From b3411a22f651c0dbca34dca87df92f8d3d130e1a Mon Sep 17 00:00:00 2001 From: couchot Date: Fri, 22 May 2015 10:14:56 +0200 Subject: [PATCH] cs --- 12TIPE.tex | 81 +-------------------- 15TSI.tex | 115 ++++++++++++++++++++++++++++++ caracgeneralise.tex | 97 ++++++++++++++++++++++++++ caracunaire.tex | 50 ++++++------- devaney.tex | 85 ++++++++++++++++++++++ main.pdf | Bin 1850414 -> 1864933 bytes main.tex | 16 +++++ preuveDistanceGeneralisee.tex | 33 +++++++++ sdd.tex | 128 ++++++++++++++++++---------------- 9 files changed, 439 insertions(+), 166 deletions(-) create mode 100644 15TSI.tex create mode 100644 caracgeneralise.tex create mode 100644 devaney.tex create mode 100644 preuveDistanceGeneralisee.tex diff --git a/12TIPE.tex b/12TIPE.tex index 4171437..e735c46 100644 --- a/12TIPE.tex +++ b/12TIPE.tex @@ -1,84 +1,5 @@ -La première section rappelle ce que sont les systèmes dynamiques chaotiques. -\section{Systèmes dynamiques chaotiques selon Devaney} -\label{subsec:Devaney} -Dans cette partie, les définitions fondamentales liées au chaos -dans les systèmes booléens sont rappelées. - - -Soit un espace topologique $(\mathcal{X},\tau)$ et une fonction continue $f : -\mathcal{X} \rightarrow \mathcal{X}$. - - - - -\begin{Def}[Chaos selon Devaney~\cite{Devaney}] -La fonction $f$ \emph{chaotique} sur $(\mathcal{X},\tau)$ -si elles est régulière et topologiquement transitive. -\end{Def} - - - -\begin{Def}[Transitivite topologique] -La fonction $f$ est dite \emph{topologiquement transitive} si, -pour chaque paire d'ensembles ouverts -$U,V \subset \mathcal{X}$, -il existe $k>0$ tel que $f^k(U) \cap V \neq -\varnothing$. -\end{Def} - -\begin{Def}[Point périodique] - Un point $P \in \mathcal{X}$ est dit \emph{périodique} de période $t$ pour - une fonction $k$ si $t$ est un entier naturel non nul tel que $k^t(P) = P$ et - pour tout $n$, $0 < n \le t-1$, on a $k^n(P) \neq P$. - Par la suite, $\emph{Per(k)}$ dénote l'ensemble des points périodiques de - $k$ dans $\mathcal{X}$ de période quelconque. -\end{Def} - - - -\begin{Def}[Régularité] -La fonction $f$ est dite \emph{régulière} -sur $(\mathcal{X}, \tau)$ si l'ensemble des points périodiques - de $f$ is dense in $\mathcal{X}$: -pour chaque point $x \in \mathcal{X}$, chaque voisin -de $x$ contient au moins un point périodique -(sans que la période soit nécessairement constante). -\end{Def} - - - - - - - - - -La propriété de chaos est souvent associée à la notion de -\og sensibilité aux conditions initiales\fg{}, notion définie elle -sur un espace métrique $(\mathcal{X},d)$ par: - - -\begin{Def}[Forte sensibilité aux conditions initiales] -Une fonction $k$ définie sur $(\mathcal{X},\tau)$ -est \emph{fortement sensible aux conditions initiales} -s'il existe une valeur $\epsilon> 0$ telle que -pour tout $X \in \mathcal{X}$ et pour tout - $\delta > 0$, il existe $Y \in \mathcal{X}$ et -$t \in \Nats$ qui vérifient $d(X,Y) < \delta$ et -$d(k^t(X) , k^t(Y)) > \epsilon$. - -La constante $\delta$ est appelée \emph{constante de sensibilité} de $f$. -\end{Def} - -John Banks et ses collègues ont cependant -démontré que la sensibilité aux conditions initiales est une conséquence -de la régularité et de la transitivité topologique~\cite{Banks92}. - - - -\section{Schéma unaire} Soit ${\mathsf{N}}$ un entier naturel et $f$ une fonction de $\Bool^{{\mathsf{N}}}$ dans lui-même. @@ -221,6 +142,6 @@ si et seulement si $\Gamma(f)$ est fortement connexe. -\section{Schéma généralisé} + diff --git a/15TSI.tex b/15TSI.tex new file mode 100644 index 0000000..47ecd36 --- /dev/null +++ b/15TSI.tex @@ -0,0 +1,115 @@ +On reprend ici le même plan que dans la section précédente. + +Soit ${\mathsf{N}}$ un entier naturel et $f$ une fonction de +$\Bool^{{\mathsf{N}}}$ dans lui-même. + + +\subsection{Des itérations généralisées aux itérations parallèles} + +Dans le schéma généralisé, à la $t^{\textrm{ème}}$ itération, +c'est l'ensemble +des $s_{t}^{\textrm{ème}}$ éléments (inclus dans $[n]$) qui +sont mis à jour (c.f. équation~(\ref{eq:schema:generalise})). +On redéfinit la fonction la fonction + $F_f: \Bool^{\mathsf{N}} \times \mathcal{P}(\{1, \ldots, \mathsf{N}\}) + \rightarrow \Bool^{\mathsf{N}}$ par + \[ + F_f(x,s)_i=\left\{ + \begin{array}{l} + f_i(x) \textrm{ si $i \in s$;}\\ + x_i \textrm{ sinon.