X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/chaos1.git/blobdiff_plain/8c453843ba5ef7f01a7a202686584c36cc7201c5..e2b5fffa698ba0b8084b270892f0d20ed6cf64c0:/main.tex

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
