\begin{document}
-\title[Recurrent Neural Networks and Chaos]{Recurrent Neural Networks
-and Chaos: Construction, Evaluation, \\
-and Prediction Ability}
+\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}
\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
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 chaotic 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.
+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.
+% Classification Scheme.
%%\keywords{Suggested keywords}%Use showkeys class option if keyword
- %display desired
+%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 chaotic
+chaos is introduced. Starting with a relationship between discrete
iterations and Devaney's chaos, we firstly show how to build a
-recurrent neural networks 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
+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 feedforward structure, of
-chaotic behaviors represented by data sets obtained from chaotic maps,
-is far more difficult than non chaotic behaviors.
+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 run chaotic neural networks
-these last years. The complex dynamics of such a networks leads to
+Several research works have proposed or used chaotic neural networks
+these last years. The complex dynamics of such networks lead to
various potential application areas: associative
memories~\cite{Crook2007267} and digital security tools like hash
functions~\cite{Xiao10}, digital
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. Thus, investigating new
-concepts is crucial in this 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.
+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
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. 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
+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}. Sometime 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
+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 investigation the converse problem is the second contribution:
-we indeed study the ability for
-classical MultiLayer Perceptrons to learn a particular family of
-discrete chaotic dynamical systems. This family, called chaotic
-iterations, is defined by a Boolean vector, an update function, and a
-sequence giving which component to update at each iteration. It has
-been previously established that such dynamical systems is
-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.
+\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 investigate
+the ability of classical multilayer perceptrons to learn a particular
+family of discrete chaotic dynamical systems. This family is defined
+by a Boolean vector, an update function, and a sequence defining the
+component to update at each iteration. It has been previously
+established that such dynamical systems are chaotically iterated (as
+it is defined by Devaney) when the chosen function has a strongly
+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
+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
\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
+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.
+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).
+$f \circ f \circ \hdots \circ f$ ($n$ 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}
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 the differences are initially small,
+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
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:
+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.
-Topological transitivity is checked when, for any point, any
+{\bf Topological transitivity} is checked when, for any point, any
neighborhood of its future evolution eventually overlap with any other
-given region. More precisely,
-
-\begin{definition} \label{def2}
-A continuous function $f$ on a topological space $(\mathcal{X},\tau)$
-is defined to be {\bf topologically transitive} if for any pair of
-open sets $U$, $V \in \mathcal{X}$ there exists
-$k \in
-\mathds{N}^{\ast}$
- such that
-$f^k(U) \cap V \neq \emptyset$.
-\end{definition}
-
-This property implies that a dynamical system cannot be broken into
-simpler subsystems.
-Intuitively, its complexity does not allow any simplification.
-On the contrary, a dense set of periodic points is an
-element of regularity that a chaotic dynamical system has to exhibit.
-
-\begin{definition} \label{def3}
-A point $x$ is called a {\bf periodic point} for $f$ of period~$n \in
-\mathds{N}^{\ast}$ if $f^{n}(x)=x$.
-\end{definition}
-
-\begin{definition} \label{def4}
-$f$ is said to be {\bf regular} on $(\mathcal{X},\tau)$ if the set of
- periodic points for $f$ is dense in $\mathcal{X}$ ( for any $x \in
- \mathcal{X}$, we can find at least one periodic point in any of its
- neighborhood).
-\end{definition}
-
-This regularity ``counteracts'' the effects of transitivity. Thus,
-due to these two properties, two points close to each other can behave
-in a completely different manner, leading to unpredictability for the
-whole system. Then,
-
-\begin{definition} \label{sensitivity}
-$f$ has {\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 $f$.
-\end{definition}
-
-Finally,
-
-\begin{definition} \label{def5}
-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.
-\end{definition}
-
-Let us notice that for a metric space the last condition follows from
-the two first ones~\cite{Banks92}.
+given region. This property implies that a dynamical system cannot be
+broken into simpler subsystems. Intuitively, its complexity does not
+allow any simplification.
