+% ****** Start of file chaos-paper.tex ******
+%
+\documentclass[%
+aip,
+cha,% long,
+numerical bibliography, %(default)
+%jcp,% short, numerical bibliography,
+%jmp,
+amsmath,amssymb,
+preprint,%
+%reprint,%
+%author-year,%
+%author-numerical,%
+]{revtex4-1}
+
+\usepackage{graphicx}% Include figure files
+\usepackage{dcolumn}% Align table columns on decimal point
+\usepackage{bm}% bold math
+\usepackage{color}
+\usepackage{ulem}
+\usepackage[mathlines]{lineno}% Enable numbering of text and display math
+\linenumbers\relax % Commence numbering lines
+% The amssymb package provides various useful mathematical symbols
+\usepackage{amssymb}
+% The amsthm package provides various useful mathematical fonts
+\usepackage{amsfonts}
+% Various mathematical packages
+\PassOptionsToPackage{cmex10draft}{graphics}
+\usepackage[standard]{ntheorem}
+% Various packages
+\usepackage{stmaryrd} % llbracket
+\usepackage{dsfont} % mathds
+\usepackage{multirow} % tables
+
+\newcommand{\ie}{\textit{i.e.}}
+\newcommand{\Nats}[0]{\ensuremath{\mathds{N}}}
+\newcommand{\R}[0]{\ensuremath{\mathds{R}}}
+\newcommand{\Bool}[0]{\ensuremath{\mathds{B}}}
+
+\begin{document}
+
+\title[Neural Networks and Chaos]{Neural Networks and Chaos:
+Construction, Evaluation of Chaotic Networks \\
+and Prediction of Chaos with Multilayer Feedforward Networks
+}
+
+\author{Jacques M. Bahi}
+\author{Jean-Fran\c{c}ois Couchot}
+\author{Christophe Guyeux}
+\email{christophe.guyeux@univ-fcomte.fr.}
+\author{Michel Salomon}
+\altaffiliation[Authors in ]{alphabetic order}
+\affiliation{
+Computer Science Laboratory (LIFC), University of Franche-Comt\'e, \\
+IUT de Belfort-Montb\'eliard, BP 527, \\
+90016 Belfort Cedex, France
+}
+
+\date{\today}
+
+\newcommand{\CG}[1]{\begin{color}{red}\textit{#1}\end{color}}
+\newcommand{\JFC}[1]{\begin{color}{blue}\textit{#1}\end{color}}
+\newcommand{\MS}[1]{\begin{color}{green}\textit{#1}\end{color}}
+
+\begin{abstract}
+%% Text of abstract
+Many research works deal with chaotic neural networks for various
+fields of application. Unfortunately, up to now these networks are
+usually claimed to be chaotic without any mathematical proof. The
+purpose of this paper is to establish, based on a rigorous theoretical
+framework, an equivalence between chaotic iterations according to
+Devaney and a particular class of neural networks. On the one hand we
+show how to build such a network, on the other hand we provide a
+method to check if a neural network is a chaotic one. Finally, the
+ability of classical feedforward multilayer perceptrons to learn sets
+of data obtained from a dynamical system is regarded. Various Boolean
+functions are iterated on finite states. Iterations of some of them
+are proven to be chaotic as it is defined by Devaney. In that
+context, important differences occur in the training process,
+establishing with various neural networks that chaotic behaviors are
+far more difficult to learn.
+\end{abstract}
+
+%%\pacs{Valid PACS appear here}% PACS, the Physics and Astronomy
+% Classification Scheme.
+%%\keywords{Suggested keywords}%Use showkeys class option if keyword
+%display desired
+\maketitle
+
+\begin{quotation}
+
+Chaotic neural networks have received a lot of attention due to the
+appealing properties of deterministic chaos (unpredictability,
+sensitivity, and so on). However, such networks are often claimed
+chaotic without any rigorous mathematical proof. Therefore, in this
+work a theoretical framework based on the Devaney's definition of
+chaos is introduced. Starting with a relationship between discrete
+iterations and Devaney's chaos, we firstly show how to build a
+recurrent neural network that is equivalent to a chaotic map and
+secondly a way to check whether an already available network, is
+chaotic or not. We also study different topological properties of
+these truly chaotic neural networks. Finally, we show that the
+learning, with neural networks having a classical feedforward
+structure, of chaotic behaviors represented by data sets obtained from
+chaotic maps, is far more difficult than non chaotic behaviors.
+\end{quotation}
+
+%% Main text
+\section{Introduction}
+\label{S1}
+
+Several research works have proposed or used chaotic neural networks
+these last years. The complex dynamics of such a network leads to
+various potential application areas: associative
+memories~\cite{Crook2007267} and digital security tools like hash
+functions~\cite{Xiao10}, digital
+watermarking~\cite{1309431,Zhang2005759}, or cipher
+schemes~\cite{Lian20091296}. In the former case, the background idea
+is to control chaotic dynamics in order to store patterns, with the
+key advantage of offering a large storage capacity. For the latter
+case, the use of chaotic dynamics is motivated by their
+unpredictability and random-like behaviors. Indeed, investigating new
+concepts is crucial for the computer security field, because new
+threats are constantly emerging. As an illustrative example, the
+former standard in hash functions, namely the SHA-1 algorithm, has
+been recently weakened after flaws were discovered.
+
+Chaotic neural networks have been built with different approaches. In
+the context of associative memory, chaotic neurons like the nonlinear
+dynamic state neuron \cite{Crook2007267} frequently constitute the
+nodes of the network. These neurons have an inherent chaotic behavior,
+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
+their universal approximator capacity
+\cite{Cybenko89,DBLP:journals/nn/HornikSW89}. 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 chaotic, is itself claimed to be chaotic
+\cite{springerlink:10.1007/s00521-010-0432-2}.
