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\title{Efficient Generation of Pseudo-Random Numbers based on Chaotic Iterations
on GPU}
\begin{document}

\author{Jacques M. Bahi, Rapha\"{e}l Couturier, and Christophe
Guyeux\thanks{Authors in alphabetic order}}

\maketitle

\begin{abstract}

\end{abstract}

\section{Introduction}

Random  numbers are  used in  many scientific  applications and  simulations. On
finite  state machines,  as computers,  it is  not possible  to  generate random
numbers but only pseudo-random numbers. In practice, a good pseudo-random number
generator (PRNG) needs  to verify some features to be used  by scientists. It is
important  to  be  able  to  generate  pseudo-random  numbers  efficiently,  the
generation  needs to  be reproducible  and a  PRNG needs  to satisfy  many usual
statistical properties. Finally, from our point a view, it is essential to prove
that  a PRNG  is  chaotic.  Concerning  the  statistical tests,  TestU01 is  the
best-known public-domain statistical testing package.   So we use it for all our
PRNGs, especially the {\it BigCrush}  which provides the largest serie of tests.
Concerning  the  chaotic properties,  Devaney~\cite{Devaney}  proposed a  common
mathematical formulation of chaotic dynamical systems.

In a  previous work~\cite{bgw09:ip}  we have proposed  a new familly  of chaotic
PRNG  based on  chaotic iterations  (IC). We  have proven  that these  PRNGs are
chaotic in the Devaney's sense.  In this paper we propose a faster version which
is also proven to be chaotic.

Although graphics  processing units (GPU)  was initially designed  to accelerate
the manipulation of  images, they are nowadays commonly  used in many scientific
applications. Therefore,  it is important  to be able to  generate pseudo-random
numbers inside a GPU when a scientific application runs in a GPU. That is why we
also provide  an efficient  PRNG for  GPU respecting based  on IC.  Such devices
allows us to generated almost 20 billions of random numbers per second.

In order  to establish  that our  PRNGs are chaotic  according to  the Devaney's
formulation, we extend what we have proposed in~\cite{guyeux10}. Moreover,  we define a new distance to measure the disorder in the chaos and we prove some interesting properties with this distance.

The rest of this paper  is organised as follows. In Section~\ref{section:related
  works}  we review  some GPU  implementions of  PRNG.  Section~\ref{section:BASIC RECALLS}  gives some  basic recalls  on  Devanay's formation  of chaos  and
chaotic iterations. In Section~\ref{sec:pseudo-random} the proof of chaos of our
PRNGs  is  studied.   Section~\ref{sec:efficient  prng}  presents  an  efficient
implementation of  our chaotic PRNG  on a CPU.   Section~\ref{sec:efficient prng
  gpu}   describes   the  GPU   implementation   of   our   chaotic  PRNG.    In
Section~\ref{sec:experiments}     some    experimentations     are    presented.
Section~\ref{sec:de  la  relativité du  désordre}  describes  the relativity  of
disorder.  In Section~\ref{sec:  chaos order  topology} the  proof  that chaotic
iterations can be described by iterations on a real interval is established. Finally, we give a conclusion and some perspectives.




\section{Related works on GPU based PRNGs}
\label{section:related works}
In the litterature many authors have work on defining GPU based PRNGs. We do not
want to be exhaustive and we just give the most significant works from our point
of view. When authors mention the  number of random numbers generated per second
we mention  it. We  consider that  a million numbers  per second  corresponds to
1MSample/s and than a billion numbers per second corresponds to 1GSample/s.

In \cite{Pang:2008:cec},  the authors define  a PRNG based on  cellular automata
which  does   not  require  high  precision  integer   arithmetics  nor  bitwise
operations. There is no mention of statistical tests nor proof that this PRNG is
chaotic.  Concerning   the  speed  of   generation,  they  can   generate  about
3.2MSample/s on a GeForce 7800 GTX GPU (which is quite old now).

In \cite{ZRKB10}, the authors propose  different versions of efficient GPU PRNGs
based on  Lagged Fibonacci, Hybrid  Taus or Hybrid  Taus.  They have  used these
PRNGs   for  Langevin   simulations   of  biomolecules   fully  implemented   on
GPU. Performance of  the GPU versions are far better than  those obtained with a
CPU and these PRNGs succeed to pass the {\it BigCrush} test of TestU01. There is
no mention that their PRNGs have chaos mathematical properties.


Authors of~\cite{conf/fpga/ThomasHL09}  have studied the  implementation of some
PRNGs on  diferrent computing architectures: CPU,  field-programmable gate array
(FPGA), GPU and massively parallel  processor. This study is interesting because
it  shows the  performance  of the  same  PRNGs on  different architeture.   For
example,  the FPGA  is globally  the  fastest architecture  and it  is also  the
efficient one because it provides the fastest number of generated random numbers
per joule. Concerning the GPU,  authors can generate betweend 11 and 16GSample/s
with a GTX 280  GPU. The drawback of this work is  that those PRNGs only succeed
the {\it Crush} test which is easier than the {\it Big Crush} test.
\newline
\newline
To the best of our knowledge no GPU implementation have been proven to have chaotic properties.

\section{Basic Recalls}
\label{section:BASIC RECALLS}
This section is devoted to basic definitions and terminologies in the fields of
topological chaos and chaotic iterations.
\subsection{Devaney's Chaotic Dynamical Systems}

In the sequel $S^{n}$ denotes the $n^{th}$ term of a sequence $S$ and $V_{i}$
denotes the $i^{th}$ component of a vector $V$. $f^{k}=f\circ ...\circ f$
is for the $k^{th}$ composition of a function $f$. Finally, the following
notation is used: $\llbracket1;N\rrbracket=\{1,2,\hdots,N\}$.


Consider a topological space $(\mathcal{X},\tau)$ and a continuous function $f :
\mathcal{X} \rightarrow \mathcal{X}$.

\begin{definition}
$f$ is said to be \emph{topologically transitive} if, for any pair of open sets
$U,V \subset \mathcal{X}$, there exists $k>0$ such that $f^k(U) \cap V \neq
\varnothing$.
\end{definition}

\begin{definition}
An element $x$ is a \emph{periodic point} for $f$ of period $n\in \mathds{N}^*$
if $f^{n}(x)=x$.% The set of periodic points of $f$ is denoted $Per(f).$
\end{definition}

\begin{definition}
$f$ 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 point $x$ in $\mathcal{X}$,
any neighborhood of $x$ contains at least one periodic point (without
necessarily the same period).
\end{definition}


\begin{definition}[Devaney's formulation of chaos~\cite{Devaney}]
$f$ is said to be \emph{chaotic} on $(\mathcal{X},\tau)$ if $f$ is regular and
topologically transitive.
\end{definition}

The chaos property is strongly linked to the notion of ``sensitivity'', defined
on a metric space $(\mathcal{X},d)$ by:

\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 $.

$\delta$ is called the \emph{constant of sensitivity} of $f$.
\end{definition}

Indeed, Banks \emph{et al.} have proven in~\cite{Banks92} that when $f$ is
chaotic and $(\mathcal{X}, d)$ is a metric space, then $f$ has the property of
sensitive dependence on initial conditions (this property was formerly an
element of the definition of chaos). To sum up, quoting Devaney
in~\cite{Devaney}, a chaotic dynamical system ``is unpredictable because of the
sensitive dependence on initial conditions. It cannot be broken down or
simplified into two subsystems which do not interact because of topological
transitivity. And in the midst of this random behavior, we nevertheless have an
element of regularity''. Fundamentally different behaviors are consequently
possible and occur in an unpredictable way.



\subsection{Chaotic Iterations}
\label{sec:chaotic iterations}


Let us consider  a \emph{system} with a finite  number $\mathsf{N} \in
\mathds{N}^*$ of elements  (or \emph{cells}), so that each  cell has a
Boolean  \emph{state}. Having $\mathsf{N}$ Boolean values for these
 cells  leads to the definition of a particular \emph{state  of the
system}. A sequence which  elements belong to $\llbracket 1;\mathsf{N}
\rrbracket $ is called a \emph{strategy}. The set of all strategies is
denoted by $\llbracket 1, \mathsf{N} \rrbracket^\mathds{N}.$

\begin{definition}
\label{Def:chaotic iterations}
The      set       $\mathds{B}$      denoting      $\{0,1\}$,      let
$f:\mathds{B}^{\mathsf{N}}\longrightarrow  \mathds{B}^{\mathsf{N}}$ be
a  function  and  $S\in  \llbracket 1, \mathsf{N} \rrbracket^\mathds{N}$  be  a  ``strategy''.  The  so-called
\emph{chaotic      iterations}     are     defined      by     $x^0\in
\mathds{B}^{\mathsf{N}}$ and
\begin{equation}
\forall    n\in     \mathds{N}^{\ast     },    \forall     i\in
\llbracket1;\mathsf{N}\rrbracket ,x_i^n=\left\{
\begin{array}{ll}
  x_i^{n-1} &  \text{ if  }S^n\neq i \\
  \left(f(x^{n-1})\right)_{S^n} & \text{ if }S^n=i.
\end{array}\right.
\end{equation}
\end{definition}

In other words, at the $n^{th}$ iteration, only the $S^{n}-$th cell is
\textquotedblleft  iterated\textquotedblright .  Note  that in  a more
general  formulation,  $S^n$  can   be  a  subset  of  components  and
$\left(f(x^{n-1})\right)_{S^{n}}$      can     be      replaced     by
$\left(f(x^{k})\right)_{S^{n}}$, where  $k<n$, describing for example,
delays  transmission~\cite{Robert1986,guyeux10}.  Finally,  let us  remark that
the term  ``chaotic'', in  the name of  these iterations,  has \emph{a
priori} no link with the mathematical theory of chaos, presented above.