+
+However, chaos needs some regularity to ``counteracts'' the effects of
+transitivity. % Thus, two points close to each other can behave in a completely different manner, the first one visiting the whole space whereas the second one has a regular orbit.
+In the Devaney's formulation, a dense set of periodic
+points is the element of regularity that a chaotic dynamical system has
+to exhibit.
+%\begin{definition} \label{def3}
+We recall that a point $x$ is a {\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 to Devaney) when the corresponding dynamical system is
+chaotic, as it is defined in the Devaney's formulation.
+
+%Let us notice that for a metric space the last condition follows from
+%the two first ones~\cite{Banks92}.
\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.
+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 that section, a system
-under consideration iteratively modifies a collection of
-$n$~components. Each component $i \in \llbracket 1; n \rrbracket$
-takes its value $x_i$ among the domain $\Bool=\{0,1\}$. A~{\it
- configuration} of the system at discrete time $t$ is the vector
+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*}
% 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
+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 $N(i,x)$.
+$i$ 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 clasically refered as \emph{asynchronous iterations}.
-Next section shows the link between asynchronous iterations and
-Devaney's Chaos.
+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 usually referred as {\it asynchronous
+ iterations}. More precisely, we have for any $i$, $1\le i \le n$,
+\begin{equation}
+\left\{ \begin{array}{l}
+x^{0} \in \Bool^n \\
+x^{t+1}_i = \left\{
+\begin{array}{l}
+ f_i(x^t) \textrm{ if $S^t = i$}\enspace , \\
+ x_i^t \textrm{ otherwise}\enspace .
+ \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}.
+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
+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}
\right.
\end{equation}
-\noindent With such a notation, configurations are defined for times
-$t=0,1,2,\ldots$ by:
+\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}\\
\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 means that only one
-component of the system is updated at an iteration: the $S^t$-th
-element. But it can be extended by considering subsets for $S^t$.
-
-
-
-
-The {\bf distance} $d$ between two points $(S,x)$ and
-$(\check{S},\check{x})\in \mathcal{X} = \llbracket1;n\rrbracket^\Nats
-\times \Bool^{n}$ is defined by
+where $\sigma$ is the so-called shift function that removes the first
+term of the strategy ({\it i.e.},~$S^0$). This definition allows to
+link asynchronous iterations with classical iterations of a dynamical
+system. Note that it can be extended by considering subsets for $S^t$.
+
+To study topological properties of these iterations, we are then left
+to introduce a {\bf distance} $d$ between two points $(S,x)$ and
+$(\check{S},\check{x})$ in $\mathcal{X} =
+\llbracket1;n\rrbracket^\Nats \times \Bool^{n}$. Let $\Delta(x,y) = 0$
+if $x=y$, and $\Delta(x,y) = 1$ else, be a distance on
+$\mathds{B}$. The distance $d$ is defined by
\begin{equation}
d((S,x);(\check{S},\check{x}))=d_{e}(x,\check{x})+d_{s}(S,\check{S})
\enspace ,
This distance is defined to reflect the following
information. Firstly, the more two systems have different components,
-the larger the distance between them is. Secondly, two systems with
+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 fulfill
+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
+The relation between $\Gamma(f)$ and $G_f$ is obvious: there exists a
path from $x$ to $x'$ in $\Gamma(f)$ if and only if there exists a
-strategy $s$ such that the parallel iteration of $G_f$ from the
-initial point $(s,x)$ reaches the configuration $x'$. Using this
-link, Guyeux~\cite{GuyeuxThese10} has proven that,
+strategy $s$ such that iterations of $G_f$ from the initial point
+$(s,x)$ reach the configuration $x'$. Using this link,
+Guyeux~\cite{GuyeuxThese10} has proven that,
\begin{theorem}%[Characterization of $\mathcal{C}$]
\label{Th:Caracterisation des IC chaotiques}
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. For
-example, consider the function $f_1\left(x_1,\dots,x_n\right)=\left(
-\overline{x_1},x_1,x_2,\dots,x_{n-1}\right)$. As $\Gamma(f_1)$ is
-obviously strongly connected, then $G_{f_1}$ is a chaotic map.