+
+What all of these chaotic neural networks have in common is that they
+are claimed to be chaotic despite a lack of any rigorous mathematical
+proof. The first contribution of this paper is to fill this gap,
+using a theoretical framework based on the Devaney's definition of
+chaos \cite{Devaney}. This mathematical theory of chaos provides both
+qualitative and quantitative tools to evaluate the complex behavior of
+a dynamical system: ergodicity, expansivity, and so on. More
+precisely, in this paper, which is an extension of a previous work
+\cite{bgs11:ip}, we establish the equivalence between chaotic
+iterations and a class of globally recurrent MLP. The second
+contribution is a study of the converse problem, indeed we study 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
+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
+connected iterations graph. In this document, we experiment several
+MLPs and try to learn some iterations of this kind. We show that
+non-chaotic iterations can be learned, whereas it is far more
+difficult for chaotic ones. That is to say, we have discovered at
+least one family of problems with a reasonable size, such that
+artificial neural networks should not be applied due to their
+inability to learn chaotic behaviors in this context.
+
+The remainder of this research work is organized as follows. The next
+section is devoted to the basics of Devaney's chaos. Section~\ref{S2}
+formally describes how to build a neural network that operates
+chaotically. Section~\ref{S3} is devoted to the dual case of checking
+whether an existing neural network is chaotic or not. Topological
+properties of chaotic neural networks are discussed in Sect.~\ref{S4}.
+The Section~\ref{section:translation} shows how to translate such
+iterations into an Artificial Neural Network (ANN), in order to
+evaluate the capability for this latter to learn chaotic behaviors.
+This ability is studied in Sect.~\ref{section:experiments}, where
+various ANNs try to learn two sets of data: the first one is obtained
+by chaotic iterations while the second one results from a non-chaotic
+system. Prediction success rates are given and discussed for the two
+sets. The paper ends with a conclusion section where our contribution
+is summed up and intended future work is exposed.
+
+\section{Chaotic Iterations according to Devaney}
+
+In this section, the well-established notion of Devaney's mathematical
+chaos is firstly recalled. Preservation of the unpredictability of
+such dynamical system when implemented on a computer is obtained by
+using some discrete iterations called ``asynchronous iterations'',
+which are thus introduced. The result establishing the link between
+such iterations and Devaney's chaos is finally presented at the end of
+this 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 {\bf iteration} of
+a {\bf 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}
+
+Various domains such as physics, biology, or economy, contain systems
+that exhibit a chaotic behavior, a well-known example is the weather.
+These systems are in particular highly sensitive to initial
+conditions, a concept usually presented as the butterfly effect: small
+variations in the initial conditions possibly lead to widely different
+behaviors. Theoretically speaking, a system is sensitive if for each
+point $x$ in the iteration space, one can find a point in each
+neighborhood of $x$ having a significantly different future evolution.
+Conversely, a system seeded with the same initial conditions always
+has the same evolution. In other words, chaotic systems have a
+deterministic behavior defined through a physical or mathematical
+model and a high sensitivity to the initial conditions. Besides
+mathematically this kind of unpredictability is also referred to as
+deterministic chaos. For example, many weather forecast models exist,
+but they give only suitable predictions for about a week, because they
+are initialized with conditions that reflect only a partial knowledge
+of the current weather. Even if the differences are initially small,
+they are amplified in the course of time, and thus make difficult a
+long-term prediction. In fact, in a chaotic system, an approximation
+of the current state is a quite useless indicator for predicting
+future states.
+
+From mathematical point of view, deterministic chaos has been
+thoroughly studied these last decades, with different research works
+that have provide various definitions of chaos. Among these
+definitions, the one given by Devaney~\cite{Devaney} is
+well-established. This definition consists of three conditions:
+topological transitivity, density of periodic points, and sensitive
+point dependence on initial conditions.
+
+{\bf Topological transitivity} is checked when, for any point, any
+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.
+
+However, chaos needs some regularity to ``counteracts'' the effects of
+transitivity.
+%\begin{definition} \label{def3}
+We recall that a point $x$ is {\bf 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 {\bf regular} on the topological space $(\mathcal{X},\tau)$ 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}
+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 us recall that $f$ has {\bf 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 {\bf constant of sensitivity} of $f$.
+
+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.
+
+%Let us notice that for a metric space the last condition follows from
+%the two first ones~\cite{Banks92}.
+
+\subsection{Asynchronous Iterations}
+
+%This section presents some basics on topological chaotic iterations.
+Let us firstly discuss about the domain of iteration. As far as we
+know, no result rules that the chaotic behavior of a dynamical system
+that has been theoretically proven on $\R$ remains valid on the
+floating-point numbers, which is the implementation domain. Thus, to
+avoid loss of chaos this work presents an alternative, that is to
+iterate Boolean maps: results that are theoretically obtained in that
+domain are preserved in implementations.
+
+Let us denote by $\llbracket a ; b \rrbracket$ the following interval
+of integers: $\{a, a+1, \hdots, b\}$, where $a~<~b$. In this section,
+a {\it 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
+%\begin{equation*}
+$x^{t}=(x_1^{t},\ldots,x_{n}^{t}) \in \Bool^n$.
+%\end{equation*}
+The dynamics of the system is described according to a function $f :
+\Bool^n \rightarrow \Bool^n$ such that
+%\begin{equation*}
+$f(x)=(f_1(x),\ldots,f_n(x))$.
+%\end{equation*}
+% Notice that $f^k$ denotes the
+% $k-$th composition $f\circ \ldots \circ f$ of the function $f$.
+
+Let be given a configuration $x$. In what follows
+$N(i,x)=(x_1,\ldots,\overline{x_i},\ldots,x_n)$ 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,
+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
+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
+ 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}
+ \end{array}
+\right.
+\end{array} \right.
