In what follows and for any function $f$, $f^n$ means the composition
$f \circ f \circ \hdots \circ f$ ($n$ times) and an \emph{iteration}
-of a \emph{dynamical system} the step that consists in
-updating the global state $x$ with respect to a function $f$ s.t.
-$x^{t+1} = f(x^t)$.
+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 {\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}
-
-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 {\emph{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 {\emph{ 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 {\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
+%\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 $f$.
-\end{definition}
-
-Finally,
-\begin{definition} \label{def5}
-The dynamical system that iterates $f$ is {\emph{ chaotic according to Devaney}}
-on $(\mathcal{X},\tau)$ if $f$ is regular, topologically transitive,
+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.
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}.
-More préciselly, we have here
+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
%element. But it can be extended by considering subsets for $S^t$.
-To study topological properties of these iteations, we are then left to
+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
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 $\mathcal{C}$]
Eq.~(\ref{eq:Gf}) are chaotic according to Devaney.
-Let us then consider two function $f_0$ and $f_1$ both in
-$\Bool^n\to\Bool^n$ defined as follows that are used all along this article.
+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(
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 built to
-model the behavior of $G_{f_0}$, which is chaotic according 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.
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}
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})
$$h(f)=\lim_{\varepsilon\to 0} \left(\limsup_{n\to \infty}
\frac{1}{n}\log H(n,\varepsilon)\right) \enspace .$$
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}