+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 functions $f_0$ and $f_1$ both in
+$\Bool^n\to\Bool^n$ that are used all along this paper. The former is
+the vectorial negation, \textit{i.e.}, $f_{0}(x_{1},\dots,x_{n})
+=(\overline{x_{1}},\dots,\overline{x_{n}})$. The latter is
+$f_1\left(x_1,\dots,x_n\right)=\left(
+\overline{x_1},x_1,x_2,\dots,x_{n-1}\right)$. 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 firstly introduce the vectorial negation
-$f_{0}(x_{1},\dots,x_{n}) =(\overline{x_{1}},\dots,\overline{x_{n}})$,
-which is such that $\Gamma(f_0)$ is strongly connected. Considering
-the map $F_{f_0}:\llbracket 1; n \rrbracket \times \mathds{B}^n \to
-\mathds{B}^n$ associated to the vectorial negation, we can build a
-multilayer perceptron neural network modeling $F_{f_0}$. Hence, 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}).
+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
\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. This new output, which represents the new state of the
- dynamical system, satisfies:
+ (\textit{i.e.}, $S^0 \in \llbracket 1;n\rrbracket$) are used to
+ compute the new output vector. This new vector, which represents
+ the new state of the dynamical system, satisfies:
\begin{equation}
x^{t+1}=F_{f_0}(S^0, x^t) \in \mathds{B}^n \enspace .
\end{equation}
given as outside input, then the sequence of successive published
output vectors $\left(x^t\right)^{t \in \mathds{N}^{\ast}}$ is exactly
the one produced by the chaotic iterations formally described in
-Eq.~(\ref{eq:CIs}). It means that mathematically if we use similar
+Eq.~(\ref{eq:Gf}). It means that mathematically if we use similar
input vectors they both generate the same successive outputs
$\left(x^t\right)^{t \in \mathds{N}^{\ast}}$, and therefore that they
are equivalent reformulations of the iterations of $G_{f_0}$ in
-$\mathcal{X}$. Finally, since the proposed neural network is build to
-model the behavior of $G_{f_0}$, which is chaotic according to
-Devaney's definition of chaos, we can conclude that the network is
-also chaotic in this sense.
+$\mathcal{X}$. Finally, since the proposed neural network is built to
+model the behavior of $G_{f_0}$, whose iterations are chaotic
+according to the 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
\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
+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 plan to study a
-generalization of this approach in a future work.
+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.
+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 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 $\left(x^{t+1}_1,\dots,x^{t+1}_n\right)$.
\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}
-Then $F=F_f$. If this recurrent neural network is seeded with
+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
+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
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 $f^n(U) \cap V \neq \emptyset$.
+\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}
-As proven in Ref.~\cite{gfb10:ip}, 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.
+\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. First of all, the topological
-transitivity property implies indecomposability.
+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
+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}
\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
+ 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}
-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
+
+\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, these recurrent neural networks exhibit the instability
property:
\begin{definition}
-A dynamical system $\left( \mathcal{X}, f\right)$ is unstable if for
+A dynamical system $\left( \mathcal{X}, f\right)$ is {\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}
-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.
+
+\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
-$$d_n(x,y)=\max\{d(f^i(x),f^i(y)): 0\leq i<n\} \enspace .$$
+\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 $n$
\varepsilon)$ the maximum cardinality of an $(n,
\varepsilon)$-separated set,
\begin{definition}
-The {\it topological entropy} of the map $f$ is defined by (see e.g.,
+The {\bf topological entropy} of the map $f$ is defined by (see e.g.,
Ref.~\cite{Adler65} or Ref.~\cite{Bowen})
$$h(f)=\lim_{\varepsilon\to 0} \left(\limsup_{n\to \infty}
\frac{1}{n}\log H(n,\varepsilon)\right) \enspace .$$
\begin{figure}
\centering
- \includegraphics[scale=0.625]{scheme}
+ \includegraphics[scale=0.5]{scheme}
\caption{Summary of addressed neural networks and chaos problems}
\label{Fig:scheme}
\end{figure}
-The Figure~\ref{Fig:scheme} is a summary of addressed neural networks and chaos problems.