Let us now recall how to define a suitable metric space where chaotic iterations
are continuous. For further explanations, see, e.g., \cite{guyeux10}.

Let $\delta $ be the \emph{discrete Boolean metric}, $\delta
(x,y)=0\Leftrightarrow x=y.$ Given a function $f$, define the function:
\begin{equation}
\begin{array}{lrll}
F_{f}: & \llbracket1;\mathsf{N}\rrbracket\times \mathds{B}^{\mathsf{N}} &
\longrightarrow & \mathds{B}^{\mathsf{N}} \\
& (k,E) & \longmapsto & \left( E_{j}.\delta (k,j)+f(E)_{k}.\overline{\delta
(k,j)}\right) _{j\in \llbracket1;\mathsf{N}\rrbracket},%
\end{array}%
\end{equation}%
\noindent where + and . are the Boolean addition and product operations.
Consider the phase space:
\begin{equation}
\mathcal{X} = \llbracket 1 ; \mathsf{N} \rrbracket^\mathds{N} \times
\mathds{B}^\mathsf{N},
\end{equation}
\noindent and the map defined on $\mathcal{X}$:
\begin{equation}
G_f\left(S,E\right) = \left(\sigma(S), F_f(i(S),E)\right), \label{Gf}
\end{equation}
\noindent where $\sigma$ is the \emph{shift} function defined by $\sigma
(S^{n})_{n\in \mathds{N}}\in \llbracket 1, \mathsf{N} \rrbracket^\mathds{N}\longrightarrow (S^{n+1})_{n\in
\mathds{N}}\in \llbracket 1, \mathsf{N} \rrbracket^\mathds{N}$ and $i$ is the \emph{initial function} 
$i:(S^{n})_{n\in \mathds{N}} \in \llbracket 1, \mathsf{N} \rrbracket^\mathds{N}\longrightarrow S^{0}\in \llbracket
1;\mathsf{N}\rrbracket$. Then the chaotic iterations proposed in
Definition \ref{Def:chaotic iterations} can be described by the following iterations:
\begin{equation}
\left\{
\begin{array}{l}
X^0 \in \mathcal{X} \\
X^{k+1}=G_{f}(X^k).%
\end{array}%
\right.
\end{equation}%

With this formulation, a shift function appears as a component of chaotic
iterations. The shift function is a famous example of a chaotic
map~\cite{Devaney} but its presence is not sufficient enough to claim $G_f$ as
chaotic. 
To study this claim, a new distance between two points $X = (S,E), Y =
(\check{S},\check{E})\in
\mathcal{X}$ has been introduced in \cite{guyeux10} as follows:
\begin{equation}
d(X,Y)=d_{e}(E,\check{E})+d_{s}(S,\check{S}),
\end{equation}
\noindent where
\begin{equation}
\left\{
\begin{array}{lll}
\displaystyle{d_{e}(E,\check{E})} & = & \displaystyle{\sum_{k=1}^{\mathsf{N}%
}\delta (E_{k},\check{E}_{k})}, \\
\displaystyle{d_{s}(S,\check{S})} & = & \displaystyle{\dfrac{9}{\mathsf{N}}%
\sum_{k=1}^{\infty }\dfrac{|S^k-\check{S}^k|}{10^{k}}}.%
\end{array}%
\right.
\end{equation}


This new distance has been introduced to satisfy the following requirements.
\begin{itemize}
\item When the number of different cells between two systems is increasing, then
their distance should increase too.
\item In addition, if two systems present the same cells and their respective
strategies start with the same terms, then the distance between these two points
must be small because the evolution of the two systems will be the same for a
while. Indeed, the two dynamical systems start with the same initial condition,
use the same update function, and as strategies are the same for a while, then
components that are updated are the same too.
\end{itemize}
The distance presented above follows these recommendations. Indeed, if the floor
value $\lfloor d(X,Y)\rfloor $ is equal to $n$, then the systems $E, \check{E}$
differ in $n$ cells ($d_e$ is indeed the Hamming distance). In addition, $d(X,Y) - \lfloor d(X,Y) \rfloor $ is a
measure of the differences between strategies $S$ and $\check{S}$. More
precisely, this floating part is less than $10^{-k}$ if and only if the first
$k$ terms of the two strategies are equal. Moreover, if the $k^{th}$ digit is
nonzero, then the $k^{th}$ terms of the two strategies are different.
The impact of this choice for a distance will be investigate at the end of the document.

Finally, it has been established in \cite{guyeux10} that,

\begin{proposition}
Let $f$ be a map from $\mathds{B}^\mathsf{N}$ to itself. Then $G_{f}$ is continuous in
the metric space $(\mathcal{X},d)$.
\end{proposition}

The chaotic property of $G_f$ has been firstly established for the vectorial
Boolean negation $f(x_1,\hdots, x_\mathsf{N}) =  (\overline{x_1},\hdots, \overline{x_\mathsf{N}})$ \cite{guyeux10}. To obtain a characterization, we have secondly
introduced the notion of asynchronous iteration graph recalled bellow.

Let $f$ be a map from $\mathds{B}^\mathsf{N}$ to itself. The
{\emph{asynchronous iteration graph}} associated with $f$ is the
directed graph $\Gamma(f)$ defined by: the set of vertices is
$\mathds{B}^\mathsf{N}$; for all $x\in\mathds{B}^\mathsf{N}$ and 
$i\in \llbracket1;\mathsf{N}\rrbracket$,
the graph $\Gamma(f)$ contains an arc from $x$ to $F_f(i,x)$. 
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
initial point $(s,x)$ reaches the point $x'$.

We have finally proven in \cite{bcgr11:ip} that,


\begin{theorem}
\label{Th:Caractérisation   des   IC   chaotiques}  
Let $f:\mathds{B}^\mathsf{N}\to\mathds{B}^\mathsf{N}$. $G_f$ is chaotic  (according to  Devaney) 
if and only if $\Gamma(f)$ is strongly connected.
\end{theorem}

This result of chaos has lead us to study the possibility to build a
pseudo-random number generator (PRNG) based on the chaotic iterations. 
As $G_f$, defined on the domain   $\llbracket 1 ;  \mathsf{N} \rrbracket^{\mathds{N}} 
\times \mathds{B}^\mathsf{N}$, is build from Boolean networks $f : \mathds{B}^\mathsf{N}
\rightarrow \mathds{B}^\mathsf{N}$, we can preserve the theoretical properties on $G_f$
during implementations (due to the discrete nature of $f$). It is as if
$\mathds{B}^\mathsf{N}$ represents the memory of the computer whereas $\llbracket 1 ;  \mathsf{N}
\rrbracket^{\mathds{N}}$ is its input stream (the seeds, for instance).

\section{Application to Pseudo-Randomness}
\label{sec:pseudo-random}
\subsection{A First Pseudo-Random Number Generator}

We have proposed in~\cite{bgw09:ip} a new family of generators that receives 
two PRNGs as inputs. These two generators are mixed with chaotic iterations, 
leading thus to a new PRNG that improves the statistical properties of each
generator taken alone. Furthermore, our generator 
possesses various chaos properties that none of the generators used as input
present.

\begin{algorithm}[h!]
%\begin{scriptsize}
\KwIn{a function $f$, an iteration number $b$, an initial configuration $x^0$
($n$ bits)}
\KwOut{a configuration $x$ ($n$ bits)}
$x\leftarrow x^0$\;
$k\leftarrow b + \textit{XORshift}(b)$\;
\For{$i=0,\dots,k$}
{
$s\leftarrow{\textit{XORshift}(n)}$\;
$x\leftarrow{F_f(s,x)}$\;
}
return $x$\;
%\end{scriptsize}
\caption{PRNG with chaotic functions}
\label{CI Algorithm}
\end{algorithm}

\begin{algorithm}[h!]
\KwIn{the internal configuration $z$ (a 32-bit word)}
\KwOut{$y$ (a 32-bit word)}
$z\leftarrow{z\oplus{(z\ll13)}}$\;
$z\leftarrow{z\oplus{(z\gg17)}}$\;
$z\leftarrow{z\oplus{(z\ll5)}}$\;
$y\leftarrow{z}$\;
return $y$\;
\medskip
\caption{An arbitrary round of \textit{XORshift} algorithm}
\label{XORshift}
\end{algorithm}





This generator is synthesized in Algorithm~\ref{CI Algorithm}.
It takes as input: a function $f$;
an integer $b$, ensuring that the number of executed iterations is at least $b$
and at most $2b+1$; and an initial configuration $x^0$.
It returns the new generated configuration $x$.  Internally, it embeds two
\textit{XORshift}$(k)$ PRNGs \cite{Marsaglia2003} that returns integers
uniformly distributed
into $\llbracket 1 ; k \rrbracket$.
\textit{XORshift} is a category of very fast PRNGs designed by George Marsaglia,
which repeatedly uses the transform of exclusive or (XOR, $\oplus$) on a number
with a bit shifted version of it. This PRNG, which has a period of
$2^{32}-1=4.29\times10^9$, is summed up in Algorithm~\ref{XORshift}. It is used
in our PRNG to compute the strategy length and the strategy elements.