+\end{equation}
+
+Next section shows the link between asynchronous iterations and
+Devaney's chaos.
+
+\subsection{On the link between asynchronous iterations and
+ Devaney's Chaos}
+
+In this subsection we recall the link we have established between
+asynchronous iterations and Devaney's chaos. The theoretical
+framework is fully described in \cite{guyeux09}.
+
+We introduce the function $F_{f}$ that is defined for any given
+application $f:\Bool^{n} \to \Bool^{n}$ by $F_{f}:
+\llbracket1;n\rrbracket\times \mathds{B}^{n} \rightarrow
+\mathds{B}^{n}$, s.t.
+\begin{equation}
+\label{eq:CIs}
+ F_{f}(s,x)_j =
+ \left\{
+ \begin{array}{l}
+ f_j(x) \textrm{ if } j= s \enspace , \\
+ x_{j} \textrm{ otherwise} \enspace .
+ \end{array}
+ \right.
+\end{equation}
+
+\noindent With such a notation, asynchronously obtained configurations
+are defined for times \linebreak $t=0,1,2,\ldots$ by:
+\begin{equation}\label{eq:sync}
+\left\{\begin{array}{l}
+ x^{0}\in \mathds{B}^{n} \textrm{ and}\\
+ x^{t+1}=F_{f}(S^t,x^{t}) \enspace .
+\end{array}\right.
+\end{equation}
+
+\noindent Finally, iterations defined in Eq.~(\ref{eq:sync}) can be
+described by the following system:
+\begin{equation}
+\left\{
+\begin{array}{lll}
+X^{0} & = & ((S^t)^{t \in \Nats},x^0) \in
+\llbracket1;n\rrbracket^\Nats \times \Bool^{n}\\
+X^{k+1}& = & G_{f}(X^{k})\\
+\multicolumn{3}{c}{\textrm{where } G_{f}\left(((S^t)^{t \in \Nats},x)\right)
+= \left(\sigma((S^t)^{t \in \Nats}),F_{f}(S^0,x)\right) \enspace ,}
+\end{array}
+\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
+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
+\begin{equation}
+d((S,x);(\check{S},\check{x}))=d_{e}(x,\check{x})+d_{s}(S,\check{S})
+\enspace ,
+\end{equation}
+where
+\begin{equation}
+d_{e}(x,\check{x})=\sum_{j=1}^{n}\Delta
+(x_{j},\check{x}_{j}) \in \llbracket 0 ; n \rrbracket
+\end{equation}
+\noindent and
+\begin{equation}
+d_{s}(S,\check{S})=\frac{9}{2n}\sum_{t=0}^{\infty
+}\frac{|S^{t}-\check{S}^{t}|}{10^{t+1}} \in [0 ; 1] \enspace .
+\end{equation}
+
+This distance is defined to reflect the following
+information. Firstly, the more two systems have different components,
+the larger the distance between them. Secondly, two systems with
+similar components and strategies, which have the same starting terms,
+must induce only a small distance. The proposed distance fulfills
+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
+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,
+Guyeux~\cite{GuyeuxThese10} has proven that,
+\begin{theorem}%[Characterization of $\mathcal{C}$]
+\label{Th:Caracterisation des IC chaotiques}
+Let $f:\Bool^n\to\Bool^n$. Iterations of $G_f$ are chaotic according
+to Devaney if and only if $\Gamma(f)$ is strongly connected.
+\end{theorem}
+
+Checking if a graph is strongly connected is not difficult (by the
+Tarjan's algorithm for instance). Let be given a strategy $S$ and a
+function $f$ such that $\Gamma(f)$ is strongly connected. In that
+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
+$\Bool^n\to\Bool^n$ 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
+$f_1\left(x_1,\dots,x_n\right)=\left(
+\overline{x_1},x_1,x_2,\dots,x_{n-1}\right)$. It is not hard to check
+that $\Gamma(f_0)$ and $\Gamma(f_1)$ are both strongly connected, then
+iterations of $G_{f_0}$ and of $G_{f_1}$ are chaotic according to
+Devaney.
+
+With this material, we are now able to build a first chaotic neural
+network, as defined in the Devaney's formulation.
+
+\section{A chaotic neural network in the sense of Devaney}
+\label{S2}
+
+Let us build a multilayer perceptron neural network modeling
+$F_{f_0}:\llbracket 1; n \rrbracket \times \mathds{B}^n \to
+\mathds{B}^n$ associated to the vectorial negation. More precisely,
+for all inputs $(s,x) \in \llbracket 1;n\rrbracket \times
+\mathds{B}^n$, the output layer will produce $F_{f_0}(s,x)$. It is
+then possible to link the output layer and the input one, in order to
+model the dependence between two successive iterations. As a result
+we obtain a global recurrent neural network that behaves as follows
+(see Fig.~\ref{Fig:perceptron}).
+
+\begin{itemize}
+\item The network is initialized with the input vector
+ $\left(S^0,x^0\right) \in \llbracket 1;n\rrbracket \times
+ \mathds{B}^n$ and computes the output vector
+ $x^1=F_{f_0}\left(S^0,x^0\right)$. This last vector is published as
+ an output one of the chaotic neural network and is sent back to the
+ input layer through the feedback links.
+\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
+ output vector. This new vector, which represents the new state of
+ the dynamical system, satisfies:
+ \begin{equation}
+ x^{t+1}=F_{f_0}(S^0, x^t) \in \mathds{B}^n \enspace .
+ \end{equation}
+\end{itemize}
+
+\begin{figure}
+ \centering
+ \includegraphics[scale=0.625]{perceptron}
+ \caption{A perceptron equivalent to chaotic iterations}
+ \label{Fig:perceptron}
+\end{figure}
+
+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
+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 $G_{f_0}$, whose iterations are
+ chaotic according to
+Devaney's definition of chaos, we can conclude that the network is
+also chaotic in this sense.
