\date{\today}
\newcommand{\CG}[1]{\begin{color}{red}\textit{#1}\end{color}}
+\newcommand{\JFC}[1]{\begin{color}{blue}\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 the
-Devaney's formulation of chaos and a particular class of neural
+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, some of them are proven to be chaotic as it is defined by
+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.
\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
\section{Introduction}
\label{S1}
+REVOIR TOUT L'INTRO et l'ABSTRACT en fonction d'asynchrone, chaotic
+
+
Several research works have proposed or run chaotic neural networks
these last years. The complex dynamics of such a networks leads to
various potential application areas: associative
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
+their universal approximator capacity.
+\JFC{Michel, peux-tu donner une ref la dessus}
+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
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
+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
+\cite{bgs11:ip}, we establish the equivalence between asynchronous
iterations and a class of globally recurrent MLP.
The investigation the converse problem is the second contribution:
we indeed study the ability for
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 can behave
-chaotically, as it is defined by Devaney, when the chosen function 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
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 chaotic iterations and Devaney's
+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
sets. The paper ends with a conclusion section where our contribution
is summed up and intended future work is exposed.
-\section{Link between Chaotic Iterations and Devaney's Chaos}
+\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 ``chaotic iterations'', which
-are thus introduced. The result establishing the link between chaotic
+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 \emph{iteration}
+of a \emph{dynamical system} is the step that consists in
+updating the global state $x^t$ at time $t$ with respect to a function $f$
+s.t. $x^{t+1} = f(x^t)$.
\subsection{Devaney's chaotic dynamical systems}
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.
+given region. This property implies that a dynamical system
+cannot be broken into simpler subsystems.
Intuitively, its complexity does not allow any simplification.
On the contrary, a dense set of periodic points is an
element of regularity that a chaotic dynamical system has to exhibit.
-\begin{definition} \label{def3}
-A point $x$ is called a {\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
+However, chaos need some regularity to ``counteracts''
+the effects of transitivity.
+%\begin{definition} \label{def3}
+We recall that a point $x$ is {\emph{periodic point}} for $f$ of
+period~$n \in \mathds{N}^{\ast}$ if $f^{n}(x)=x$.
+%\end{definition}
+Then, the map
+%\begin{definition} \label{def4}
+$f$ is {\emph{ regular}} on $(\mathcal{X},\tau)$ if the set of
+ periodic points for $f$ is dense in $\mathcal{X}$ (for any $x \in
\mathcal{X}$, we can find at least one periodic point in any of its
neighborhood).
-\end{definition}
+%\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.
-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}
+Let we recall that $f$
+has {\emph{ sensitive dependence on initial conditions}} if there
+exists $\delta >0$ such that, for any $x\in \mathcal{X}$ and any
+neighborhood $V$ of $x$, there exist $y\in V$ and $n > 0$ such that
+$d\left(f^{n}(x), f^{n}(y)\right) >\delta $. The value $\delta$ is called the
+ {\emph{constant of sensitivity}} of $f$.
-Finally,
+Finally, The dynamical system that iterates $f$ is {\emph{ chaotic according to Devaney}} on $(\mathcal{X},\tau)$ if $f$ is regular, topologically transitive,
+and has sensitive dependence to its initial conditions.
+In what follows, iterations are said to be \emph{chaotic according Devaney}
+when corresponding dynamical system is chaotic according Devaney.
-\begin{definition} \label{def5}
-$f$ 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}.
+%Let us notice that for a metric space the last condition follows from
+%the two first ones~\cite{Banks92}.
-\subsection{Chaotic Iterations}
+\subsection{Asynchronous Iterations}
-This section presents some basics on topological chaotic 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 function that has
-been theoretically proven on $\R$ remains valid on the floating-point
+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 $a~<~b$. A {\it system}
+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 $t$ (also said at {\it
- iteration} $t$) is the vector
+takes its value $x_i$ among the domain $\Bool=\{0,1\}$.
+A \emph{configuration} of the system at discrete time $t$ is the vector
%\begin{equation*}
$x^{t}=(x_1^{t},\ldots,x_{n}^{t}) \in \Bool^n$.