-Section~\ref{S2} has 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 $A$ 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 Artificial Neural Networks
-for Predicting Chaotic Behaviors}
+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 sense 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
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
+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
The problem of deciding whether classical feedforward ANNs are
suitable to approximate topological chaotic iterations may then be
-reduced to evaluate ANNs 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
-Perceptron.
+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
$1110$: 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. Next section shows how to translate iterations of
-such functions into a model amenable to be learned by an ANN.
-
-This section presents how (not) chaotic iterations of $G_f$ are
-translated into another model more suited to artificial neural
-networks. 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}).
+$\Gamma(g)$ 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
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 closed in a
+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.
-Let us secondly 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
+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
\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. 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
+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
are compared.
Thereafter we give, for the different learning setups and data sets,
-the mean prediction success rate obtained for each output. 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.
+the mean prediction success rate obtained for each output. Such a rate
+represents 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.}
\centering {\small
\begin{tabular}{|c|c||c|c|c|}
\hline
-\multicolumn{5}{|c|}{Networks topology: 6~inputs, 5~outputs and one hidden layer} \\
+\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} \\
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 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.
-
-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.
+configurations. For all these 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 feedforward networks are unable to learn them.
+
+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.
+Figures~\ref{Fig:chaotic_predictions} and
+\ref{Fig:non-chaotic_predictions} present the predictions given by two
+feedforward multilayer perceptrons that were respectively trained to
+learn chaotic and non-chaotic data, using the second coding scheme.
+Each figure shows for each sample of the test subset (577~samples,
+representing 25\% of the 2304~samples) the configuration that should
+have been predicted and the one given by the multilayer perceptron. It
+can be seen that for the chaotic data the predictions are far away
+from the expected configurations. Obviously, the better predictions
+for the non-chaotic data reflect their regularity.
Let us now compare the two coding schemes. Firstly, the second scheme
disturbs the learning process. In fact in this scheme the
configuration is always expressed as a natural number, whereas in the
-first one the number of inputs follows the increase of the boolean
+first one the number of inputs follows the increase of the Boolean
vectors coding configurations. In this latter case, the coding gives a
finer information on configuration evolution.
-
\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} \\
+\multicolumn{5}{|c|}{Networks topology: 3~inputs, 2~outputs, and one hidden layer} \\
\hline
\hline
& Hidden neurons & \multicolumn{3}{c|}{10 neurons} \\
\end{tabular}
\end{table}
+\begin{figure}
+ \centering
+ \includegraphics[scale=0.5]{chaotic_trace2}
+ \caption {Second coding scheme - Predictions obtained for a chaotic test subset.}
+ \label{Fig:chaotic_predictions}
+\end{figure}
+
+\begin{figure}
+ \centering
+ \includegraphics[scale=0.5]{non-chaotic_trace2}
+ \caption{Second coding scheme - Predictions obtained for a non-chaotic test subset.}
+ \label{Fig:non-chaotic_predictions}
+\end{figure}
+
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
-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.
+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.}
\centering
\begin{tabular}{|c||c|c|c|}
\hline
-\multicolumn{4}{|c|}{Networks topology: 3~inputs, 1~output and one hidden layer} \\
+\multicolumn{4}{|c|}{Networks topology: 3~inputs, 1~output, and one hidden layer} \\
\hline
\hline
Epochs & 125 & 250 & 500 \\
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
+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. Then, 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
+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
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. Furthermore, iterations such that not all
-of the components are updated at each step are very common in
-biological and physics mechanisms. Therefore, one can reasonably
-wonder whether neural networks should be applied in these contexts.
+information hiding schemes.
In future work we intend to enlarge the comparison between the
learning of truly chaotic and non-chaotic behaviors. Other
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. Lastly, thresholds separating systems depending on
-the ability to learn their dynamics will be established.
+will then 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.