} + \end{array}\right. + \] + +Dans ce schéma d'itérations généralisées, +pour une configuration initiale +$x^0\in\Bool^{\mathsf{N}}$ et une stratégie $S = \left(s_t\right)_{t \in \mathds{N}} +\in \mathcal{P}(\{1, \ldots, {\mathsf{N}}\})^{\Nats}$, +les +configurations $x^t$ sont définies par la récurrence +\begin{equation}\label{eq:asyn} + x^{t+1}=F_f(s_t,x^t). + \end{equation} + Soit alors $G_f$ une fonction de $\Bool^{\mathsf{N}} \times \mathcal{P}(\{1, \ldots, {\mathsf{N}}\})^{\Nats}$ + dans lui-même définie par + \[ + G_f(S,x)=(\sigma(S),F_f(s_0,x)), + \] + où la fonction $\sigma$ est définit comme à la section précédente. + A nouveau, les itérations généralisées + de $f$ induites par $x^0$ et la stratégie $S$. + décrivent la même orbite que les + itérations parallèles de $G_f$ depuis un point initial + $X^0=(x^0,S)$ + + + +%%%%%%%%%%%%%%%%%%%% +On peut alors construire l'espace +$\mathcal{X} = \Bool^{\mathsf{N}} \times +\mathcal{P}(\{1, \ldots, {\mathsf{N}}\})^{\Nats}$ + +\subsection{Une métrique pour $\mathcal{X}$} + +Cette nouvelle distance va comparer des ensembles. +On rappelle pour quelques notions ensemblistes. +Pour $A$ et $B$ deux ensembles de l'univers $\Omega$, +on rappelle la définition de l'opérateur +de \emph{différence ensembliste} symétrique : +\[ +A \Delta B = (A \cap \overline{B}) \cup (\overline{A} \cap B) +\] +où $\overline{B}$ désigne le complémentaire de $B$ dans $\Omega$. + +On considère l'espace $\mathcal{X}=\mathcal{P}(\{1, \ldots, {\mathsf{N}}\})^{\Nats}\times +\Bool^{\mathsf{N}}$ et +on définit la distance $d$ entre les points $X=(S,x)$ et +$X'=(S',x')$ de $\mathcal{X}$ par +\[ +d(X,X')= d_H(x,x')+d_S(S,S'),~\textrm{où}~ +\left\{ +\begin{array}{l} +\displaystyle{d_H(x,x')=\sum_{i=1}^{\mathsf{N}} |x_i-x'_i|}\\[5mm] +\displaystyle{d_S(S,S')=\frac{9}{{\mathsf{N}}}\sum_{t\in\Nats}\frac{|S_t \Delta S'_t|}{10^{t+1}}}. +\end{array} +\right.\,. +\] + +La fonction $d$ est une somme de deux fonctions. +La fonction $d_H$ est la distance de Hamming; il est aussi établi que la +somme de deux distances est une distance. +Ainsi, pour montrer que $d$ est aussi une distance, il suffit +de montrer que $d_S$ en une aussi, ce qui est fait en annexe~\ref{anx:distance:generalise}. + +La section suivante caractérise les fonctions $f$ qui sont +chaotiques pour le schéma généralisées. + + +\subsection{Caractérisation des fonctions chaotiques +pour le schéma généralisé} +On a les théorèmes suivants dont les preuves sont données en +annexe~\ref{anx:chaos:generalise}. + +\begin{theorem} $G_f$ est transitive si et seulement si + $\Gamma(f)$ est fortement connexe. +\end{theorem} + +\begin{theorem} +\label{Prop: T est dans R} $\mathcal{T} \subset \mathcal{R}$. +\end{theorem} + + +\begin{theorem}%[Characterization of $\mathcal{C}$] +\label{Th:CaracIC} +Soit $f:\Bool^{\mathsf{N}}\to\Bool^{\mathsf{N}}$. La fonction $G_f$ est chaotique +si et seulement si $\Gamma(f)$ est fortement connexe. +\end{theorem} + + + + + + + + + diff --git a/caracgeneralise.tex b/caracgeneralise.tex new file mode 100644 index 0000000..e1c825a --- /dev/null +++ b/caracgeneralise.tex @@ -0,0 +1,97 @@ + +Commençons par caractériser l'ensemble $\mathcal{T}$ des fonctions transitives: + +\begin{theorem} $G_f$ est transitive si et seulement si + $\Gamma(f)$ est fortement connexe. +\end{theorem} + +\begin{Proof} + +$\Longleftarrow$ Supposons que $\Gamma(f)$ soit fortement connexe. +Soient $(x,S)$ et $(x',S')$ deux points de $\mathcal{X}$ et $\varepsilon >0$. +On