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\vspace{-0.5cm}\hspace{-2cm}Computer Science Laboratory, LIFC

\hspace{-2cm}University of Franche-Comt\'e

\hspace{-2cm}25030 Besan\c{c}on, France.

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\begin{center}
Detailed changes and addressed issues in the revision of the paper

"Recurrent Neural Networks and Chaos: Construction, \\
Evaluation, and Prediction Ability" \\
renamed in \\
Neural Networks and Chaos: Construction, Evaluation of Chaotic Networks, \\
and Prediction of Chaos with Multilayer Feedforward Networks"

by Jacques M. Bahi, Jean-Fran\c{c}ois Couchot, \\
Christophe Guyeux, and Michel Salomon

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Please, find below the detailed changes and issues we addressed in the
revision of our above mentioned paper that we resubmit.

All the remarks  and recommendations of the three  reviewers have been
considered and have led to modifications in the paper.

\begin{itemize}
\item Concerning the remarks of reviewer No. 1:
\end{itemize}

\begin{enumerate}
\item Please  explain the topic "Recurrent Neural  Networks and Chaos:
  Construction,  Evaluation, and  Prediction Ability"  which  makes me
  confused. Does  the paper discuss the  construction, evaluation, and
  prediction ability of chaotic neural networks? And in the paper, the
  chaotic neural networks are mainly done research on.
  
  {\it Our response}

  Indeed, our  contributions are twofold.  Firstly, we present  how to
  construct chaotic neural networks, according to Devaney's definition
  of chaos,  and we discuss  their evaluation. Secondly, we  study the
  capacity  of  classical  multilayer  feedfoward neural  networks  to
  predict chaotic data. To highlight clearly these contributions we
  have changed the title of our paper, as said above.
\item Please  explain the Line  242( "than chaotic iterations  Ff with
  initial  condition............" ) and  Line 298(  "investigate, when
  comparing neural networks and Devaney's chaos").

  {\it Our response}

  To be completed
\item  Section  VI  analyzes  the  suitability  of  artificial  neural
  networks for  predicting chaotic behaviors.  So, I  think section VI
  don't correspond to the topic. And homoplastically, lines 30-32("the
  learning, with  neural networks  having a feedforward  structure, of
  chaotic  behaviors represented  by data  sets obtained  from chaotic
  maps, is  far more difficult  than non chaotic behaviors")  refer to
  the learning of neural  networks having a feedforward structure. But
  lines 402-405 refer to all the network topologies.
 
  {\it Our response}

  We  have corrected  lines 402-405  (now lines  419-422)  as follows:
  ``For all  those feedforward network topologies and  all outputs the
  obtained  results for  the non-chaotic  case outperform  the chaotic
  ones. Finally, the rates for  the strategies show that the different
  feedforward networks are unable  to learn them.'' More generally, in
  the whole paper the term feedforward has been added, when needed, to
  clarify our discussion.
\item Please  explain the  meanings of the  percentage in TABLE  I and
  TABLE II.

  {\it Our response}

  We  have explained how  the success  rates (percentage)  are computed
  lines~405-407.
\item Except  for prediction  success rates, in  order to  reflect the
  prediction ability, please add the analysis of the prediction errors
  for data sequence and diagram it.

 {\it Our response}

 We have added two figures (Figures~3 and 4) and some comments about them
 lines~425-431. We hope that we have answered the reviewer remark.
\end{enumerate}

\begin{itemize}
\item Concerning the remarks of reviewer No. 2:
\end{itemize}

\begin{enumerate}
\item On  the basis of the  result botained by Guyeux  in Ref[12], the
  authors dealed  with chaotic neural  networks for various  fields of
  appli1cation.  Firstly, the authors  described how to build a neural
  network that can  be trained to learn a  given chaotic map function,
  then 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.  As  the authors said that this  work is different from
  most of prviews works,  this manuscript gave a rigorous mathematical
  proof  for  chaos  of  chaotic  neural  networks.  This  is  a  very
  interesting work.  On the other hand, I think, the conclusion is too
  long,  and some Definitions  such as  Definition 1-Definition  5 are
  wellknow Definition, it's no necessary to be presented any more.

  {\it Our response}

  As suggested by the reviewer the conclusion has been shortened and we
  have presented those definitions differently.
\end{enumerate}

We are very grateful to the reviewers who, by their recommendations,
allowed us to improve our paper.

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