%\end{equation*}
obtained by switching the $i-$th component of $x$ ($\overline{x_i}$ is
indeed the negation of $x_i$). Intuitively, $x$ and $N(i,x)$ are
neighbors. The discrete iterations of $f$ are represented by the
-oriented {\it graph of iterations} $\Gamma(f)$. In such a graph,
+oriented \emph{graph of iterations} $\Gamma(f)$. In such a graph,
vertices are configurations of $\Bool^n$ and there is an arc labeled
$i$ from $x$ to $N(i,x)$ if and only if $f_i(x)$ is $N(i,x)$.
-In the sequel, the {\it strategy} $S=(S^{t})^{t \in \Nats}$ is the
-sequence defining the component to update at time $t$ and $S^{t}$
-denotes its $t-$th term. We introduce the function $F_{f}$ that is
+In the sequel, the \emph{strategy} $S=(S^{t})^{t \in \Nats}$ is the
+sequence defining which component to update at time $t$ and $S^{t}$
+denotes its $t-$th term.
+This iteration scheme that only modifies one element at each iteration
+is classically referred as \emph{asynchronous iterations}.
+More precisely, we have here for any $i$, $1\le i \le n$,
+$$
+\left\{ \begin{array}{l}
+x^{0} \in \Bool^n \\
+x^{t+1}_i = \left\{
+\begin{array}{l}
+ f_i(x^t) \textrm{ if $S^t = i$} \\
+ x_i^t \textrm{ otherwise}
+ \end{array}
+\right.
+\end{array} \right.
+$$
+
+Next section shows the link between asynchronous iterations and
+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.
\right.
\end{equation}
-\noindent With such a notation, configurations are defined for times
+\noindent With such a notation, configurations
+asynchronously obtained are defined for times
$t=0,1,2,\ldots$ by:
\begin{equation}\label{eq:sync}
\left\{\begin{array}{l}
\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 $S^t$.
+strategy ({\it i.e.},~$S^0$).
+This definition allows to links asynchronous iterations with
+classical iterations of a dynamical system.
-Let us finally remark that, despite the use of the term {\it chaotic},
-there is {\it priori} no connection between these iterations and the
-mathematical theory of chaos presented previously.
-\subsection{Chaotic Iterations and Devaney's Chaos}
+%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 $S^t$.
-In this subsection we recall the link we have established between
-chaotic iterations and Devaney's chaos. The theoretical framework is
-fully described in \cite{guyeux09}.
-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
+To study topological properties of these iterations, we are then left to
+introduce a {\emph{ distance}} $d$ between two points $(S,x)$ and
+$(\check{S},\check{x})\in \mathcal{X} = \llbracket1;n\rrbracket^\Nats.
+\times \Bool^{n}$. It is defined by
\begin{equation}
d((S,x);(\check{S},\check{x}))=d_{e}(x,\check{x})+d_{s}(S,\check{S})
\enspace ,
}\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,
+Notice that the more two systems have different components,
the larger the distance between them is. Secondly, two systems with
similar components and strategies, which have the same starting terms,
must induce only a small distance. The proposed distance fulfill
The relation between $\Gamma(f)$ and $G_f$ is clear: there exists a
path from $x$ to $x'$ in $\Gamma(f)$ if and only if there exists a
-strategy $s$ such that the parallel iteration of $G_f$ from the
+strategy $s$ such that iterations of $G_f$ from the
initial point $(s,x)$ reaches the configuration $x'$. Using this
link, Guyeux~\cite{GuyeuxThese10} has proven that,
\begin{theorem}%[Characterization of $\mathcal{C}$]
\label{Th:Caracterisation des IC chaotiques}
-Let $f:\Bool^n\to\Bool^n$. $G_f$ is chaotic (according to Devaney)
-if and only if $\Gamma(f)$ is strongly connected.
+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 function $f_0$ and $f_1$ both in
+$\Bool^n\to\Bool^n$ that are used all along this article.
+The former is the vectorial negation, \textit{i.e.},
+$f_{0}(x_{1},\dots,x_{n}) =(\overline{x_{1}},\dots,\overline{x_{n}})$.
+The latter is $f_1\left(x_1,\dots,x_n\right)=\left(
+\overline{x_1},x_1,x_2,\dots,x_{n-1}\right)$.
