From e2b5fffa698ba0b8084b270892f0d20ed6cf64c0 Mon Sep 17 00:00:00 2001 From: guyeux Date: Sun, 9 Oct 2011 15:03:54 +0200 Subject: [PATCH] =?utf8?q?Avanc=C3=A9es=20dans=20la=20relecture?= MIME-Version: 1.0 Content-Type: text/plain; charset=utf8 Content-Transfer-Encoding: 8bit --- main.tex | 12 ++++++------ 1 file changed, 6 insertions(+), 6 deletions(-) diff --git a/main.tex b/main.tex index 0ce8df0..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 -- 2.39.5