+
+The previous construction scheme is not restricted to function $f_0$.
+It can be extended to any function $f$ such that $G_f$ is a chaotic
+map by training the network to model $F_{f}:\llbracket 1; n \rrbracket
+\times \mathds{B}^n \to \mathds{B}^n$. Due to
+Theorem~\ref{Th:Caracterisation des IC chaotiques}, we can find
+alternative functions $f$ for $f_0$ through a simple check of their
+graph of iterations $\Gamma(f)$. For example, we can build another
+chaotic neural network by using $f_1$ instead of $f_0$.
+
+\section{Checking whether a neural network is chaotic or not}
+\label{S3}
+
+We focus now on the case where a neural network is already available,
+and for which we want to know if it is chaotic. Typically, in many
+research papers neural network are usually claimed to be chaotic
+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
+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 $n$ bits.
+Moreover, each binary output is connected with a feedback connection
+to an input one.
+
+\begin{itemize}
+\item During initialization, the network is seeded with $n$~bits
+ denoted $\left(x^0_1,\dots,x^0_n\right)$ and an integer value $S^0$
+ that belongs to $\llbracket1;n\rrbracket$.
+\item At iteration~$t$, the last output vector
+ $\left(x^t_1,\dots,x^t_n\right)$ defines the $n$~bits used to
+ 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
+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
+$\left(s,\left(x_1,\dots,x_n\right)\right) \in \llbracket 1;n
+\rrbracket \times \mathds{B}^n$ 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
+$\left(s,\left(x_1,\dots, x_n\right)\right)$. 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}
+\left(F\left(1,\left(x_1,x_2,\dots,x_n\right)\right),\dots,
+ F\left(n,\left(x_1,x_2,\dots,x_n\right)\right)\right) \enspace .
+\end{equation}
+Thus, for any $j$, $1 \le j \le n$, we have
+$f_j\left(x_1,x_2,\dots,x_n\right) =
+F\left(j,\left(x_1,x_2,\dots,x_n\right)\right)$.
+If this recurrent neural network is seeded with
+$\left(x_1^0,\dots,x_n^0\right)$ and $S \in \llbracket 1;n
+\rrbracket^{\mathds{N}}$, it produces exactly the
+same output vectors than the
+chaotic iterations of $F_f$ with initial
+condition $\left(S,(x_1^0,\dots, x_n^0)\right) \in \llbracket 1;n
+\rrbracket^{\mathds{N}} \times \mathds{B}^n$.
+Theoretically speaking, such iterations of $F_f$ are thus a formal
+model of these kind of recurrent neural networks. In the rest of this
+paper, we will call such multilayer perceptrons CI-MLP($f$), which
+stands for ``Chaotic Iterations based MultiLayer Perceptron''.
+
+Checking if CI-MLP($f$) behaves chaotically according to Devaney's
+definition of chaos is simple: we need just to verify if the
+associated graph of iterations $\Gamma(f)$ is strongly connected or
+not. As an incidental consequence, we finally obtain an equivalence
+between chaotic iterations and CI-MLP($f$). Therefore, we can
+obviously study such multilayer perceptrons with mathematical tools
+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.
+
+\section{Topological properties of chaotic neural networks}
+\label{S4}
+
+Let us first recall two fundamental definitions from the mathematical
+theory of chaos.
+
+\begin{definition} \label{def8}
+A function $f$ is said to be {\bf expansive} if $\exists
+\varepsilon>0$, $\forall x \neq y$, $\exists n \in \mathds{N}$ such
+that $d\left(f^n(x),f^n(y)\right) \geq \varepsilon$.
+\end{definition}
+
+\noindent In other words, a small error on any initial condition is
+always amplified until $\varepsilon$, which denotes the constant of
+expansivity of $f$.
+
+\begin{definition} \label{def9}
+A discrete dynamical system is said to be {\bf topologically mixing}
+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}
+
+\noindent Topologically mixing means that the dynamical system evolves
+in time such that any given region of its topological space might
+overlap with any other region.
+
+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 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 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 {\bf not
+ decomposable} if it is not the union of two closed sets $A, B
+\subset \mathcal{X}$ such that $f(A) \subset A, f(B) \subset B$.
+\end{definition}
+
+\noindent Hence, reducing the set of outputs generated by CI-MLP($f$),
+in order to simplify its complexity, is impossible if $\Gamma(f)$ is
+strongly connected. Moreover, under this hypothesis CI-MLPs($f$) are
+strongly transitive:
+
+\begin{definition} \label{def11}
+A dynamical system $\left( \mathcal{X}, f\right)$ is {\bf strongly
+ transitive} if $\forall x,y \in \mathcal{X}$, $\forall r>0$,
+$\exists z \in \mathcal{X}$, $d(z,x)~\leq~r \Rightarrow \exists n \in
+\mathds{N}^{\ast}$, $f^n(z)=y$.
+\end{definition}
+
+\noindent According to this definition, for all pairs of points $(x,
+y)$ in the phase space, a point $z$ can be found in the neighborhood
+of $x$ such that one of its iterates $f^n(z)$ is $y$. Indeed, this
+result has been established during the proof of the transitivity
+presented in Ref.~\cite{guyeux09}. Among other things, the strong
+transitivity leads to the fact that without the knowledge of the
+initial input layer, all outputs are possible. Additionally, no point
+of the output space can be discarded when studying CI-MLPs: this space
+is 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 {\bf unstable}
+if for
+all $x \in \mathcal{X}$, the orbit $\gamma_x:n \in \mathds{N}
+\longmapsto f^n(x)$ is unstable, that means: $\exists \varepsilon >
+0$, $\forall \delta>0$, $\exists y \in \mathcal{X}$, $\exists n \in
+\mathds{N}$, such that $d(x,y)<\delta$ and
+$d\left(\gamma_x(n),\gamma_y(n)\right) \geq \varepsilon$.
