X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/chaos1.git/blobdiff_plain/9e6c76e39059d7f227f7ed48ad71195374a2f2a5..e2b5fffa698ba0b8084b270892f0d20ed6cf64c0:/main.tex diff --git a/main.tex b/main.tex index de92eaa..82e99bf 100644 --- a/main.tex +++ b/main.tex @@ -40,7 +40,7 @@ preprint,% \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 } @@ -97,7 +97,7 @@ work a theoretical framework based on the Devaney's definition of 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 @@ -110,7 +110,7 @@ chaotic maps, is far more difficult than non chaotic behaviors. \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 @@ -136,8 +136,8 @@ are suitable to model nonlinear relationships between data, due to 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}. @@ -151,7 +151,7 @@ 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 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 @@ -1072,13 +1072,13 @@ Chaotic/non chaotic & \multicolumn{3}{c|}{Output = Strategy} \\ 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 @@ -1092,10 +1092,7 @@ implemented in a new steganographic method \cite{guyeux10ter}. As 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 @@ -1104,8 +1101,7 @@ be investigated too, to discover which tools are the most relevant 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{}