X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/chaos1.git/blobdiff_plain/9e6c76e39059d7f227f7ed48ad71195374a2f2a5..e2b5fffa698ba0b8084b270892f0d20ed6cf64c0:/main.tex?ds=sidebyside

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{}
 