+It is not hard to see 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
+Firstly, 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 produces $F_{f_0}(s,x)$. 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
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, then the sequence of successive published
+given as outside input,
+\JFC{en dire davantage sur l'outside world}
+ 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
+$\mathcal{X}$. 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.
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 $n$
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
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
+\rrbracket \times \mathds{B}^n$
+\JFC{ici, cela devait etre $S^t$ et pas $s$, nn ?}
+ 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 $f:
+$\left(s,\left(x_1,\dots, x_n\right)\right)$.
+\JFC{ici, cela devait etre $S^t$ et pas $s$, nn ?}
+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}
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 speakig, such iterations of $F_f$ are thus a formal model of
+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''.
contrarily, their unpredictable behavior. An example of such a study
is given in the next section.
+\JFC{Ce qui suit est davantage qu'un exemple.Il faudrait
+motiver davantage, non?}
+
+
\section{Topological properties of chaotic neural networks}
\label{S4}
theory of chaos.
\begin{definition} \label{def8}
-A function $f$ is said to be {\bf expansive} if $\exists
+A function $f$ is said to be {\emph{ 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}
\begin{definition} \label{def9}
-A discrete dynamical system is said to be {\bf topologically mixing}
+A discrete dynamical system is said to be {\emph{ 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}
+\JFC{Donner un sens à ces definitions}
+
-As proven in Ref.~\cite{gfb10:ip}, chaotic iterations are expansive
-and topologically mixing when $f$ is the vectorial negation $f_0$.
+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
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.
+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
-indecomposable} if it is not the union of two closed sets $A, B
+A dynamical system $\left( \mathcal{X}, f\right)$ is
+{\emph{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}
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
+A dynamical system $\left( \mathcal{X}, f\right)$ is {\emph{ 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}
Furthermore, those recurrent neural networks exhibit the instability
property:
\begin{definition}
-A dynamical system $\left( \mathcal{X}, f\right)$ is unstable if for
+A dynamical system $\left( \mathcal{X}, f\right)$ is \emph{unstable}
+if for
all $x \in \mathcal{X}$, the orbit $\gamma_x:n \in \mathds{N}
\longmapsto f^n(x)$ is unstable, that means: $\exists \varepsilon >
0$, $\forall \delta>0$, $\exists y \in \mathcal{X}$, $\exists n \in
\varepsilon)$ the maximum cardinality of an $(n,
\varepsilon)$-separated set,
\begin{definition}
-The {\it topological entropy} of the map $f$ is defined by (see e.g.,
+The \emph{topological entropy} of the map $f$ is defined by (see e.g.,
Ref.~\cite{Adler65} or Ref.~\cite{Bowen})
$$h(f)=\lim_{\varepsilon\to 0} \left(\limsup_{n\to \infty}
\frac{1}{n}\log H(n,\varepsilon)\right) \enspace .$$
\begin{figure}
\centering
\includegraphics[scale=0.625]{scheme}
- \caption{Summary of addressed membership problems}
+ \caption{Summary of addressed neural networks and chaos problems}
\label{Fig:scheme}
\end{figure}
-The Figure~\ref{Fig:scheme} is a summary of the addressed problems.
+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
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~$((S^t)^{t \in
+networks.
+\JFC{détailler le more suited}
+Formally, input and output vectors are pairs~$((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}).
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 backpropagation, can learn successfully the
+with Bayesian Regulation back-propagation, can learn successfully the
dynamics of Chua's circuit.
-In these experimentations we consider MLPs having one hidden layer of
+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
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}
will be stated. Lastly, thresholds separating systems depending on
the ability to learn their dynamics will be established.
+% \appendix{}
+
+
+
+% \begin{definition} \label{def2}
+% A continuous function $f$ on a topological space $(\mathcal{X},\tau)$
+% is defined to be {\emph{topologically transitive}} if for any pair of
+% open sets $U$, $V \in \mathcal{X}$ there exists
+% $k \in
+% \mathds{N}^{\ast}$
+% such that
+% $f^k(U) \cap V \neq \emptyset$.
+% \end{definition}
+
+
\bibliography{chaos-paper}% Produces the bibliography via BibTeX.
\end{document}