construit la stratégie $\tilde S$ telle que la distance +entre $(x,\tilde S)$ et $(x,S)$ est inférieure à +$\varepsilon$ et telle que les itérations parallèles de $G_f$ depuis +$(x,\tilde S)$ mènent au point $(x',S')$. + +Pour cela, on pose $t_1 =-\lfloor\log_{10}(\varepsilon)\rfloor$ et $x''$ la +configuration de $\Bool^{\mathsf{N}}$ obtenue depuis $(x,S)$ +après $t_1$ itérations +parallèles de $G_f$. +Comme $\Gamma(f)$ est fortement connexe, il existe une +stratégie $S''$ et un entier $t_2$ tels que $x'$ est atteint depuis +$(x'',S'')$ après $t_2$ itérations de $G_f$. + +Considérons à présent la stratégie +$\tilde S=(s_0,\dots,s_{t_1-1},s''_0,\dots,s''_{t_2-1},s'_0,s'_1,s'_2,s'_3\dots)$. +Il est évident que $(x',s')$ est atteint depuis $(x,\tilde S)$ après +$t_1+t_2$ itérations parallèles de $G_f$. Puisque +$\tilde s_t=s_t$ pour $t0$. Pour +prouver que $f$ appartient à $\mathcal{R}$, il suffit de prouver +qu'il existe une stratégie $\tilde S$ telle que la distance entre +$(x,\tilde S)$ et $(x,S)$ est inférieure à $\varepsilon$ et telle que +$(x,\tilde S)$ est un point périodique. + +Soit $t_1=-\lfloor \log_{10}(\varepsilon)\rfloor$ et soit $x'$ la +configuration obtenue après $t_1$ itérations de $G_f$ depuis $(x,S)$. +D'après la proposition précédente, $\Gamma(f)$ est fortement connexe. +Ainsi, il existe une stratégie $S'$ et un nombre $t_2\in\Nats$ tels +que $x$ est atteint depuis $(x',S')$ après $t_2$ itérations de $G_f$. + +Soit alors la stratégie $\tilde S$ qui alterne les $t_1$ premiers termes +de $S$ avec les $t_2$ premiers termes de $S'$. +Ainsi $\tilde S$ est définie par +\begin{equation*} +(s_0,\dots,s_{t_1-1},s'_0,\dots,s'_{t_2-1},s_0,\dots,s_{t_1-1},s'_0,\dots,s'_{t_2-1},s_0,\dots). +\end{equation*} +Il est évident que $(x,\tilde S)$ s'obtient à partir de $(x,\tilde S)$ après +$t_1+t_2$ itérations parallèles de $G_f$. Ainsi $(x,\tilde S)$ est un point +périodique. Puisque $\tilde s_t$ est égal à $s_t$ pour $t0$. +Soient $(x,S)$ et $(x',S')$ deux points de $\mathcal{X}$ et $\varepsilon >0$. On construit la stratégie $\tilde S$ telle que la distance -entre $(\tilde S,x)$ et $(S,x)$ est inférieure à +entre $(x,\tilde S)$ et $(x,S)$ est inférieure à $\varepsilon$ et telle que les itérations parallèles de $G_f$ depuis -$(\tilde S,x)$ mènent au point $(S',x')$. +$(x,\tilde S)$ mènent au point $(x',S')$. Pour cela, on pose $t_1 =-\lfloor\log_{10}(\varepsilon)\rfloor$ et $x''$ la -configuration de $\Bool^n$ obtenue depuis $(S,x)$ après $t_1$ itérations +configuration de $\Bool^{\mathsf{N}}$ obtenue depuis $(x,S)$ après $t_1$ +itérations parallèles de $G_f$. Comme $\Gamma(f)$ est fortement connexe, il existe une stratégie $S''$ et un entier $t_2$ tels que $x'$ est atteint depuis -$(S'',x'')$ après $t_2$ itérations de $G_f$. +$(x'',S'')$ après $t_2$ itérations de $G_f$. Considérons à présent la stratégie $\tilde S=(s_0,\dots,s_{t_1-1},s''_0,\dots,s''_{t_2-1},s'_0,s'_1,s'_2,s'_3\dots)$. -Il est évident que $(s',x')$ est atteint depuis $(\tilde S,x)$ après +Il est évident que $(x',s')$ est atteint depuis $(x,\tilde S)$ après $t_1+t_2$ itérations parallèles de $G_f$. Puisque $\tilde s_t=s_t$ pour $t0$. Pour +Soit $(x,S) \in\mathcal{X}$ et $\varepsilon >0$. Pour prouver que $f$ appartient à $\mathcal{R}$, il suffit de prouver qu'il existe une stratégie $\tilde S$ telle que la distance entre -$(\tilde S,x)$ et $(S,x)$ est inférieure à $\varepsilon$ et telle que -$(\tilde S,x)$ est un point périodique. +$(x,\tilde S)$ et $(x,S)$ est inférieure à $\varepsilon$ et telle que +$(x,\tilde S)$ est un point périodique. Soit $t_1=-\lfloor \log_{10}(\varepsilon)\rfloor$ et soit $x'$ la -configuration obtenue après $t_1$ itérations de $G_f$ depuis $(S,x)$. +configuration obtenue après $t_1$ itérations de $G_f$ depuis $(x,S)$. D'après la proposition précédente, $\Gamma(f)$ est fortement connexe. Ainsi, il existe une stratégie $S'$ et un nombre $t_2\in\Nats$ tels -que $x$ est atteint depuis $(S',x')$ après $t_2$ itérations de $G_f$. +que $x$ est atteint depuis $(x',S')$ après $t_2$ itérations de $G_f$. Soit alors la stratégie $\tilde S$ qui alterne les $t_1$ premiers termes de $S$ avec les $t_2$ premiers termes de $S'$. @@ -84,10 +84,10 @@ Ainsi $\tilde S$ est définie par \begin{equation*} (s_0,\dots,s_{t_1-1},s'_0,\dots,s'_{t_2-1},s_0,\dots,s_{t_1-1},s'_0,\dots,s'_{t_2-1},s_0,\dots). \end{equation*} -Il est évident que $(\tilde S,x)$ s'obtient à partir de $(\tilde S,x)$ après -$t_1+t_2$ itérations parallèles de $G_f$. Ainsi $(\tilde S,x)$ est un point +Il est évident que $(x,\tilde S)$ s'obtient à partir de $(x,\tilde S)$ après +$t_1+t_2$ itérations parallèles de $G_f$. Ainsi $(x,\tilde S)$ est un point périodique. Puisque $\tilde s_t$ est égal à $s_t$ pour $t0$ tel que $f^k(U) \cap V \neq +\varnothing$. +\end{Def} + +\begin{Def}[Point périodique] + Un point $P \in \mathcal{X}$ est dit \emph{périodique} de période $t$ pour + une fonction $k$ si $t$ est un entier naturel non nul tel que $k^t(P) = P$ et + pour tout $n$, $0 < n \le t-1$, on a $k^n(P) \neq P$. + Par la suite, $\emph{Per(k)}$ dénote l'ensemble des points périodiques de + $k$ dans $\mathcal{X}$ de période quelconque. +\end{Def} + + + +\begin{Def}[Régularité] +La fonction $f$ est dite \emph{régulière} +sur $(\mathcal{X}, \tau)$ si l'ensemble des points périodiques + de $f$ is dense in $\mathcal{X}$: +pour chaque point $x \in \mathcal{X}$, chaque voisin +de $x$ contient au moins un point périodique +(sans que la période soit nécessairement constante). +\end{Def} + + + + + + + + + +La propriété de chaos est souvent associée à la notion de +\og sensibilité aux conditions initiales\fg{}, notion définie elle +sur un espace métrique $(\mathcal{X},d)$ par: + + +\begin{Def}[Forte sensibilité aux conditions initiales] +Une fonction $k$ définie sur $(\mathcal{X},\tau)$ +est \emph{fortement sensible aux conditions initiales} +s'il existe une valeur $\epsilon> 0$ telle que +pour tout $X \in \mathcal{X}$ et pour tout + $\delta > 0$, il existe $Y \in \mathcal{X}$ et +$t \in \Nats$ qui vérifient $d(X,Y) < \delta$ et +$d(k^t(X) , k^t(Y)) > \epsilon$. + +La constante $\delta$ est appelée \emph{constante de sensibilité} de $f$. +\end{Def} + +John Banks et ses collègues ont cependant +démontré que la sensibilité aux conditions initiales est une conséquence +de la régularité et de la transitivité topologique~\cite{Banks92}. + + + + + + + + + + + diff --git a/main.pdf b/main.pdf index 1d276aa13712aa4af2d05cb3a9d6036e9c1b50ec..b151af3ab5af982ddb05a45ad0287eb834c8c5c1 100644 GIT binary patch delta 101596 zcmZs?V{~Rs(>9zlwry)-+cqY)J;5ZI_>3mD&e*nX+sVXsW@2mJT-SX+&-4BGe)QV4 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a/main.tex +++ b/main.tex @@ -173,14 +173,23 @@ au chaos} \chapter{Characterisation des systèmes discrets chaotiques} + +La première section rappelle ce que sont les systèmes dynamiques chaotiques. Dire que cette caractérisation dépend du type de stratégie : unaire (TIPE), généralisée (TSI). Pour chacune d'elle, on introduit une distance différente. On montre qu'on a des résultats similaires. +\section{Systèmes dynamiques chaotiques selon Devaney} +\label{subsec:Devaney} +\input{devaney} + +\section{Schéma unaire} \input{12TIPE} +\section{Schéma généralisé} +\input{15TSI} générer des fonctions vérifiant ceci (TIPE12 juste sur le résultat d'adrien). @@ -225,11 +234,18 @@ générer des fonctions vérifiant ceci (TIPE12 juste sur le résultat d'adrien) \input{annexecontinuite.tex} + + \section{Caractérisation des fonctions $f$ rendant chaotique $G_f$ dans $(\mathcal{X},d)$}\label{anx:chaos:unaire} \input{caracunaire.tex} +\section{Preuve que $d$ est une distance sur $\mathcal{X}$}\label{anx:distance:generalise} +\input{preuveDistanceGeneralisee} + +\section{Caractérisation des fonctions $f$ rendant chaotique $G_f$ dans $(\mathcal{X},d)$}\label{anx:chaos:generalise} +\input{caracgeneralise.tex} diff --git a/preuveDistanceGeneralisee.tex b/preuveDistanceGeneralisee.tex new file mode 100644 index 0000000..9b7f3b6 --- /dev/null +++ b/preuveDistanceGeneralisee.tex @@ -0,0 +1,33 @@ +Pour $S,S' \in \mathcal{P}(\{1, \ldots, {\mathsf{N}}\})$, +on définit +\[ +\displaystyle{d_S(S,S')=\frac{9}{{\mathsf{N}}}\sum_{t\in\Nats}\frac{|S_t \Delta S'_t|}{10^{t+1}}}. +\] +Montrons que $d_S$ est une distance sur $\mathcal{P}(\{1, \ldots, {\mathsf{N}}\})$. + + + +Soit $S$, $S'$ et $S''$ trois parties de $[{\mathsf{N}}]$. +\begin{itemize} +\item De manière évidente, $d_s(S,S')$ est positive ou bien nulle si + et seulement si $S$ et $S'$ sont égales. +\item Comme la différence symétrique est commutative, la valeur de + $d_S(S,S')$ est égale à celle de $d_S(S',S)$. +\item On a enfin la succession d'éléments suivants: + $$ + \begin{array}{rcl} + S \Delta S' & = & (S \cap \overline{S'}) \cup (\overline{S} \cap S')\\ + & = & (S \cap \overline{S'} \cap S'' ) \cup (S \cap \overline{S'} \cap \overline{S''} ) \cup (\overline{S} \cap S'\cap S'') \cup (\overline{S} \cap S'\cap \overline{S''})\\ + & \subseteq & (S \cap \overline{S'} \cap S'' ) \cup (S \cap \overline{S'} \cap \overline{S''} ) \cup (\overline{S} \cap S'\cap S'') \cup (\overline{S} \cap S'\cap \overline{S''}) \cup \\ + & & (\overline{S} \cap \overline{S'} \cap S'') \cup (S \cap S' \cap \overline{S''} ) \cup (\overline{S} \cap \overline{S'} \cap S'') \cup (S \cap S' \cap \overline{S''})\\ + & = & (\overline{S'} \cap S'' ) \cup (S \cap \overline{S''} ) \cup (\overline{S} \cap S'') \cup (S'\cap \overline{S''}) \\ + & = & (S \Delta S'') \cup (S'' \Delta S') + \end{array} + $$ + On en déduit ainsi que +$|S \Delta S'| \le |S \Delta S''| + |S'' \Delta S'|$ et donc que +l'égalité triangulaire $d_S(S,S') \le d_S(S,S'') + d_S(S'',S')$ est établie. +\end{itemize} + + + diff --git a/sdd.tex b/sdd.tex index 530ef34..eb51e97 100644 --- a/sdd.tex +++ b/sdd.tex @@ -23,32 +23,32 @@ de conjonction \og . \fg{} et unaire de négation \og $\overline{\mathstrut\enskip}$ \fg{}. -Soit $n$ un entier naturel. -On introduit quelques notations à propos d'éléments de $\Bool^n$. -L'ensemble $\{1,\dots, n\}$ sera par la suite noté $[n]$. +Soit ${\mathsf{N}}$ un entier naturel. +On introduit quelques notations à propos d'éléments de $\Bool^{\mathsf{N}}$. +L'ensemble $\{1,\dots, {\mathsf{N}}\}$ sera par la suite noté $[{\mathsf{N}}]$. Le $i^{\textrm{ème}}$ composant d'un élément -$x \in \Bool^n$ s'écrit $x_i$. -Si l'ensemble $I$ est une partie de $[n]$, alors -$\overline{x}^I$ est l'élément $y\in \Bool^n$ +$x \in \Bool^{\mathsf{N}}$ s'écrit $x_i$. +Si l'ensemble $I$ est une partie de $[{\mathsf{N}}]$, alors +$\overline{x}^I$ est l'élément $y\in \Bool^{\mathsf{N}}$ tel que $y_i = 1 - x_i$ si $i\in I$ et $y_i = x_i$ sinon. -On considère les deux abréviations $\overline{x}$ pour $\overline{x}^{[n]}$ +On considère les deux abréviations $\overline{x}$ pour $\overline{x}^{[{\mathsf{N}}]}$ (chaque composant de $\overline{x}$ est nié: c'est une négation composante à composante) -et $\overline{x}^i$ pour $\overline{x}^{\{i\}}$ pour $i \in [n]$ +et $\overline{x}^i$ pour $\overline{x}^{\{i\}}$ pour $i \in [{\mathsf{N}}]$ (seul $x_i$ est nié dans $\overline{x}$). -Pour tout $x$ et $y$ dans $\Bool^n$, l'ensemble -$\Delta(x, y)$, contient les $i \in [n]$ +Pour tout $x$ et $y$ dans $\Bool^{\mathsf{N}}$, l'ensemble +$\Delta(x, y)$, contient les $i \in [{\mathsf{N}}]$ tels que $x_i \neq y_i$. -Soit enfin $f : \Bool^n \rightarrow \Bool^n$. Son $i^{\textrm{ème}}$ composant -est nommé $f_i$ qui est une fonction de $\Bool^n$ dans $\Bool$. +Soit enfin $f : \Bool^n \rightarrow \Bool^{\mathsf{N}}$. Son $i^{\textrm{ème}}$ composant +est nommé $f_i$ qui est une fonction de $\Bool^{\mathsf{N}}$ dans $\Bool$. Pour chaque -$x$ dans $\Bool^n$, l'ensemble +$x$ dans $\Bool^{\mathsf{N}}$, l'ensemble $\Delta f(x)$ est défini par $\Delta f(x) = \Delta(x,f(x))$. On peut admettre que $f (x) = \overline{x}^{\Delta f(x)}$ . \begin{xpl}\label{xpl:1} -On considère $n= 3$ et $f:\Bool^3 \rightarrow \Bool^3$ telle que +On considère ${\mathsf{N}}= 3$ et $f:\Bool^3 \rightarrow \Bool^3$ telle que $f(x)=(f_1(x),f_2(x),f_3(x))$ avec $$\begin{array}{rcl} f_1(x_1, x_2, x_3) &=& (\overline{x_1} + \overline{x_2}).x_3 \textrm{, }\\ @@ -110,58 +110,64 @@ d'éléments étudiés (gènes, protéines,\ldots). Un réseau booléen est défini à partir d'une fonction booléenne: \[ -f:\Bool^n\to\Bool^n,\qquad x=(x_1,\dots,x_n)\mapsto f(x)=(f_1(x),\dots,f_n(x)), +f:\Bool^{\mathsf{N}}\to\Bool^{\mathsf{N}},\qquad x=(x_1,\dots,x_{\mathsf{N}})\mapsto f(x)=(f_1(x),\dots,f_{\mathsf{N}}(x)), \] -et un {\emph{schéma itératif}} ou encore \emph{mode de mise à jour}. À partir d'une configuration initiale $x^0\in\Bool^n$, la suite $(x^{t})^{t +et un {\emph{schéma itératif}} ou encore \emph{mode de mise à jour}. À partir d'une configuration initiale $x^0\in\Bool^{\mathsf{N}}$, la suite $(x^{t})^{t \in \Nats}$ des configurations du système est construite selon l'un des schémas suivants : \begin{itemize} \item \textbf{Schéma parallèle synchrone :} basé sur la relation de récurrence - $x^{t+1}=f(x^t)$. Tous les $x_i$, $1 \le i \le n$, sont ainsi mis à jour à + $x^{t+1}=f(x^t)$. Tous les $x_i$, $1 \le i \le {\mathsf{N}}$, sont ainsi mis à jour à chaque itération en utilisant l'état global précédent du système $x^t$. \item \textbf{Schéma unaire :} ce schéma est parfois qualifié de chaotique dans la littérature. Il consiste à modifier la valeur - d'un unique élément $i$, $1 \le i \le n$, à + d'un unique élément $i$, $1 \le i \le {\mathsf{N}}$, à chaque itération. Le choix de l'élément qui est modifié à chaque itération est défini par une suite $S = \left(s^t\right)^{t \in \mathds{N}}$ qui est une séquence - d'indices dans $[n]$. Cette suite est appelée \emph{stratégie unaire}. - Il est basé sur la relation définie pour tout $i \in [n]$ par - $$ + d'indices dans $[{\mathsf{N}}]$. Cette suite est appelée \emph{stratégie unaire}. + Il est basé sur la relation définie pour tout $i \in [{\mathsf{N}}]$ par + +\begin{equation} x^{t+1}_i= \left\{ \begin{array}{l} f_i(x^t) \textrm{ si } i=s^t, \\ x^t_i\textrm{ sinon.} \end{array} - \right.$$ + \right. +\label{eq:schema:unaire} +\end{equation} \item \textbf{Schéma généralisé:} dans ce schéma, ce sont les valeurs - d'un ensemble d'éléments de $[n]$ qui sont modifiées à chaque itération. + d'un ensemble d'éléments de $[{\mathsf{N}}]$ qui sont modifiées à chaque itération. Dans le cas particulier où c'est la valeur d'un singleton - $\{k\}$, $1 \le k \le n$, qui est modifiée à + $\{k\}$, $1 \le k \le {\mathsf{N}}$, qui est modifiée à chaque itération, on retrouve le mode unaire. Dans le second cas particulier où ce sont les valeurs de - tous les éléments de $\{1, \ldots, n\}$ qui sont modifiées + tous les éléments de $\{1, \ldots, {\mathsf{N}}\}$ qui sont modifiées à chaque itération, on retrouve le mode parallèle. Ce mode généralise donc les deux modes précédents. Plus formellement, à la $t^{\textrm{ème}}$ itération, seuls les éléments de la partie - $s^{t} \in \mathcal{P}([n])$ sont mis à + $s^{t} \in \mathcal{P}([{\mathsf{N}}])$ sont mis à jour. La suite $S = \left(s^t\right)^{t \in \mathds{N}}$ est une séquence de sous-ensembles - de $[n]$ appelée \emph{stratégie généralisée}. - Il est basé sur la relation définie pour tout $i \in [n]$ par - $$ + de $[{\mathsf{N}}]$ appelée \emph{stratégie généralisée}. + Il est basé sur la relation définie pour tout $i \in [{\mathsf{N}}]$ par + \begin{equation} x^{t+1}_i= \left\{ \begin{array}{l} f_i(x^t) \textrm{ si } i \in s^t, \\ x^t_i\textrm{ sinon.} \end{array} - \right.