From 44232560a3cd3ab96e3c35d7e4445f5185fb52d3 Mon Sep 17 00:00:00 2001 From: Christophe Guyeux Date: Fri, 13 Mar 2015 17:02:11 +0100 Subject: [PATCH 1/1] =?utf8?q?avanc=C3=A9es?= MIME-Version: 1.0 Content-Type: text/plain; charset=utf8 Content-Transfer-Encoding: 8bit --- intro.tex | 22 +++++++++------------- 1 file changed, 9 insertions(+), 13 deletions(-) diff --git a/intro.tex b/intro.tex index b2cdd1c..b3e6754 100644 --- a/intro.tex +++ b/intro.tex @@ -1,6 +1,6 @@ The exploitation of chaotic systems to generate pseudorandom sequences is an hot topic~\cite{915396,915385,5376454}. Such systems are fundamentally -chosen due to their unpredictable character and their sensibility to initial conditions. +chosen due to their unpredictable character and their sensitiveness to initial conditions. In most cases, these generators simply consist in iterating a chaotic function like the logistic map~\cite{915396,915385} or the Arnold's one~\cite{5376454}\ldots It thus remains to find optimal parameters in such functions so that attractors are @@ -9,18 +9,14 @@ In order to check the quality of the produced outputs, it is usual to test the PRNGs (Pseudo-Random Number Generators) with statistical batteries like the so-called DieHARD~\cite{Marsaglia1996}, NIST~\cite{Nist10}, or TestU01~\cite{LEcuyerS07}. -% -% Dans son acception vulgarisée, -% la notion de chaos est souvent réduite à celle de forte sensibilité -% aux conditions initiales (le fameux \og \emph{effet papillon}\fg{}): -% une fonction continue $k$ définie sur un espace métrique -% est dite \emph{fortement sensible aux conditions initiales} si pour tout -% point $x$ et pour toute valeur positive $\epsilon$ -% il est possible de trouver un point $y$, arbitrairement proche -% de $x$, et un entier $t$ tels que la distance entre les -% $t^{\textrm{ièmes}}$ itérés de $x$ et de $y$ -% -- notés $k^t(x)$ et $k^t(y)$ -% -- est supérieure à $\epsilon$. +In its general understanding, the chaos notion is often reduced to the strong +sensitiveness to the initial conditions (the well known ``butterfly effect''): +a continuous function $k$ defined on a metrical space is said +\emph{strongly sensitive to the initial conditions} if for all point +$x$ and all positive value $\epsilon$, it is possible to find another +point $y$, as close as possible to $x$, and an integer $t$ such that the distance +between the $t$-th iterates of $x$ and $y$, denoted by $k^t(x)$ and $k^t(y)$, +are larger than $\epsilon$. % Cependant, dans sa définition du chaos, % Devaney~\cite{Devaney} impose à la fonction chaotique deux autres propriétés % appelées \emph{transitivité} et \emph{régularité}, -- 2.39.5