+\end{definition}
+
+\noindent This property, which is implied by the sensitive point
+dependence on initial conditions, leads to the fact that in all
+neighborhoods of any point $x$, there are points that can be apart by
+$\varepsilon$ in the future through iterations of the
+CI-MLP($f$). Thus, we can claim that the behavior of these MLPs is
+unstable when $\Gamma (f)$ is strongly connected.
+
+Let us now consider a compact metric space $(M, d)$ and $f: M
+\rightarrow M$ a continuous map. For each natural number $n$, a new
+metric $d_n$ is defined on $M$ by
+\begin{equation}
+d_n(x,y)=\max\{d(f^i(x),f^i(y)): 0\leq i<n\} \enspace .
+\end{equation}
+
+Given any $\varepsilon > 0$ and $n \geqslant 1$, two points of $M$ are
+$\varepsilon$-close with respect to this metric if their first $n$
+iterates are $\varepsilon$-close.
+
+This metric allows one to distinguish in a neighborhood of an orbit
+the points that move away from each other during the iteration from
+the points that travel together. A subset $E$ of $M$ is said to be
+$(n, \varepsilon)$-separated if each pair of distinct points of $E$ is
+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 {\bf topological entropy} of the map $f$ is defined by (see e.g.,
+Ref.~\cite{Adler65} or Ref.~\cite{Bowen})
+$$h(f)=\lim_{\varepsilon\to 0} \left(\limsup_{n\to \infty}
+\frac{1}{n}\log H(n,\varepsilon)\right) \enspace .$$
+\end{definition}
+
+Then we have the following result \cite{GuyeuxThese10},
+\begin{theorem}
+$\left( \mathcal{X},d\right)$ is compact and the topological entropy
+of $(\mathcal{X},G_{f_0})$ is infinite.
+\end{theorem}
+
+\begin{figure}
+ \centering
+ \includegraphics[scale=0.5]{scheme}
+ \caption{Summary of addressed neural networks and chaos problems}
+ \label{Fig:scheme}
+\end{figure}
+
+Figure~\ref{Fig:scheme} is a summary of addressed neural networks and
+chaos problems. In Section~\ref{S2} we have explained how to
+construct a truly chaotic neural networks, $A$ for
+instance. Section~\ref{S3} has shown how to check whether a given MLP
+$A$ or $C$ is chaotic or not in the sens of Devaney, and how to study
+its topological behavior. The last thing to investigate, when
+comparing neural networks and Devaney's chaos, is to determine whether
+an artificial neural network $C$ is able to learn or predict some
+chaotic behaviors of $B$, as it is defined in the Devaney's
+formulation (when they are not specifically constructed for this
+purpose). This statement is studied in the next section.
+
+\section{Suitability of Feedforward Neural Networks
+for Predicting Chaotic and Non-chaotic Behaviors}
+
+In the context of computer science different topic areas have an
+interest in chaos, as for steganographic
+techniques~\cite{1309431,Zhang2005759}. Steganography consists in
+embedding a secret message within an ordinary one, while the secret
+extraction takes place once at destination. The reverse ({\it i.e.},
+automatically detecting the presence of hidden messages inside media)
+is called steganalysis. Among the deployed strategies inside
+detectors, there are support vectors
+machines~\cite{Qiao:2009:SM:1704555.1704664}, neural
+networks~\cite{10.1109/ICME.2003.1221665,10.1109/CIMSiM.2010.36}, and
+Markov chains~\cite{Sullivan06steganalysisfor}. Most of these
+detectors give quite good results and are rather competitive when
+facing steganographic tools. However, to the best of our knowledge
+none of the considered information hiding schemes fulfills the Devaney
+definition of chaos~\cite{Devaney}. Indeed, one can wonder whether
+detectors continue to give good results when facing truly chaotic
+schemes. More generally, there remains the open problem of deciding
+whether artificial intelligence is suitable for predicting topological
+chaotic behaviors.
+
+\subsection{Representing Chaotic Iterations for Neural Networks}
+\label{section:translation}
+
+The problem of deciding whether classical feedforward ANNs are
+suitable to approximate topological chaotic iterations may then be
+reduced to evaluate such neural networks on iterations of functions
+with Strongly Connected Component (SCC)~graph of iterations. To
+compare with non-chaotic iterations, the experiments detailed in the
+following sections are carried out using both kinds of function
+(chaotic and non-chaotic). Let us emphasize on the difference between
+this kind of neural networks and the Chaotic Iterations based
+multilayer peceptron.
+
+We are then left to compute two disjoint function sets that contain
+either functions with topological chaos properties or not, depending
+on the strong connectivity of their iterations graph. This can be
+achieved for instance by removing a set of edges from the iteration
+graph $\Gamma(f_0)$ of the vectorial negation function~$f_0$. One can
+deduce whether a function verifies the topological chaos property or
+not by checking the strong connectivity of the resulting graph of
+iterations.
+
+For instance let us consider the functions $f$ and $g$ from $\Bool^4$
+to $\Bool^4$ respectively defined by the following lists:
+$$[0, 0, 2, 3, 13, 13, 6, 3, 8, 9, 10, 11, 8, 13, 14,
+ 15]$$ $$\mbox{and } [11, 14, 13, 14, 11, 10, 1, 8, 7, 6, 5, 4, 3, 2,
+ 1, 0] \enspace.$$ In other words, the image of $0011$ by $g$ is
+$1110$: it is obtained as the binary value of the fourth element in
+the second list (namely~14). It is not hard to verify that
+$\Gamma(f)$ is not SCC (\textit{e.g.}, $f(1111)$ is $1111$) whereas
+$\Gamma(g)$ is. The remaining of this section shows how to translate
+iterations of such functions into a model amenable to be learned by an
+ANN. 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}).