$$ + \right. +\label{eq:schema:generalise} +\end{equation} + @@ -176,22 +182,22 @@ La section suivante détaille comment représenter graphiquement les évolutions -Pour un entier naturel $n$ et une -fonction $f : B^n \rightarrow B^n$, plusieurs évolutions sont possibles +Pour un entier naturel ${\mathsf{N}}$ et une +fonction $f : B^{\mathsf{N}} \rightarrow B^{\mathsf{N}}$, plusieurs évolutions sont possibles en fonction du schéma itératif retenu. Celles-ci sont représentées par un graphe orienté dont les noeuds -sont les éléments de $\Bool^n$ (voir \textsc{Figure}~\ref{fig:xpl:graphs}). +sont les éléments de $\Bool^{\mathsf{N}}$ (voir \textsc{Figure}~\ref{fig:xpl:graphs}). \begin{itemize} \item Le \emph{graphe des itérations synchrones} de $f$, noté $\textsc{gis}(f)$ -est le graphe orienté de $\Bool^n$ qui contient un arc $x \rightarrow y$ si +est le graphe orienté de $\Bool^{\mathsf{N}}$ qui contient un arc $x \rightarrow y$ si et seulement si $y=f(x)$. \item Le \emph{graphe des itérations unaires} de $f$, noté $\textsc{giu}(f)$ -est le graphe orienté de $\Bool^n$ qui contient un arc $x \rightarrow y$ si +est le graphe orienté de $\Bool^{\mathsf{N}}$ qui contient un arc $x \rightarrow y$ si et seulement s'il existe $x \in \Delta f(x)$ tel que $y = \overline{x}^i$. \item Le \emph{graphe des itérations généralisées} de $f$, noté $\textsc{gig}(f)$ -est le graphe orienté de $\Bool^n$ qui contient un arc $x \rightarrow y$ si +est le graphe orienté de $\Bool^{\mathsf{N}}$ qui contient un arc $x \rightarrow y$ si et seulement s'il existe un ensemble $I\subseteq \Delta f(x)$ tel que $y = \overline{x}^I$. On peut remarquer que ce graphe contient comme sous-graphe à la fois celui des itérations synchrones et celui @@ -251,7 +257,7 @@ On remarque le cycle $((101,111),(111,011),(011,101))$ \subsection{Attracteurs} On dit que le point -$x \in \Bool^n$ est un \emph{point fixe} de $f$ si $x = f (x)$. +$x \in \Bool^{\mathsf{N}}$ est un \emph{point fixe} de $f$ si $x = f (x)$. Les points fixes sont particulièrement intéressants car ils correspondent aux états stables: dans chaque graphe d'itérations, le point $x$ est un point fixe @@ -259,12 +265,12 @@ si et seulement si il est son seul successeur. -Soit $\Gamma$ un graphe d'itérations (synchrones, unaires ou généralisées) +Soit un graphe d'itérations (synchrones, unaires ou généralisées) de $f$. -Les \emph{attracteurs} de $\Gamma$ sont les +Les \emph{attracteurs} de ce graphe sont les plus petits sous-ensembles (au sens de l'inclusion) non vides -$A \subseteq \Bool^n$ tels que pour tout arc -$x \rightarrow y$ de $\Gamma$, si $x$ est un élément de $A$, alors +$A \subseteq \Bool^{\mathsf{N}}$ tels que pour tout arc +$x \rightarrow y$, si $x$ est un élément de $A$, alors $y$ aussi. Un attracteur qui contient au moins deux éléments est dit \emph{cyclique}. On en déduit qu'un attracteur cyclique ne contient pas de point fixe. @@ -277,12 +283,12 @@ On a la proposition suivante: \begin{theorem}\label{Prop:attracteur} Le point $x$ est un point fixe si et seulement si -$\{x\}$ est un attracteur de $\Gamma$. -En d'autres termes, les attracteurs non cycliques de $\Gamma$ +$\{x\}$ est un attracteur du graphe d'itération (synchrone, unaire, généralisé). +En d'autres termes, les attracteurs non cycliques de celui-ci sont les points fixes de $f$. -Ainsi pour chaque $x\in \Bool^n$, il existe au moins un chemin +Ainsi pour chaque $x\in \Bool^{\mathsf{N}}$, il existe au moins un chemin depuis $x$ qui atteint un attracteur. -Ainsi $\Gamma$ contient toujours au moins un attracteur. +Ainsi tout graphe d'itérations contient toujours au moins un attracteur. \end{theorem} @@ -300,7 +306,7 @@ Les interactions entre les composants du système peuvent être mémorisées dans la {\emph{matrice jacobienne discrète}} $f'$. Celle-ci est définie comme étant la fonction qui à chaque -configuration $x\in\Bool^n$ associe la matrice de taille +configuration $x\in\Bool^{\mathsf{N}}$ associe la matrice de taille $n\times n$ telle que \begin{equation} f'(x)=(f'_{ij}(x)),\qquad @@ -314,7 +320,7 @@ $\overline{x}^j_j$ et $x_j$ sont considérés comme des entiers naturels dans $\Z$. Lorsqu'on supprime les signes dans la matrice jacobienne discrète, on obtient une matrice notée $B(f)$ aussi de taille -$n\times n$. +${\mathsf{N}}\times {\mathsf{N}}$. Celle-ci mémorise uniquement l'existence d'une dépendance de tel élément vis à vis de tel élément. @@ -323,7 +329,7 @@ les uns par rapport aux autres. Cette matrice est nommée \emph{matrice d'incidence}. \begin{theorem} -Si $f_i$ ne dépend pas de $x_j$, alors pour tout $x\in [n]$, +Si $f_i$ ne dépend pas de $x_j$, alors pour tout $x\in [{\mathsf{N}}]$, $f_i(\overline{x}^j)$ est égal à $f_i(x)$, \textit{i.e}, $f'_{ij}(x)=0$. Ainsi $B(f)_{ij}$ est nulle. La réciproque est aussi vraie. \end{theorem} @@ -333,10 +339,10 @@ $f'_{ij}(x)=0$. Ainsi $B(f)_{ij}$ est nulle. La réciproque est aussi vraie. En outre, les interactions peuvent se représenter à l'aide d'un graphe $\Gamma(f)$ orienté et signé défini ainsi: -l'ensemble des sommets est -$[n]$ et il existe un arc de $j$ à $i$ de signe +l'ensemble des sommet %s est +$[{\mathsf{N}}]$ et il existe un arc de $j$ à $i$ de signe $s\in\{-1,1\}$, noté $(j,s,i)$, si $f_{ij}(x)=s$ pour au moins -un $x\in\Bool^n$. +un $x\in\Bool^{\mathsf{N}}$. On note que la présence de deux arcs de signes opposés entre deux sommets donnés @@ -515,28 +521,28 @@ arc positif (resp. un arc négatif) de $i$ vers lui-même. \subsection{Conditions de convergence}\label{sec:Robert:async} Parmi les itérations unaires caractérisées par leurs stratégies -$S=(s^t)^{t \in \Nats}$ d'éléments appartenant à $[n]$, +$S=(s^t)^{t \in \Nats}$ d'éléments appartenant à $[{\mathsf{N}}]$, sont jugées intéressantes celles qui activent au moins une fois -chacun des $i\in[n]$. Dans le cas contraire, un élément n'est jamais modifié. +chacun des $i\in[{\mathsf{N}}]$. Dans le cas contraire, un élément n'est jamais modifié. Plus formellement, une séquence finie $S=(s^t)^{t \in \Nats}$ -est dite \emph{complète} relativement à $[n]$ si -tout indice de $[n]$ +est dite \emph{complète} relativement à $[{\mathsf{N}}]$ si +tout indice de $[{\mathsf{N}}]$ s'y retrouve au moins une fois. Parmi toutes les stratégies unaires de -$[n]^{\Nats}$, on qualifie de: +$[{\mathsf{N}}]^{\Nats}$, on qualifie de: \begin{itemize} \item \emph{périodiques} celles qui sont constituées par une répétition indéfinie -d'une même séquence $S$ complète relativement à $[n]$. +d'une même séquence $S$ complète relativement à $[{\mathsf{N}}]$. En particulier toute séquence périodique est complète. \item \emph{pseudo-périodiques} celles qui sont constituées par une succession indéfinie de séquences (de longueur éventuellement variable non supposée bornée) complètes. Autrement dit dans chaque stratégie pseudo-périodique, chaque indice de -$1$ à $n$ revient indéfiniment. +$1$ à ${\mathsf{N}}$ revient indéfiniment. \end{itemize} @@ -547,14 +553,14 @@ dans le mode des itérations unaires. \begin{theorem}\label{Th:conv:GIU} Si le graphe $\Gamma(f)$ n'a pas de cycle et si la stratégie unaire est pseudo-périodique, alors tout chemin de $\textsc{giu}(f)$ atteint -l'unique point fixe $\zeta$ en au plus $n$ pseudo-périodes. +l'unique point fixe $\zeta$ en au plus ${\mathsf{N}}$ pseudo-périodes. \end{theorem} Le qualificatif \emph{pseudo-périodique} peut aisément s'étendre aux stratégies généralisées comme suit. Lorsqu'une stratégie généralisée est constituée d'une -succession indéfinie de séquences de parties de $[n]$ -dont l'union est $[n]$, cette stratégie est {pseudo-périodique}. +succession indéfinie de séquences de parties de $[{\mathsf{N}}]$ +dont l'union est $[{\mathsf{N}}]$, cette stratégie est {pseudo-périodique}. J. Bahi~\cite{Bah00} a démontré le théorème suivant: \begin{theorem}\label{Th:Bahi} -- 2.39.5