We have proven in \cite{bcgr11:ip} that,
\begin{theorem}
  Let $f: \mathds{B}^{n} \rightarrow \mathds{B}^{n}$, $\Gamma(f)$ its
  iteration graph, $\check{M}$ its adjacency
  matrix and $M$ a $n\times n$ matrix defined as in the previous lemma.
  If $\Gamma(f)$ is strongly connected, then 
  the output of the PRNG detailed in Algorithm~\ref{CI Algorithm} follows 
  a law that tends to the uniform distribution 
  if and only if $M$ is a double stochastic matrix.
\end{theorem} 

This former generator as successively passed various batteries of statistical tests, as the NIST tests~\cite{bcgr11:ip}.

\subsection{Improving the Speed of the Former Generator}

Instead of updating only one cell at each iteration, we can try to choose a
subset of components and to update them together. Such an attempt leads
to a kind of merger of the two sequences used in Algorithm 
\ref{CI Algorithm}. When the updating function is the vectorial negation,
this algorithm can be rewritten as follows:

\begin{equation}
\left\{
\begin{array}{l}
x^0 \in \llbracket 0, 2^\mathsf{N}-1 \rrbracket, S \in \llbracket 0, 2^\mathsf{N}-1 \rrbracket^\mathds{N} \\
\forall n \in \mathds{N}^*, x^n = x^{n-1} \oplus S^n,
\end{array}
\right.
\label{equation Oplus}
\end{equation}
where $\oplus$ is for the bitwise exclusive or between two integers. 
This rewritten can be understood as follows. The $n-$th term $S^n$ of the
sequence $S$, which is an integer of $\mathsf{N}$ binary digits, presents
the list of cells to update in the state $x^n$ of the system (represented
as an integer having $\mathsf{N}$ bits too). More precisely, the $k-$th 
component of this state (a binary digit) changes if and only if the $k-$th 
digit in the binary decomposition of $S^n$ is 1.

The single basic component presented in Eq.~\ref{equation Oplus} is of 
ordinary use as a good elementary brick in various PRNGs. It corresponds
to the following discrete dynamical system in chaotic iterations:

\begin{equation}
\forall    n\in     \mathds{N}^{\ast     },    \forall     i\in
\llbracket1;\mathsf{N}\rrbracket ,x_i^n=\left\{
\begin{array}{ll}
  x_i^{n-1} &  \text{ if  } i \notin \mathcal{S}^n \\
  \left(f(x^{n-1})\right)_{S^n} & \text{ if }i \in \mathcal{S}^n.
\end{array}\right.
\label{eq:generalIC}
\end{equation}
where $f$ is the vectorial negation and $\forall n \in \mathds{N}$, 
$\mathcal{S}^n \subset \llbracket 1, \mathsf{N} \rrbracket$ is such that
$k \in \mathcal{S}^n$ if and only if the $k-$th digit in the binary
decomposition of $S^n$ is 1. Such chaotic iterations are more general
than the ones presented in Definition \ref{Def:chaotic iterations} for 
the fact that, instead of updating only one term at each iteration,
we select a subset of components to change.


Obviously, replacing Algorithm~\ref{CI Algorithm} by 
Equation~\ref{equation Oplus}, possible when the iteration function is
the vectorial negation, leads to a speed improvement. However, proofs
of chaos obtained in~\cite{bg10:ij} have been established
only for chaotic iterations of the form presented in Definition 
\ref{Def:chaotic iterations}. The question is now to determine whether the
use of more general chaotic iterations to generate pseudo-random numbers 
faster, does not deflate their topological chaos properties.

\subsection{Proofs of Chaos of the General Formulation of the Chaotic Iterations}
\label{deuxième def}
Let us consider the discrete dynamical systems in chaotic iterations having 
the general form:

\begin{equation}
\forall    n\in     \mathds{N}^{\ast     },    \forall     i\in
\llbracket1;\mathsf{N}\rrbracket ,x_i^n=\left\{
\begin{array}{ll}
  x_i^{n-1} &  \text{ if  } i \notin \mathcal{S}^n \\
  \left(f(x^{n-1})\right)_{S^n} & \text{ if }i \in \mathcal{S}^n.
\end{array}\right.
\label{general CIs}
\end{equation}

In other words, at the $n^{th}$ iteration, only the cells whose id is
contained into the set $S^{n}$ are iterated.

Let us now rewrite these general chaotic iterations as usual discrete dynamical
system of the form $X^{n+1}=f(X^n)$ on an ad hoc metric space. Such a formulation
is required in order to study the topological behavior of the system.

Let us introduce the following function:
\begin{equation}
\begin{array}{cccc}
 \chi: & \llbracket 1; \mathsf{N} \rrbracket \times \mathcal{P}\left(\llbracket 1; \mathsf{N} \rrbracket\right) & \longrightarrow & \mathds{B}\\
         & (i,X) & \longmapsto  & \left\{ \begin{array}{ll} 0 & \textrm{if }i \notin X, \\ 1 & \textrm{if }i \in X,  \end{array}\right.
\end{array} 
\end{equation}
where $\mathcal{P}\left(X\right)$ is for the powerset of the set $X$, that is, $Y \in \mathcal{P}\left(X\right) \Longleftrightarrow Y \subset X$.

Given a function $f:\mathds{B}^\mathsf{N} \longrightarrow \mathds{B}^\mathsf{N} $, define the function:
\begin{equation}
\begin{array}{lrll}
F_{f}: & \mathcal{P}\left(\llbracket1;\mathsf{N}\rrbracket \right) \times \mathds{B}^{\mathsf{N}} &
\longrightarrow & \mathds{B}^{\mathsf{N}} \\
& (P,E) & \longmapsto & \left( E_{j}.\chi (j,P)+f(E)_{j}.\overline{\chi
(j,P)}\right) _{j\in \llbracket1;\mathsf{N}\rrbracket},%
\end{array}%
\end{equation}%
where + and . are the Boolean addition and product operations, and $\overline{x}$ 
is the negation of the Boolean $x$.
Consider the phase space:
\begin{equation}
\mathcal{X} = \mathcal{P}\left(\llbracket 1 ; \mathsf{N} \rrbracket\right)^\mathds{N} \times
\mathds{B}^\mathsf{N},
\end{equation}
\noindent and the map defined on $\mathcal{X}$:
\begin{equation}
G_f\left(S,E\right) = \left(\sigma(S), F_f(i(S),E)\right), \label{Gf}
\end{equation}
\noindent where $\sigma$ is the \emph{shift} function defined by $\sigma
(S^{n})_{n\in \mathds{N}}\in \mathcal{P}\left(\llbracket 1 ; \mathsf{N} \rrbracket\right)^\mathds{N}\longrightarrow (S^{n+1})_{n\in
\mathds{N}}\in \mathcal{P}\left(\llbracket 1 ; \mathsf{N} \rrbracket\right)^\mathds{N}$ and $i$ is the \emph{initial function} 
$i:(S^{n})_{n\in \mathds{N}} \in \mathcal{P}\left(\llbracket 1 ; \mathsf{N} \rrbracket\right)^\mathds{N}\longrightarrow S^{0}\in \mathcal{P}\left(\llbracket 1 ; \mathsf{N} \rrbracket\right)$. 
Then the general chaotic iterations defined in Equation \ref{general CIs} can 
be described by the following discrete dynamical system:
\begin{equation}
\left\{
\begin{array}{l}
X^0 \in \mathcal{X} \\
X^{k+1}=G_{f}(X^k).%
\end{array}%
\right.
\end{equation}%

Another time, a shift function appears as a component of these general chaotic 
iterations. 