+
+Firstly, let us focus on how to memorize configurations. Two distinct
+translations are proposed. In the first case, we take one input in
+$\Bool$ per component; in the second case, configurations are
+memorized as natural numbers. A coarse attempt to memorize
+configuration as natural number could consist in labeling each
+configuration with its translation into decimal numeral system.
+However, such a representation induces too many changes between a
+configuration labeled by a power of two and its direct previous
+configuration: for instance, 16~(10000) and 15~(01111) are close in a
+decimal ordering, but their Hamming distance is 5. This is why Gray
+codes~\cite{Gray47} have been preferred.
+
+Secondly, let us detail how to deal with strategies. Obviously, it is
+not possible to translate in a finite way an infinite strategy, even
+if both $(S^t)^{t \in \Nats}$ and $\sigma((S^t)^{t \in \Nats})$ belong
+to $\{1,\ldots,n\}^{\Nats}$. Input strategies are then reduced to
+have a length of size $l \in \llbracket 2,k\rrbracket$, where $k$ is a
+parameter of the evaluation. Notice that $l$ is greater than or equal
+to $2$ since we do not want the shift $\sigma$~function to return an
+empty strategy. Strategies are memorized as natural numbers expressed
+in base $n+1$. At each iteration, either none or one component is
+modified (among the $n$ components) leading to a radix with $n+1$
+entries. Finally, we give an other input, namely $m \in \llbracket
+1,l-1\rrbracket$, which is the number of successive iterations that
+are applied starting from $x$. Outputs are translated with the same
+rules.
+
+To address the complexity issue of the problem, let us compute the
+size of the data set an ANN has to deal with. Each input vector of an
+input-output pair is composed of a configuration~$x$, an excerpt $S$
+of the strategy to iterate of size $l \in \llbracket 2, k\rrbracket$,
+and a number $m \in \llbracket 1, l-1\rrbracket$ of iterations that
+are executed.
+
+Firstly, there are $2^n$ configurations $x$, with $n^l$ strategies of
+size $l$ for each of them. Secondly, for a given configuration there
+are $\omega = 1 \times n^2 + 2 \times n^3 + \ldots+ (k-1) \times n^k$
+ways of writing the pair $(m,S)$. Furthermore, it is not hard to
+establish that
+\begin{equation}
+\displaystyle{(n-1) \times \omega = (k-1)\times n^{k+1} - \sum_{i=2}^k n^i} \nonumber
+\end{equation}
+then
+\begin{equation}
+\omega =
+\dfrac{(k-1)\times n^{k+1}}{n-1} - \dfrac{n^{k+1}-n^2}{(n-1)^2} \enspace . \nonumber
+\end{equation}
+\noindent And then, finally, the number of input-output pairs for our
+ANNs is
+$$
+2^n \times \left(\dfrac{(k-1)\times n^{k+1}}{n-1} - \dfrac{n^{k+1}-n^2}{(n-1)^2}\right) \enspace .
+$$
+For instance, for $4$ binary components and a strategy of at most
+$3$~terms we obtain 2304~input-output pairs.
+
+\subsection{Experiments}
+\label{section:experiments}
+
+To study if chaotic iterations can be predicted, we choose to train
+the multiLayer perceptron. As stated before, this kind of network is
+in particular well-known for its universal approximation property
+\cite{Cybenko89,DBLP:journals/nn/HornikSW89}. Furthermore, MLPs have
+been already considered for chaotic time series prediction. For
+example, in~\cite{dalkiran10} the authors have shown that a
+feedforward MLP with two hidden layers, and trained with Bayesian
+Regulation back-propagation, can learn successfully the dynamics of
+Chua's circuit.
+
+In these experiments we consider MLPs having one hidden layer of
+sigmoidal neurons and output neurons with a linear activation
+function. They are trained using the Limited-memory
+Broyden-Fletcher-Goldfarb-Shanno quasi-newton algorithm in combination
+with the Wolfe linear search. The training process is performed until
+a maximum number of epochs is reached. To prevent overfitting and to
+estimate the generalization performance we use holdout validation by
+splitting the data set into learning, validation, and test subsets.
+These subsets are obtained through random selection such that their
+respective size represents 65\%, 10\%, and 25\% of the whole data set.
+
+Several neural networks are trained for both iterations coding
+schemes. In both cases iterations have the following layout:
+configurations of four components and strategies with at most three
+terms. Thus, for the first coding scheme a data set pair is composed
+of 6~inputs and 5~outputs, while for the second one it is respectively
+3~inputs and 2~outputs. As noticed at the end of the previous section,
+this leads to data sets that consist of 2304~pairs. The networks
+differ in the size of the hidden layer and the maximum number of
+training epochs. We remember that to evaluate the ability of neural
+networks to predict a chaotic behavior for each coding scheme, the
+trainings of two data sets, one of them describing chaotic iterations,
+are compared.
+
+Thereafter we give, for the different learning setups and data sets,
+the mean prediction success rate obtained for each output. A such rate
+represent the percentage of input-output pairs belonging to the test
+subset for which the corresponding output value was correctly
+predicted. These values are computed considering 10~trainings with
+random subsets construction, weights and biases initialization.
+Firstly, neural networks having 10 and 25~hidden neurons are trained,
+with a maximum number of epochs that takes its value in
+$\{125,250,500\}$ (see Tables~\ref{tab1} and \ref{tab2}). Secondly,
+we refine the second coding scheme by splitting the output vector such
+that each output is learned by a specific neural network
+(Table~\ref{tab3}). In this last case, we increase the size of the
+hidden layer up to 40~neurons and we consider larger number of epochs.
+
+\begin{table}[htbp!]
