\begin{document}
\title[Neural Networks and Chaos]{Neural Networks and Chaos:
-Construction, Evaluation of Chaotic Networks \\
+Construction, Evaluation of Chaotic Networks, \\
and Prediction of Chaos with Multilayer Feedforward Networks
}
chaos is introduced. Starting with a relationship between discrete
iterations and Devaney's chaos, we firstly show how to build a
recurrent neural network that is equivalent to a chaotic map and
-secondly a way to check whether an already available network, is
+secondly a way to check whether an already available network is
chaotic or not. We also study different topological properties of
these truly chaotic neural networks. Finally, we show that the
learning, with neural networks having a classical feedforward
\label{S1}
Several research works have proposed or used chaotic neural networks
-these last years. The complex dynamics of such a network leads to
+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
their universal approximator capacity
\cite{Cybenko89,DBLP:journals/nn/HornikSW89}. Thus, this kind of
networks can be trained to model a physical phenomenon known to be
-chaotic such as Chua's circuit \cite{dalkiran10}. Sometimes, a neural
-network which is build by combining transfer functions and initial
+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}.
precisely, in this paper, which is an extension of a previous work
\cite{bgs11:ip}, we establish the equivalence between chaotic
iterations and a class of globally recurrent MLP. The second
-contribution is a study of the converse problem, indeed we study the
+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 which
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 be stated.
% \appendix{}