To study the Devaney's chaos property, a distance between two points 
$X = (S,E), Y = (\check{S},\check{E})$ of $\mathcal{X}$ must be defined.
Let us introduce:
\begin{equation}
d(X,Y)=d_{e}(E,\check{E})+d_{s}(S,\check{S}),
\label{nouveau d}
\end{equation}
\noindent where
\begin{equation}
\left\{
\begin{array}{lll}
\displaystyle{d_{e}(E,\check{E})} & = & \displaystyle{\sum_{k=1}^{\mathsf{N}%
}\delta (E_{k},\check{E}_{k})}\textrm{ is another time the Hamming distance}, \\
\displaystyle{d_{s}(S,\check{S})} & = & \displaystyle{\dfrac{9}{\mathsf{N}}%
\sum_{k=1}^{\infty }\dfrac{|S^k\Delta {S}^k|}{10^{k}}}.%
\end{array}%
\right.
\end{equation}
where $|X|$ is the cardinality of a set $X$ and $A\Delta B$ is for the symmetric difference, defined for sets A, B as
$A\,\Delta\,B = (A \setminus B) \cup (B \setminus A)$.


\begin{proposition}
The function $d$ defined in Eq.~\ref{nouveau d} is a metric on $\mathcal{X}$.
\end{proposition}

\begin{proof}
 $d_e$ is the Hamming distance. We will prove that $d_s$ is a distance
too, thus $d$ will be a distance as sum of two distances.
 \begin{itemize}
\item Obviously, $d_s(S,\check{S})\geqslant 0$, and if $S=\check{S}$, then 
$d_s(S,\check{S})=0$. Conversely, if $d_s(S,\check{S})=0$, then 
$\forall k \in \mathds{N}, |S^k\Delta {S}^k|=0$, and so $\forall k, S^k=\check{S}^k$.
 \item $d_s$ is symmetric 
($d_s(S,\check{S})=d_s(\check{S},S)$) due to the commutative property
of the symmetric difference. 
\item Finally, $|S \Delta S''| = |(S \Delta \varnothing) \Delta S''|= |S \Delta (S'\Delta S') \Delta S''|= |(S \Delta S') \Delta (S' \Delta S'')|\leqslant |S \Delta S'| + |S' \Delta S''|$, 
and so for all subsets $S,S',$ and $S''$ of $\llbracket 1, \mathsf{N} \rrbracket$, 
we have $d_s(S,S'') \leqslant d_e(S,S')+d_s(S',S'')$, and the triangle
inequality is obtained.
 \end{itemize}
\end{proof}


Before being able to study the topological behavior of the general 
chaotic iterations, we must firstly establish that:

\begin{proposition}
 For all $f:\mathds{B}^\mathsf{N} \longrightarrow \mathds{B}^\mathsf{N} $, the function $G_f$ is continuous on 
$\left( \mathcal{X},d\right)$.
\end{proposition}


\begin{proof}
We use the sequential continuity.
Let $(S^n,E^n)_{n\in \mathds{N}}$ be a sequence of the phase space $%
\mathcal{X}$, which converges to $(S,E)$. We will prove that $\left(
G_{f}(S^n,E^n)\right) _{n\in \mathds{N}}$ converges to $\left(
G_{f}(S,E)\right) $. Let us remark that for all $n$, $S^n$ is a strategy,
thus, we consider a sequence of strategies (\emph{i.e.}, a sequence of
sequences).\newline
As $d((S^n,E^n);(S,E))$ converges to 0, each distance $d_{e}(E^n,E)$ and $d_{s}(S^n,S)$ converges
to 0. But $d_{e}(E^n,E)$ is an integer, so $\exists n_{0}\in \mathds{N},$ $%
d_{e}(E^n,E)=0$ for any $n\geqslant n_{0}$.\newline
In other words, there exists a threshold $n_{0}\in \mathds{N}$ after which no
cell will change its state:
$\exists n_{0}\in \mathds{N},n\geqslant n_{0}\Rightarrow E^n = E.$

In addition, $d_{s}(S^n,S)\longrightarrow 0,$ so $\exists n_{1}\in %
\mathds{N},d_{s}(S^n,S)<10^{-1}$ for all indexes greater than or equal to $%
n_{1}$. This means that for $n\geqslant n_{1}$, all the $S^n$ have the same
first term, which is $S^0$: $\forall n\geqslant n_{1},S_0^n=S_0.$

Thus, after the $max(n_{0},n_{1})^{th}$ term, states of $E^n$ and $E$ are
identical and strategies $S^n$ and $S$ start with the same first term.\newline
Consequently, states of $G_{f}(S^n,E^n)$ and $G_{f}(S,E)$ are equal,
so, after the $max(n_0, n_1)^{th}$ term, the distance $d$ between these two points is strictly less than 1.\newline
\noindent We now prove that the distance between $\left(
G_{f}(S^n,E^n)\right) $ and $\left( G_{f}(S,E)\right) $ is convergent to
0. Let $\varepsilon >0$. \medskip
\begin{itemize}
\item If $\varepsilon \geqslant 1$, we see that distance
between $\left( G_{f}(S^n,E^n)\right) $ and $\left( G_{f}(S,E)\right) $ is
strictly less than 1 after the $max(n_{0},n_{1})^{th}$ term (same state).
\medskip
\item If $\varepsilon <1$, then $\exists k\in \mathds{N},10^{-k}\geqslant
\varepsilon > 10^{-(k+1)}$. But $d_{s}(S^n,S)$ converges to 0, so
\begin{equation*}
\exists n_{2}\in \mathds{N},\forall n\geqslant
n_{2},d_{s}(S^n,S)<10^{-(k+2)},
\end{equation*}%
thus after $n_{2}$, the $k+2$ first terms of $S^n$ and $S$ are equal.
\end{itemize}
\noindent As a consequence, the $k+1$ first entries of the strategies of $%
G_{f}(S^n,E^n)$ and $G_{f}(S,E)$ are the same ($G_{f}$ is a shift of strategies) and due to the definition of $d_{s}$, the floating part of
the distance between $(S^n,E^n)$ and $(S,E)$ is strictly less than $%
10^{-(k+1)}\leqslant \varepsilon $.\bigskip \newline
In conclusion,
$$
\forall \varepsilon >0,\exists N_{0}=max(n_{0},n_{1},n_{2})\in \mathds{N}%
,\forall n\geqslant N_{0},
 d\left( G_{f}(S^n,E^n);G_{f}(S,E)\right)
\leqslant \varepsilon .
$$
$G_{f}$ is consequently continuous.
\end{proof}


It is now possible to study the topological behavior of the general chaotic
iterations. We will prove that,

\begin{theorem}
\label{t:chaos des general}
 The general chaotic iterations defined on Equation~\ref{general CIs} satisfy
the Devaney's property of chaos.
\end{theorem}

Let us firstly prove the following lemma.

\begin{lemma}[Strong transitivity]
\label{strongTrans}
 For all couples $X,Y \in \mathcal{X}$ and any neighborhood $V$ of $X$, we can 
find $n \in \mathds{N}^*$ and $X' \in V$ such that $G^n(X')=Y$.
\end{lemma}

\begin{proof}
 Let $X=(S,E)$, $\varepsilon>0$, and $k_0 = \lfloor log_{10}(\varepsilon)+1 \rfloor$. 
Any point $X'=(S',E')$ such that $E'=E$ and $\forall k \leqslant k_0, S'^k=S^k$, 
are in the open ball $\mathcal{B}\left(X,\varepsilon\right)$. Let us define 
$\check{X} = \left(\check{S},\check{E}\right)$, where $\check{X}= G^{k_0}(X)$.
We denote by $s\subset \llbracket 1; \mathsf{N} \rrbracket$ the set of coordinates
that are different between $\check{E}$ and the state of $Y$. Thus each point $X'$ of
the form $(S',E')$ where $E'=E$ and $S'$ starts with 
$(S^0, S^1, \hdots, S^{k_0},s,\hdots)$, verifies the following properties:
\begin{itemize}
 \item $X'$ is in $\mathcal{B}\left(X,\varepsilon\right)$,
 \item the state of $G_f^{k_0+1}(X')$ is the state of $Y$.
\end{itemize}
Finally the point $\left(\left(S^0, S^1, \hdots, S^{k_0},s,s^0, s^1, \hdots\right); E\right)$, 
where $(s^0,s^1, \hdots)$ is the strategy of $Y$, satisfies the properties
claimed in the lemma.
\end{proof}

We can now prove the Theorem~\ref{t:chaos des general}...

\begin{proof}[Theorem~\ref{t:chaos des general}]
Firstly, strong transitivity implies transitivity.

Let $(S,E) \in\mathcal{X}$ and $\varepsilon >0$. To
prove that $G_f$ is regular, it is sufficient to prove that
there exists a strategy $\tilde S$ such that the distance between
$(\tilde S,E)$ and $(S,E)$ is less than $\varepsilon$, and such that
$(\tilde S,E)$ is a periodic point.

Let $t_1=\lfloor-\log_{10}(\varepsilon)\rfloor$, and let $E'$ be the
configuration that we obtain from $(S,E)$ after $t_1$ iterations of
$G_f$. As $G_f$ is strongly transitive, there exists a strategy $S'$ 
and $t_2\in\mathds{N}$ such
that $E$ is reached from $(S',E')$ after $t_2$ iterations of $G_f$.