+\caption{Prediction success rates for configurations expressed as boolean vectors.}
+\label{tab1}
+\centering {\small
+\begin{tabular}{|c|c||c|c|c|}
+\hline
+\multicolumn{5}{|c|}{Networks topology: 6~inputs, 5~outputs and one hidden layer} \\
+\hline
+\hline
+\multicolumn{2}{|c||}{Hidden neurons} & \multicolumn{3}{c|}{10 neurons} \\
+\cline{3-5}
+\multicolumn{2}{|c||}{Epochs} & 125 & 250 & 500 \\
+\hline
+\multirow{6}{*}{\rotatebox{90}{Chaotic}}&Output~(1) & 90.92\% & 91.75\% & 91.82\% \\
+& Output~(2) & 69.32\% & 78.46\% & 82.15\% \\
+& Output~(3) & 68.47\% & 78.49\% & 82.22\% \\
+& Output~(4) & 91.53\% & 92.37\% & 93.4\% \\
+& Config. & 36.10\% & 51.35\% & 56.85\% \\
+& Strategy~(5) & 1.91\% & 3.38\% & 2.43\% \\
+\hline
+\multirow{6}{*}{\rotatebox{90}{Non-chaotic}}&Output~(1) & 97.64\% & 98.10\% & 98.20\% \\
+& Output~(2) & 95.15\% & 95.39\% & 95.46\% \\
+& Output~(3) & 100\% & 100\% & 100\% \\
+& Output~(4) & 97.47\% & 97.90\% & 97.99\% \\
+& Config. & 90.52\% & 91.59\% & 91.73\% \\
+& Strategy~(5) & 3.41\% & 3.40\% & 3.47\% \\
+\hline
+\hline
+\multicolumn{2}{|c||}{Hidden neurons} & \multicolumn{3}{c|}{25 neurons} \\ %& \multicolumn{3}{|c|}{40 neurons} \\
+\cline{3-5}
+\multicolumn{2}{|c||}{Epochs} & 125 & 250 & 500 \\ %& 125 & 250 & 500 \\
+\hline
+\multirow{6}{*}{\rotatebox{90}{Chaotic}}&Output~(1) & 91.65\% & 92.69\% & 93.93\% \\ %& 91.94\% & 92.89\% & 94.00\% \\
+& Output~(2) & 72.06\% & 88.46\% & 90.5\% \\ %& 74.97\% & 89.83\% & 91.14\% \\
+& Output~(3) & 79.19\% & 89.83\% & 91.59\% \\ %& 76.69\% & 89.58\% & 91.84\% \\
+& Output~(4) & 91.61\% & 92.34\% & 93.47\% \\% & 82.77\% & 92.93\% & 93.48\% \\
+& Config. & 48.82\% & 67.80\% & 70.97\% \\%& 49.46\% & 68.94\% & 71.11\% \\
+& Strategy~(5) & 2.62\% & 3.43\% & 3.78\% \\% & 3.10\% & 3.10\% & 3.03\% \\
+\hline
+\multirow{6}{*}{\rotatebox{90}{Non-chaotic}}&Output~(1) & 97.87\% & 97.99\% & 98.03\% \\ %& 98.16\% \\
+& Output~(2) & 95.46\% & 95.84\% & 96.75\% \\ % & 97.4\% \\
+& Output~(3) & 100\% & 100\% & 100\% \\%& 100\% \\
+& Output~(4) & 97.77\% & 97.82\% & 98.06\% \\%& 98.31\% \\
+& Config. & 91.36\% & 91.99\% & 93.03\% \\%& 93.98\% \\
+& Strategy~(5) & 3.37\% & 3.44\% & 3.29\% \\%& 3.23\% \\
+\hline
+\end{tabular}
+}
+\end{table}
+
+Table~\ref{tab1} presents the rates obtained for the first coding
+scheme. For the chaotic data, it can be seen that as expected
+configuration prediction becomes better when the number of hidden
+neurons and maximum epochs increases: an improvement by a factor two
+is observed (from 36.10\% for 10~neurons and 125~epochs to 70.97\% for
+25~neurons and 500~epochs). We also notice that the learning of
+outputs~(2) and~(3) is more difficult. Conversely, for the
+non-chaotic case the simplest training setup is enough to predict
+configurations. For all those feedforward network topologies and all
+outputs the obtained results for the non-chaotic case outperform the
+chaotic ones. Finally, the rates for the strategies show that the
+different networks are unable to learn them.
+
+\newpage
+
+For the second coding scheme (\textit{i.e.}, with Gray Codes)
+Table~\ref{tab2} shows that any network learns about five times more
+non-chaotic configurations than chaotic ones. As in the previous
+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
+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}
+\centering
+\begin{tabular}{|c|c||c|c|c|}
+\hline
+\multicolumn{5}{|c|}{Networks topology: 3~inputs, 2~outputs and one hidden layer} \\
+\hline
+\hline
+& Hidden neurons & \multicolumn{3}{c|}{10 neurons} \\
+\cline{2-5}
+& Epochs & 125 & 250 & 500 \\ %& 1000
+\hline
+\multirow{2}{*}{Chaotic}& Config.~(1) & 13.29\% & 13.55\% & 13.08\% \\ %& 12.5\%
+& Strategy~(2) & 0.50\% & 0.52\% & 1.32\% \\ %& 1.42\%
+\hline
+\multirow{2}{*}{Non-Chaotic}&Config.~(1) & 77.12\% & 74.00\% & 72.60\% \\ %& 75.81\%
+& Strategy~(2) & 0.42\% & 0.80\% & 1.16\% \\ %& 1.42\%
+\hline
+\hline
+& Hidden neurons & \multicolumn{3}{c|}{25 neurons} \\
+\cline{2-5}
+& Epochs & 125 & 250 & 500 \\ %& 1000
+\hline
+\multirow{2}{*}{Chaotic}& Config.~(1) & 12.27\% & 13.15\% & 13.05\% \\ %& 15.44\%
+& Strategy~(2) & 0.71\% & 0.66\% & 0.88\% \\ %& 1.73\%
+\hline
+\multirow{2}{*}{Non-Chaotic}&Config.~(1) & 73.60\% & 74.70\% & 75.89\% \\ %& 68.32\%
+& Strategy~(2) & 0.64\% & 0.97\% & 1.23\% \\ %& 1.80\%
+\hline
+\end{tabular}
+\end{table}
+
+Unfortunately, in practical applications the number of components is
+usually unknown. Hence, the first coding scheme cannot be used
+systematically. Therefore, we provide a refinement of the second
+scheme: each output is learned by a different ANN. Table~\ref{tab3}
+presents the results for this approach. In any case, whatever the
+considered feedforward network topologies, the maximum epoch number
+and the kind of iterations, the configuration success rate is slightly
+improved. Moreover, the strategies predictions rates reach almost
+12\%, whereas in Table~\ref{tab2} they never exceed 1.5\%. Despite of
+this improvement, a long term prediction of chaotic iterations still
+appear to be an open issue.
