\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
\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
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
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}
-
-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,
+%\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 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, 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.
-\end{definition}
+In what follows, iterations are said to be \emph{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}.
+%Let us notice that for a metric space the last condition follows from
+%the two first ones~\cite{Banks92}.
\subsection{Asynchronous Iterations}
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
+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
+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 clasically refered as \emph{asynchronous iterations}.
+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.
\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.
+%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
+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 $\mathcal{C}$]
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}
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
-not decomposable} 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})
$$h(f)=\lim_{\varepsilon\to 0} \left(\limsup_{n\to \infty}
\frac{1}{n}\log H(n,\varepsilon)\right) \enspace .$$
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}).
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}