Consider the strategy $\tilde S$ that alternates the first $t_1$ terms
of $S$ and the first $t_2$ terms of $S'$: $$\tilde
S=(S_0,\dots,S_{t_1-1},S'_0,\dots,S'_{t_2-1},S_0,\dots,S_{t_1-1},S'_0,\dots,S'_{t_2-1},S_0,\dots).$$ It
is clear that $(\tilde S,E)$ is obtained from $(\tilde S,E)$ after
$t_1+t_2$ iterations of $G_f$. So $(\tilde S,E)$ is a periodic
point. Since $\tilde S_t=S_t$ for $t<t_1$, by the choice of $t_1$, we
have $d((S,E),(\tilde S,E))<\epsilon$.
\end{proof}



\section{Efficient PRNG based on Chaotic Iterations}
\label{sec:efficient prng}

In  order to  implement efficiently  a PRNG  based on  chaotic iterations  it is
possible to improve  previous works [ref]. One solution  consists in considering
that the  strategy used contains all the  bits for which the  negation is
achieved out. Then in order to apply  the negation on these bits we can simply
apply the  xor operator between  the current number  and the strategy. In
order to obtain the strategy we also use a classical PRNG.

Here  is an  example with  16-bits numbers  showing how  the bitwise  operations
are
applied.  Suppose  that $x$ and the  strategy $S^i$ are defined  in binary mode.
Then the following table shows the result of $x$ xor $S^i$.
$$
\begin{array}{|cc|cccccccccccccccc|}
\hline
x      &=&1&0&1&1&1&0&1&0&1&0&0&1&0&0&1&0\\
\hline
S^i      &=&0&1&1&0&0&1&1&0&1&1&1&0&0&1&1&1\\
\hline
x \oplus S^i&=&1&1&0&1&1&1&0&0&0&1&1&1&0&1&0&1\\
\hline

\hline
 \end{array}
$$

%% \begin{figure}[htbp]
%% \begin{center}
%% \fbox{
%% \begin{minipage}{14cm}
%% unsigned int CIprng() \{\\
%%   static unsigned int x = 123123123;\\
%%   unsigned long t1 = xorshift();\\
%%   unsigned long t2 = xor128();\\
%%   unsigned long t3 = xorwow();\\
%%   x = x\textasciicircum (unsigned int)t1;\\
%%   x = x\textasciicircum (unsigned int)(t2$>>$32);\\
%%   x = x\textasciicircum (unsigned int)(t3$>>$32);\\
%%   x = x\textasciicircum (unsigned int)t2;\\
%%   x = x\textasciicircum (unsigned int)(t1$>>$32);\\
%%   x = x\textasciicircum (unsigned int)t3;\\
%%   return x;\\
%% \}
%% \end{minipage}
%% }
%% \end{center}
%% \caption{sequential Chaotic Iteration PRNG}
%% \label{algo:seqCIprng}
%% \end{figure}



\lstset{language=C,caption={C code of the sequential chaotic iterations based
PRNG},label=algo:seqCIprng}
\begin{lstlisting}
unsigned int CIprng() {
  static unsigned int x = 123123123;
  unsigned long t1 = xorshift();
  unsigned long t2 = xor128();
  unsigned long t3 = xorwow();
  x = x^(unsigned int)t1;
  x = x^(unsigned int)(t2>>32);
  x = x^(unsigned int)(t3>>32);
  x = x^(unsigned int)t2;
  x = x^(unsigned int)(t1>>32);
  x = x^(unsigned int)t3;
  return x;
}
\end{lstlisting}





In listing~\ref{algo:seqCIprng}  a sequential version of  our chaotic iterations
based PRNG is  presented.  The xor operator is  represented by \textasciicircum.
This  function uses  three classical  64-bits PRNG:  the  \texttt{xorshift}, the
\texttt{xor128}  and  the  \texttt{xorwow}.   In  the following,  we  call  them
xor-like PRNGSs.   These three PRNGs are  presented in~\cite{Marsaglia2003}.  As
each xor-like PRNG  used works with 64-bits and as our  PRNG works with 32-bits,
the use of \texttt{(unsigned int)} selects the 32 least significant bits whereas
\texttt{(unsigned int)(t3$>>$32)}  selects the 32 most significants  bits of the
variable \texttt{t}.   So to produce a  random number realizes  6 xor operations
with 6 32-bits  numbers produced by 3 64-bits PRNG.   This version successes the
BigCrush of the TestU01 battery~\cite{LEcuyerS07}.

\section{Efficient PRNGs based on chaotic iterations on GPU}
\label{sec:efficient prng gpu}

In  order to benefit  from computing  power of  GPU, a  program needs  to define
independent blocks of threads which  can be computed simultaneously. In general,
the larger the number of threads is,  the more local memory is used and the less
branching  instructions are  used (if,  while, ...),  the better  performance is
obtained  on  GPU.  So  with  algorithm  \ref{algo:seqCIprng}  presented in  the
previous section, it is possible to  build a similar program which computes PRNG
on   GPU.  In  the   CUDA~\cite{Nvid10}  environment,   threads  have   a  local
identificator, called \texttt{ThreadIdx} relative to the block containing them.


\subsection{Naive version for GPU}

From the CPU version, it is possible  to obtain a quite similar version for GPU.
The principe consists in assigning the computation of a PRNG as in sequential to
each thread  of the  GPU.  Of course,  it is  essential that the  three xor-like
PRNGs  used for  our computation  have different  parameters. So  we  chose them
randomly with  another PRNG. As the  initialisation is performed by  the CPU, we
have  chosen  to  use  the  ISAAC  PRNG~\ref{Jenkins96}  to  initalize  all  the
parameters for  the GPU version  of our PRNG.   The implementation of  the three
xor-like  PRNGs  is  straightforward  as  soon as  their  parameters  have  been
allocated in  the GPU memory.  Each xor-like PRNGs  used works with  an internal
number  $x$  which keeps  the  last  generated  random numbers.  Other  internal
variables  are   also  used   by  the  xor-like   PRNGs.  More   precisely,  the
implementation of the  xor128, the xorshift and the  xorwow respectively require
4, 5 and 6 unsigned long as internal variables.

\begin{algorithm}

\KwIn{InternalVarXorLikeArray: array with internal variables of the 3 xor-like
PRNGs in global memory\;
NumThreads: Number of threads\;}
\KwOut{NewNb: array containing random numbers in global memory}
\If{threadIdx is concerned by the computation} {
  retrieve data from InternalVarXorLikeArray[threadIdx] in local variables\;
  \For{i=1 to n} {
    compute a new PRNG as in Listing\ref{algo:seqCIprng}\;
    store the new PRNG in NewNb[NumThreads*threadIdx+i]\;
  }
  store internal variables in InternalVarXorLikeArray[threadIdx]\;
}

\caption{main kernel for the chaotic iterations based PRNG GPU naive version}
\label{algo:gpu_kernel}
\end{algorithm}

Algorithm~\ref{algo:gpu_kernel}  presents a naive  implementation of  PRNG using
GPU.  According  to the available  memory in the  GPU and the number  of threads
used simultenaously,  the number  of random numbers  that a thread  can generate
inside   a    kernel   is   limited,   i.e.    the    variable   \texttt{n}   in
algorithm~\ref{algo:gpu_kernel}. For example, if  $100,000$ threads are used and
if $n=100$\footnote{in fact, we need to add the initial seed (a 32-bits number)}
then   the  memory   required   to  store   internals   variables  of   xor-like
PRNGs\footnote{we multiply this number by $2$ in order to count 32-bits numbers}
and  random  number of  our  PRNG  is  equals to  $100,000\times  ((4+5+6)\times
2+(1+100))=1,310,000$ 32-bits numbers, i.e. about $52$Mb.

All the  tests performed  to pass the  BigCrush of TestU01  succeeded. Different
number of threads, called \texttt{NumThreads} in our algorithm, have been tested
upto $10$ millions.

\begin{remark}
Algorithm~\ref{algo:gpu_kernel}  has  the  advantage to  manipulate  independent
PRNGs, so this version is easily usable on a cluster of computer. The only thing
to ensure is to use a single ISAAC PRNG. For this, a simple solution consists in
using a master node for the initialization which computes the initial parameters
for all the differents nodes involves in the computation.
\end{remark}

\subsection{Improved version for GPU}

As GPU cards using CUDA have shared memory between threads of the same block, it
is possible  to use this  feature in order  to simplify the  previous algorithm,
i.e., using less  than 3 xor-like PRNGs. The solution  consists in computing only
one xor-like PRNG by thread, saving  it into shared memory and using the results
of some  other threads in the  same block of  threads. In order to  define which
thread uses the result of which other  one, we can use a permutation array which
contains  the indexes  of  all threads  and  for which  a  permutation has  been
performed.  In Algorithm~\ref{algo:gpu_kernel2}, 2 permutations arrays are used.
The    variable   \texttt{offset}    is    computed   using    the   value    of
\texttt{permutation\_size}.   Then we  can compute  \texttt{o1}  and \texttt{o2}
which represent the indexes of the  other threads for which the results are used
by the  current thread. In  the algorithm, we  consider that a  64-bits xor-like
PRNG is used, that is why both 32-bits parts are used.

