure parallel iterations, 
% delays may introduce divergence.

% For instance, let $J^t$ be constantly equal to the identity matrix and
% consider the matrix 
% \begin{equation*}
% S^t = \left(
% \begin{array}{lllll}
% t & t' & t & t & t \\
% t & t & t & t & t \\
% t & t & t & t & t \\
% t & t & t & t & t\\
% t & t & t & t & t \\
% \end{array}
% \right)
% \textrm{ where }
% t' = \left\{
% \begin{array}{ll}
% t & \textrm{ if t is even;} \\
% t-1 & \textrm{ otherwise.}
% \end{array}
% \right.
% \end{equation*}

% We have both 
% \begin{eqnarray*}
% X^{2k+1} & =& F(X^{2k}) \\
% X^{2k} & =& \left(
% \begin{array}{l}
% F_1(X_1^{2k-1},X_2^{2k-2},X_3^{2k-1},X_4^{2k-1},X_5^{2k-1}) \\
% F_2(X^{2k-1})\\
% \vdots \\
% F_5(X^{2k-1})
% \end{array}
% \right).
% \end{eqnarray*}


% Starting from $X^0 = (0,0,0,1,1)$, the system reaches 
% $X^1 = (0,1,0,1,1)$ and enters in a cycle between  these 
% two configurations (as in some chaotic iterations).  

% In
% the  next  section, a  particular  execution  mode  is described  which  enables
% asynchronism in iterations while guaranteeing the convergence.


In  what follows  and for  brevity  reasons, configurations  are represented  as
decimal numbers instead of binary numbers.   It is not hard to establish that in
parallel synchronous iterations, starting  from any configuration different from
7 (111) leads  to the fixed point 2  (010), and starting from 7 leads  to 7.  
With other strategies,
the behavior is  quite different although the  system always
reaches a stabilization.  Indeed, for the  initial states 1, 3 and 5, the system
converges towards anyone of the two fixed  points 2 and 7.  However, there is no
infinite  cycle and  the convergence  property defined  in  \Equ{eq:ltl:conv} is
verified. 
Finally,  due to the presence of 
cycle in the connexion graph, 
known   sufficient  conditions~\cite{Bah00}  establishing  convergence
results for asynchronous modes cannot be applied here. 
We address this question in what follows. 
\end{xpl}


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