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{}