This version also succeeds to the {\it BigCrush} batteries of tests.

\begin{algorithm}

\KwIn{InternalVarXorLikeArray: array with internal variables of 1 xor-like PRNGs
in global memory\;
NumThreads: Number of threads\;
tab1, tab2: Arrays containing permutations of size permutation\_size\;}

\KwOut{NewNb: array containing random numbers in global memory}
\If{threadId is concerned} {
  retrieve data from InternalVarXorLikeArray[threadId] in local variables\;
  offset = threadIdx\%permutation\_size\;
  o1 = threadIdx-offset+tab1[offset]\;
  o2 = threadIdx-offset+tab2[offset]\;
  \For{i=1 to n} {
    t=xor-like()\;
    shared\_mem[threadId]=(unsigned int)t\;
    x = x $\oplus$ (unsigned int) t\;
    x = x $\oplus$ (unsigned int) (t>>32)\;
    x = x $\oplus$ shared[o1]\;
    x = x $\oplus$ shared[o2]\;

    store the new PRNG in NewNb[NumThreads*threadId+i]\;
  }
  store internal variables in InternalVarXorLikeArray[threadId]\;
}

\caption{main kernel for the chaotic iterations based PRNG GPU efficient
version}
\label{algo:gpu_kernel2}
\end{algorithm}

\subsection{Theoretical Evaluation of the Improved Version}

A run of Algorithm~\ref{algo:gpu_kernel2} consists in four operations having 
the form of Equation~\ref{equation Oplus}, which is equivalent to the iterative
system of Eq.~\ref{eq:generalIC}. That is, four iterations of the general chaotic
iterations are realized between two stored values of the PRNG.
To be certain that we are in the framework of Theorem~\ref{t:chaos des general},
we must guarantee that this dynamical system iterates on the space 
$\mathcal{X} = \mathcal{P}\left(\llbracket 1, \mathsf{N} \rrbracket\right)^\mathds{N}\times\mathds{B}^\mathsf{N}$.
The left term $x$ obviously belongs into $\mathds{B}^ \mathsf{N}$.
To prevent from any flaws of chaotic properties, we must check that each right 
term, corresponding to terms of the strategies,  can possibly be equal to any
integer of $\llbracket 1, \mathsf{N} \rrbracket$. 

Such a result is obvious for the two first lines, as for the xor-like(), all the
integers belonging into its interval of definition can occur at each iteration.
It can be easily stated for the two last lines by an immediate mathematical
induction.

Thus Algorithm~\ref{algo:gpu_kernel2} is a concrete realization of the general
chaotic iterations presented previously, and for this reason, it satisfies the 
Devaney's formulation of a chaotic behavior.

\section{Experiments}
\label{sec:experiments}

Different experiments  have been  performed in order  to measure  the generation
speed. We have used  a computer equiped with Tesla C1060 NVidia  GPU card and an
Intel  Xeon E5530 cadenced  at 2.40  GHz for  our experiments  and we  have used
another one  equipped with  a less performant  CPU and  a GeForce GTX  280. Both
cards have 240 cores.

In Figure~\ref{fig:time_gpu}  we compare the number of  random numbers generated
per second.  In order to obtain the optimal performance we remove the storage of
random numbers  in the GPU memory. This  step is time consumming  and slows down
the random number  generation.  Moreover, if you are  interested by applications
that consome random  numbers directly when they are  generated, their storage is
completely useless. In this figure we can see that when the number of threads is
greater than approximately  30,000 upto 5 millions the  number of random numbers
generated per second is almost constant.   With the naive version, it is between
2.5 and 3GSample/s.   With the optimized version, it  is approximately equals to
20GSample/s. Finally  we can remark  that both GPU  cards are quite  similar. In
practice,  the Tesla C1060  has more  memory than  the GTX  280 and  this memory
should be of better quality.

\begin{figure}[htbp]
\begin{center}
  \includegraphics[scale=.7]{curve_time_gpu.pdf}
\end{center}
\caption{Number of random numbers generated per second}
\label{fig:time_gpu}
\end{figure}


In  comparison,   Listing~\ref{algo:seqCIprng}  allows  us   to  generate  about
138MSample/s with only one core of the Xeon E5530.





\section{The relativity of disorder}
\label{sec:de la relativité du désordre}

In the next two sections, we investigate the impact of the choices that have
lead to the definitions of measures in Sections \ref{sec:chaotic iterations} and \ref{deuxième def}.

\subsection{Impact of the topology's finenesse}

Let us firstly introduce the following notations.

\begin{notation}
$\mathcal{X}_\tau$ will denote the topological space
$\left(\mathcal{X},\tau\right)$, whereas $\mathcal{V}_\tau (x)$ will be the set
of all the neighborhoods of $x$ when considering the topology $\tau$ (or simply
$\mathcal{V} (x)$, if there is no ambiguity).
\end{notation}



\begin{theorem}
\label{Th:chaos et finesse}
Let $\mathcal{X}$ a set and $\tau, \tau'$ two topologies on $\mathcal{X}$ s.t.
$\tau'$ is finer than $\tau$. Let $f:\mathcal{X} \to \mathcal{X}$, continuous
both for $\tau$ and $\tau'$.

If $(\mathcal{X}_{\tau'},f)$ is chaotic according to Devaney, then
$(\mathcal{X}_\tau,f)$ is chaotic too.
\end{theorem}

\begin{proof}
Let us firstly establish the transitivity of $(\mathcal{X}_\tau,f)$.

Let $\omega_1, \omega_2$ two open sets of $\tau$. Then $\omega_1, \omega_2 \in
\tau'$, becaus $\tau'$ is finer than $\tau$. As $f$ is $\tau'-$transitive, we
can deduce that $\exists n \in \mathds{N}, \omega_1 \cap f^{(n)}(\omega_2) =
\varnothing$. Consequently, $f$ is $\tau-$transitive.

Let us now consider the regularity of $(\mathcal{X}_\tau,f)$, \emph{i.e.}, for
all $x \in \mathcal{X}$, and for all $\tau-$neighborhood $V$ of $x$, there is a
periodic point for $f$ into $V$.

Let $x \in \mathcal{X}$ and $V \in \mathcal{V}_\tau (x)$ a $\tau-$neighborhood
of $x$. By definition, $\exists \omega \in \tau, x \in \omega \subset V$.

But $\tau \subset \tau'$, so $\omega \in \tau'$, and then $V \in
\mathcal{V}_{\tau'} (x)$. As $(\mathcal{X}_{\tau'},f)$ is regular, there is a
periodic point for $f$ into $V$, and the regularity of $(\mathcal{X}_\tau,f)$ is
proven. 
\end{proof}

\subsection{A given system can always be claimed as chaotic}

Let $f$ an iteration function on $\mathcal{X}$ having at least a fixed point.
Then this function is chaotic (in a certain way):

\begin{theorem}
Let $\mathcal{X}$ a nonempty set and $f: \mathcal{X} \to \X$ a function having
at least a fixed point.
Then $f$ is $\tau_0-$chaotic, where $\tau_0$ is the trivial (indiscrete)
topology on $\X$.
\end{theorem}


\begin{proof}
$f$ is transitive when $\forall \omega, \omega' \in \tau_0 \setminus
\{\varnothing\}, \exists n \in \mathds{N}, f^{(n)}(\omega) \cap \omega' \neq
\varnothing$.
As $\tau_0 = \left\{ \varnothing, \X \right\}$, this is equivalent to look for
an integer $n$ s.t. $f^{(n)}\left( \X \right) \cap \X \neq \varnothing$. For
instance, $n=0$ is appropriate.

Let us now consider $x \in \X$ and $V \in \mathcal{V}_{\tau_0} (x)$. Then $V =
\mathcal{X}$, so $V$ has at least a fixed point for $f$. Consequently $f$ is
regular, and the result is established.
\end{proof}




\subsection{A given system can always be claimed as non-chaotic}

\begin{theorem}
Let $\mathcal{X}$ be a set and $f: \mathcal{X} \to \X$.
If $\X$ is infinite, then $\left( \X_{\tau_\infty}, f\right)$ is not chaotic
(for the Devaney's formulation), where $\tau_\infty$ is the discrete topology.
\end{theorem}

\begin{proof}
Let us prove it by contradiction, assuming that $\left(\X_{\tau_\infty},
f\right)$ is both transitive and regular.

Let $x \in \X$ and $\{x\}$ one of its neighborhood. This neighborhood must
contain a periodic point for $f$, if we want that $\left(\X_{\tau_\infty},
f\right)$ is regular. Then $x$ must be a periodic point of $f$.

Let $I_x = \left\{ f^{(n)}(x), n \in \mathds{N}\right\}$. This set is finite
because  $x$ is periodic, and $\mathcal{X}$ is infinite, then $\exists y \in
\mathcal{X}, y \notin I_x$.

As $\left(\X_{\tau_\infty}, f\right)$ must be transitive, for all open nonempty
sets $A$ and $B$, an integer $n$ must satisfy $f^{(n)}(A) \cap B \neq
\varnothing$. However $\{x\}$ and $\{y\}$ are open sets and $y \notin I_x
\Rightarrow \forall n, f^{(n)}\left( \{x\} \right) \cap \{y\} = \varnothing$.
\end{proof}






\section{Chaos on the order topology}
\label{sec: chaos order topology}
\subsection{The phase space is an interval of the real line}

\subsubsection{Toward a topological semiconjugacy}

In what follows, our intention is to establish, by using a topological
semiconjugacy, that chaotic iterations over $\mathcal{X}$ can be described as
iterations on a real interval. To do so, we must firstly introduce some
notations and terminologies. 