+
+\begin{table}
+\caption{Prediction success rates for split outputs.}
+\label{tab3}
+\centering
+\begin{tabular}{|c||c|c|c|}
+\hline
+\multicolumn{4}{|c|}{Networks topology: 3~inputs, 1~output and one hidden layer} \\
+\hline
+\hline
+Epochs & 125 & 250 & 500 \\
+\hline
+\hline
+Chaotic & \multicolumn{3}{c|}{Output = Configuration} \\
+\hline
+10~neurons & 12.39\% & 14.06\% & 14.32\% \\
+25~neurons & 13.00\% & 14.28\% & 14.58\% \\
+40~neurons & 11.58\% & 13.47\% & 14.23\% \\
+\hline
+\hline
+Non chaotic & \multicolumn{3}{c|}{Output = Configuration} \\
+\cline{2-4}
+%Epochs & 125 & 250 & 500 \\
+\hline
+10~neurons & 76.01\% & 74.04\% & 78.16\% \\
+25~neurons & 76.60\% & 72.13\% & 75.96\% \\
+40~neurons & 76.34\% & 75.63\% & 77.50\% \\
+\hline
+\hline
+Chaotic/non chaotic & \multicolumn{3}{c|}{Output = Strategy} \\
+\cline{2-4}
+%Epochs & 125 & 250 & 500 \\
+\hline
+10~neurons & 0.76\% & 0.97\% & 1.21\% \\
+25~neurons & 1.09\% & 0.73\% & 1.79\% \\
+40~neurons & 0.90\% & 1.02\% & 2.15\% \\
+\hline
+\multicolumn{4}{c}{} \\
+\hline
+Epochs & 1000 & 2500 & 5000 \\
+\hline
+\hline
+Chaotic & \multicolumn{3}{c|}{Output = Configuration} \\
+\hline
+10~neurons & 14.51\% & 15.22\% & 15.22\% \\
+25~neurons & 16.95\% & 17.57\% & 18.46\% \\
+40~neurons & 17.73\% & 20.75\% & 22.62\% \\
+\hline
+\hline
+Non chaotic & \multicolumn{3}{c|}{Output = Configuration} \\
+\cline{2-4}
+%Epochs & 1000 & 2500 & 5000 \\
+\hline
+10~neurons & 78.98\% & 80.02\% & 79.97\% \\
+25~neurons & 79.19\% & 81.59\% & 81.53\% \\
+40~neurons & 79.64\% & 81.37\% & 81.37\% \\
+\hline
+\hline
+Chaotic/non chaotic & \multicolumn{3}{c|}{Output = Strategy} \\
+\cline{2-4}
+%Epochs & 1000 & 2500 & 5000 \\
+\hline
+10~neurons & 3.47\% & 9.98\% & 11.66\% \\
+25~neurons & 3.92\% & 8.63\% & 10.09\% \\
+40~neurons & 3.29\% & 7.19\% & 7.18\% \\
+\hline
+\end{tabular}
+\end{table}
+
+%
+% TO BE COMPLETED
+%
+
+\section{Conclusion}
+
+In this paper, we have established an equivalence between chaotic
+iterations, according to the Devaney's definition of chaos, and a
+class of multilayer perceptron neural networks. Firstly, we have
+described how to build a neural network that can be trained to learn a
+given chaotic map function. Secondly, we found a condition that allow
+to check whether the iterations induced by a function are chaotic or
+not, and thus if a chaotic map is obtained. Thanks to this condition
+our approach is not limited to a particular function. In the dual
+case, we show that checking if a neural network is chaotic consists in
+verifying a property on an associated graph, called the graph of
+iterations. These results are valid for recurrent neural networks
+with a particular architecture. However, we believe that a similar
+work can be done for other neural network architectures. Finally, we
+have discovered at least one family of problems with a reasonable
+size, such that artificial neural networks should not be applied in
+the presence of chaos, due to their inability to learn chaotic
+behaviors in this context. Such a consideration is not reduced to a
+theoretical detail: this family of discrete iterations is concretely
+implemented in a new steganographic method \cite{guyeux10ter}. As
+steganographic detectors embed tools like neural networks to
+distinguish between original and stego contents, our studies tend to
+prove that such detectors might be unable to tackle with chaos-based
+information hiding schemes.
+
+In future work we intend to enlarge the comparison between the
+learning of truly chaotic and non-chaotic behaviors. Other
+computational intelligence tools such as support vector machines will
+be investigated too, to discover which tools are the most relevant
+when facing a truly chaotic phenomenon. A comparison between learning
+rate success and prediction quality will be realized. Concrete
+consequences in biology, physics, and computer science security fields
+will be stated.
+
+% \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}
+%
+% ****** End of file chaos-paper.tex ******