Let $\mathcal{S}_\mathsf{N}$ be the set of sequences belonging into $\llbracket
1; \mathsf{N}\rrbracket$ and $\mathcal{X}_{\mathsf{N}} = \mathcal{S}_\mathsf{N}
\times \B^\mathsf{N}$.


\begin{definition}
The function $\varphi: \mathcal{S}_{10} \times\mathds{B}^{10} \rightarrow \big[
0, 2^{10} \big[$ is defined by:
\begin{equation}
 \begin{array}{cccl}
\varphi: & \mathcal{X}_{10} = \mathcal{S}_{10} \times\mathds{B}^{10}&
\longrightarrow & \big[ 0, 2^{10} \big[ \\
 & (S,E) = \left((S^0, S^1, \hdots ); (E_0, \hdots, E_9)\right) & \longmapsto &
\varphi \left((S,E)\right)
\end{array}
\end{equation}
where $\varphi\left((S,E)\right)$ is the real number:
\begin{itemize}
\item whose integral part $e$ is $\displaystyle{\sum_{k=0}^9 2^{9-k} E_k}$, that
is, the binary digits of $e$ are $E_0 ~ E_1 ~ \hdots ~ E_9$.
\item whose decimal part $s$ is equal to $s = 0,S^0~ S^1~ S^2~ \hdots =
\sum_{k=1}^{+\infty} 10^{-k} S^{k-1}.$ 
\end{itemize}
\end{definition}



$\varphi$ realizes the association between a point of $\mathcal{X}_{10}$ and a
real number into $\big[ 0, 2^{10} \big[$. We must now translate the chaotic
iterations $\Go$ on this real interval. To do so, two intermediate functions
over $\big[ 0, 2^{10} \big[$ must be introduced:


\begin{definition}
\label{def:e et s}
Let $x \in \big[ 0, 2^{10} \big[$ and:
\begin{itemize}
\item $e_0, \hdots, e_9$ the binary digits of the integral part of $x$:
$\displaystyle{\lfloor x \rfloor = \sum_{k=0}^{9} 2^{9-k} e_k}$.
\item $(s^k)_{k\in \mathds{N}}$ the digits of $x$, where the chosen decimal
decomposition of $x$ is the one that does not have an infinite number of 9: 
$\displaystyle{x = \lfloor x \rfloor + \sum_{k=0}^{+\infty} s^k 10^{-k-1}}$.
\end{itemize}
$e$ and $s$ are thus defined as follows:
\begin{equation}
\begin{array}{cccl}
e: & \big[ 0, 2^{10} \big[ & \longrightarrow & \mathds{B}^{10} \\
 & x & \longmapsto & (e_0, \hdots, e_9)
\end{array}
\end{equation}
and
\begin{equation}
 \begin{array}{cccc}
s: & \big[ 0, 2^{10} \big[ & \longrightarrow & \llbracket 0, 9
\rrbracket^{\mathds{N}} \\
 & x & \longmapsto & (s^k)_{k \in \mathds{N}}
\end{array}
\end{equation}
\end{definition}

We are now able to define the function $g$, whose goal is to translate the
chaotic iterations $\Go$ on an interval of $\mathds{R}$.

\begin{definition}
$g:\big[ 0, 2^{10} \big[ \longrightarrow \big[ 0, 2^{10} \big[$ is defined by:
\begin{equation}
\begin{array}{cccc}
g: & \big[ 0, 2^{10} \big[ & \longrightarrow & \big[ 0, 2^{10} \big[ \\
 & x & \longmapsto & g(x)
\end{array}
\end{equation}
where g(x) is the real number of $\big[ 0, 2^{10} \big[$ defined bellow:
\begin{itemize}
\item its integral part has a binary decomposition equal to $e_0', \hdots,
e_9'$, with:
 \begin{equation}
e_i' = \left\{
\begin{array}{ll}
e(x)_i & \textrm{ if } i \neq s^0\\
e(x)_i + 1 \textrm{ (mod 2)} & \textrm{ if } i = s^0\\
\end{array}
\right.
\end{equation}
\item whose decimal part is $s(x)^1, s(x)^2, \hdots$
\end{itemize}
\end{definition}

\bigskip


In other words, if $x = \displaystyle{\sum_{k=0}^{9} 2^{9-k} e_k + 
\sum_{k=0}^{+\infty} s^{k} ~10^{-k-1}}$, then:
\begin{equation}
g(x) =
\displaystyle{\sum_{k=0}^{9} 2^{9-k} (e_k + \delta(k,s^0) \textrm{ (mod 2)}) + 
\sum_{k=0}^{+\infty} s^{k+1} 10^{-k-1}}. 
\end{equation}


\subsubsection{Defining a metric on $\big[ 0, 2^{10} \big[$}

Numerous metrics can be defined on the set $\big[ 0, 2^{10} \big[$, the most
usual one being the Euclidian distance recalled bellow:

\begin{notation}
\index{distance!euclidienne}
$\Delta$ is the Euclidian distance on $\big[ 0, 2^{10} \big[$, that is,
$\Delta(x,y) = |y-x|^2$.
\end{notation}

\medskip

This Euclidian distance does not reproduce exactly the notion of proximity
induced by our first distance $d$ on $\X$. Indeed $d$ is finer than $\Delta$.
This is the reason why we have to introduce the following metric:



\begin{definition}
Let $x,y \in \big[ 0, 2^{10} \big[$.
$D$ denotes the function from $\big[ 0, 2^{10} \big[^2$ to $\mathds{R}^+$
defined by: $D(x,y) = D_e\left(e(x),e(y)\right) + D_s\left(s(x),s(y)\right)$,
where:
\begin{center}
$\displaystyle{D_e(E,\check{E}) = \sum_{k=0}^\mathsf{9} \delta (E_k,
\check{E}_k)}$, ~~and~ $\displaystyle{D_s(S,\check{S}) = \sum_{k = 1}^\infty
\dfrac{|S^k-\check{S}^k|}{10^k}}$.
\end{center}
\end{definition}

\begin{proposition}
$D$ is a distance on $\big[ 0, 2^{10} \big[$.
\end{proposition}

\begin{proof}
The three axioms defining a distance must be checked.
\begin{itemize}
\item $D \geqslant 0$, because everything is positive in its definition. If
$D(x,y)=0$, then $D_e(x,y)=0$, so the integral parts of $x$ and $y$ are equal
(they have the same binary decomposition). Additionally, $D_s(x,y) = 0$, then
$\forall k \in \mathds{N}^*, s(x)^k = s(y)^k$. In other words, $x$ and $y$ have
the same $k-$th decimal digit, $\forall k \in \mathds{N}^*$. And so $x=y$.
\item $D(x,y)=D(y,x)$.
\item Finally, the triangular inequality is obtained due to the fact that both
$\delta$ and $\Delta(x,y)=|x-y|$ satisfy it.
\end{itemize}
\end{proof}


The convergence of sequences according to $D$ is not the same than the usual
convergence related to the Euclidian metric. For instance, if $x^n \to x$
according to $D$, then necessarily the integral part of each $x^n$ is equal to
the integral part of $x$ (at least after a given threshold), and the decimal
part of $x^n$ corresponds to the one of $x$ ``as far as required''.
To illustrate this fact, a comparison between $D$ and the Euclidian distance is
given Figure \ref{fig:comparaison de distances}. These illustrations show that
$D$ is richer and more refined than the Euclidian distance, and thus is more
precise.


\begin{figure}[t]
\begin{center}
  \subfigure[Function $x \to dist(x;1,234) $ on the interval
$(0;5)$.]{\includegraphics[scale=.35]{DvsEuclidien.pdf}}\quad
  \subfigure[Function $x \to dist(x;3) $ on the interval
$(0;5)$.]{\includegraphics[scale=.35]{DvsEuclidien2.pdf}}
\end{center}
\caption{Comparison between $D$ (in blue) and the Euclidian distane (in green).}
\label{fig:comparaison de distances}
\end{figure}




\subsubsection{The semiconjugacy}

It is now possible to define a topological semiconjugacy between $\mathcal{X}$
and an interval of $\mathds{R}$:

\begin{theorem}
Chaotic iterations on the phase space $\mathcal{X}$ are simple iterations on
$\mathds{R}$, which is illustrated by the semiconjugacy of the diagram bellow:
\begin{equation*}
\begin{CD}
\left(~\mathcal{S}_{10} \times\mathds{B}^{10}, d~\right) @>G_{f_0}>>
\left(~\mathcal{S}_{10} \times\mathds{B}^{10}, d~\right)\\
    @V{\varphi}VV                    @VV{\varphi}V\\
\left( ~\big[ 0, 2^{10} \big[, D~\right)  @>>g> \left(~\big[ 0, 2^{10} \big[,
D~\right)
\end{CD}
\end{equation*}
\end{theorem}

\begin{proof}
$\varphi$ has been constructed in order to be continuous and onto.
\end{proof}

In other words, $\mathcal{X}$ is approximately equal to $\big[ 0, 2^\mathsf{N}
\big[$.






\subsection{Study of the chaotic iterations described as a real function}


\begin{figure}[t]
\begin{center}
  \subfigure[ICs on the interval
$(0,9;1)$.]{\includegraphics[scale=.35]{ICs09a1.pdf}}\quad
  \subfigure[ICs on the interval
$(0,7;1)$.]{\includegraphics[scale=.35]{ICs07a95.pdf}}\\
  \subfigure[ICs on the interval
$(0,5;1)$.]{\includegraphics[scale=.35]{ICs05a1.pdf}}\quad
  \subfigure[ICs on the interval
$(0;1)$]{\includegraphics[scale=.35]{ICs0a1.pdf}}
\end{center}
\caption{Representation of the chaotic iterations.}
\label{fig:ICs}
\end{figure}




\begin{figure}[t]
\begin{center}
  \subfigure[ICs on the interval
$(510;514)$.]{\includegraphics[scale=.35]{ICs510a514.pdf}}\quad
  \subfigure[ICs on the interval
$(1000;1008)$]{\includegraphics[scale=.35]{ICs1000a1008.pdf}}
\end{center}
\caption{ICs on small intervals.}
\label{fig:ICs2}
\end{figure}

\begin{figure}[t]
\begin{center}
  \subfigure[ICs on the interval
$(0;16)$.]{\includegraphics[scale=.3]{ICs0a16.pdf}}\quad
  \subfigure[ICs on the interval 
$(40;70)$.]{\includegraphics[scale=.45]{ICs40a70.pdf}}\quad
\end{center}
\caption{General aspect of the chaotic iterations.}
\label{fig:ICs3}
\end{figure}


We have written a Python program to represent the chaotic iterations with the
vectorial negation on the real line $\mathds{R}$. Various representations of
these CIs are given in Figures \ref{fig:ICs}, \ref{fig:ICs2} and \ref{fig:ICs3}.
It can be remarked that the function $g$ is a piecewise linear function: it is
linear on each interval having the form $\left[ \dfrac{n}{10},
\dfrac{n+1}{10}\right[$, $n \in \llbracket 0;2^{10}\times 10 \rrbracket$ and its
slope is equal to 10. Let us justify these claims:

\begin{proposition}
\label{Prop:derivabilite des ICs}
Chaotic iterations $g$ defined on $\mathds{R}$ have derivatives of all orders on
$\big[ 0, 2^{10} \big[$, except on the 10241 points in $I$ defined by $\left\{
\dfrac{n}{10} ~\big/~ n \in \llbracket 0;2^{10}\times 10\rrbracket \right\}$.

Furthermore, on each interval of the form $\left[ \dfrac{n}{10},
\dfrac{n+1}{10}\right[$, with $n \in \llbracket 0;2^{10}\times 10 \rrbracket$,
$g$ is a linear function, having a slope equal to 10: $\forall x \notin I,
g'(x)=10$.
\end{proposition}


\begin{proof}
Let $I_n = \left[ \dfrac{n}{10}, \dfrac{n+1}{10}\right[$, with $n \in \llbracket
0;2^{10}\times 10 \rrbracket$. All the points of $I_n$ have the same integral
prat $e$ and the same decimal part $s^0$: on the set $I_n$,  functions $e(x)$
and $x \mapsto s(x)^0$ of Definition \ref{def:e et s} only depend on $n$. So all
the images $g(x)$ of these points $x$:
\begin{itemize}
\item Have the same integral part, which is $e$, except probably the bit number
$s^0$. In other words, this integer has approximately the same binary
decomposition than $e$, the sole exception being the digit $s^0$ (this number is
then either $e+2^{10-s^0}$ or $e-2^{10-s^0}$, depending on the parity of $s^0$,
\emph{i.e.}, it is equal to $e+(-1)^{s^0}\times 2^{10-s^0}$).
\item A shift to the left has been applied to the decimal part $y$, losing by
doing so the common first digit $s^0$. In other words, $y$ has been mapped into
$10\times y - s^0$.
\end{itemize}
To sum up, the action of $g$ on the points of $I$ is as follows: first, make a
multiplication by 10, and second, add the same constant to each term, which is
$\dfrac{1}{10}\left(e+(-1)^{s^0}\times 2^{10-s^0}\right)-s^0$.
\end{proof}

\begin{remark}
Finally, chaotic iterations are elements of the large family of functions that
are both chaotic and piecewise linear (like the tent map).
\end{remark}



\subsection{Comparison of the two metrics on $\big[ 0, 2^\mathsf{N} \big[$}

The two propositions bellow allow to compare our two distances on $\big[ 0,
2^\mathsf{N} \big[$:

\begin{proposition}
Id: $\left(~\big[ 0, 2^\mathsf{N} \big[,\Delta~\right) \to \left(~\big[ 0,
2^\mathsf{N} \big[, D~\right)$ is not continuous. 
\end{proposition}

\begin{proof}
The sequence $x^n = 1,999\hdots 999$ constituted by $n$ 9 as decimal part, is
such that:
\begin{itemize}
\item $\Delta (x^n,2) \to 0.$
\item But $D(x^n,2) \geqslant 1$, then $D(x^n,2)$ does not converge to 0.
\end{itemize}

The sequential characterization of the continuity concludes the demonstration.
\end{proof}



A contrario:

\begin{proposition}
Id: $\left(~\big[ 0, 2^\mathsf{N} \big[,D~\right) \to \left(~\big[ 0,
2^\mathsf{N} \big[, \Delta ~\right)$ is a continuous fonction. 
\end{proposition}

\begin{proof}
If $D(x^n,x) \to 0$, then $D_e(x^n,x) = 0$ at least for $n$ larger than a given
threshold, because $D_e$ only returns integers. So, after this threshold, the
integral parts of all the $x^n$ are equal to the integral part of $x$. 

Additionally, $D_s(x^n, x) \to 0$, then $\forall k \in \mathds{N}^*, \exists N_k
\in \mathds{N}, n \geqslant N_k \Rightarrow D_s(x^n,x) \leqslant 10^{-k}$. This
means that for all $k$, an index $N_k$ can be found such that, $\forall n
\geqslant N_k$, all the $x^n$ have the same $k$ firsts digits, which are the
digits of $x$. We can deduce the convergence $\Delta(x^n,x) \to 0$, and thus the
result.
\end{proof}

The conclusion of these propositions is that the proposed metric is more precise
than the Euclidian distance, that is:

\begin{corollary}
$D$ is finer than the Euclidian distance $\Delta$.
\end{corollary}

This corollary can be reformulated as follows:

\begin{itemize}
\item The topology produced by $\Delta$ is a subset of the topology produced by
$D$.
\item $D$ has more open sets than $\Delta$.
\item It is harder to converge for the topology $\tau_D$ inherited by $D$, than
to converge with the one inherited by $\Delta$, which is denoted here by
$\tau_\Delta$.
\end{itemize}


\subsection{Chaos of the chaotic iterations on $\mathds{R}$}
\label{chpt:Chaos des itérations chaotiques sur R}



\subsubsection{Chaos according to Devaney}

We have recalled previously that the chaotic iterations $\left(\Go,
\mathcal{X}_d\right)$ are chaotic according to the formulation of Devaney. We
can deduce that they are chaotic on $\mathds{R}$ too, when considering the order
topology, because:
\begin{itemize}
\item $\left(\Go, \mathcal{X}_d\right)$ and $\left(g, \big[ 0, 2^{10}
\big[_D\right)$ are semiconjugate by $\varphi$,
\item Then $\left(g, \big[ 0, 2^{10} \big[_D\right)$ is a system chaotic
according to Devaney, because the semiconjugacy preserve this character.
\item But the topology generated by $D$ is finer than the topology generated by
the Euclidian distance $\Delta$ -- which is the order topology.
\item According to Theorem \ref{Th:chaos et finesse}, we can deduce that the
chaotic iterations $g$ are indeed chaotic, as defined by Devaney, for the order
topology on $\mathds{R}$.
\end{itemize}

This result can be formulated as follows.

\begin{theorem}
\label{th:IC et topologie de l'ordre}
The chaotic iterations $g$ on $\mathds{R}$ are chaotic according to the
Devaney's formulation, when $\mathds{R}$ has his usual topology, which is the
order topology.
\end{theorem}

Indeed this result is weaker than the theorem establishing the chaos for the
finer topology $d$. However the Theorem \ref{th:IC et topologie de l'ordre}
still remains important. Indeed, we have studied in our previous works a set
different from the usual set of study ($\mathcal{X}$ instead of $\mathds{R}$),
in order to be as close as possible from the computer: the properties of
disorder proved theoretically will then be preserved when computing. However, we
could wonder whether this change does not lead to a disorder of a lower quality.
In other words, have we replaced a situation of a good disorder lost when
computing, to another situation of a disorder preserved but of bad quality.
Theorem \ref{th:IC et topologie de l'ordre} prove exactly the contrary.
 







\section{Conclusion}
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