\input ams \special{header=mdrlib.ps} \def\frac#1#2{{#1\over#2}} \def\dfrac#1#2{{#1\over#2}} \long\def\InsertFig#1 #2 #3 #4\EndFig{\par \hbox{\hskip #1mm$\vbox to#2mm{\vfil\special{" #3}}#4$}} \long\def\InsertPSFile#1 #2 #3 #4 #5 #6\EndFig{\par \hbox{\hskip #1mm$\vbox to#2mm{% \vfil\special{psfile=#5 hscale=#3 vscale=#4}}#6$}} \long\def\InsertImage#1 #2 #3 #4 #5 #6 #7 #8\EndFig{\par \hbox{\hskip #1mm$\vbox to#2mm{% \vfil\special{psfile="`img2eps file #7 height #4 mm width #3 mm gamma #5 angle #6}}#8$}} \long\def\LabelTeX#1 #2 #3\ELTX{\rlap{\kern#1mm\raise#2mm\hbox{#3}}} \font\tenCal=eusm10 \font\sevenCal=eusm7 \font\fiveCal=eusm5 \newfam\Calfam \textfont\Calfam=\tenCal \scriptfont\Calfam=\sevenCal \scriptscriptfont\Calfam=\fiveCal \def\Cal{\fam\Calfam\tenCal} \font\tenBbb=msbm10 \font\sevenBbb=msbm7 \font\fiveBbb=msbm5 \newfam\Bbbfam \textfont\Bbbfam=\tenBbb \scriptfont\Bbbfam=\sevenBbb \scriptscriptfont\Bbbfam=\fiveBbb \def\Bbb{\fam\Bbbfam\tenBbb} % Usual sets of numbers \def\bC{{\Bbb C}} \def\bN{{\Bbb N}} \def\bP{{\Bbb P}} \def\bQ{{\Bbb Q}} \def\bR{{\Bbb R}} \def\bZ{{\Bbb Z}} \def\?{\hbox{$\,$}} \def\cE{{\Cal E}} \def\cH{{\Cal H}} \def\diam{\mathop{\rm diam}} \def\Re{\mathop{\rm Re}} \def\Im{\mathop{\rm Im}} \magnification=1200 \vsize=20cm\hsize=13.6cm \parindent=0mm \parskip=5pt plus 1pt minus 1pt \nopagenumbers \font\seventeenbf=ecbx10 at 17.28pt \font\fourteenbf=ecbx10 at 14.4pt \def\build#1^#2_#3{\mathop{#1}\limits^{#2}_{#3}} \def\\{\hfil\break} \centerline{\seventeenbf Objets fractals :} \medskip \centerline{\seventeenbf illustration de quelques concepts} \medskip \centerline{\seventeenbf et outils mathématiques} \medskip\medskip \centerline{\it Jean-Pierre Demailly} \centerline{\it Institut Fourier, Université de Grenoble I} \vskip 2cm Conférence donnée le jeudi 15 mai dans la cadre du module du Collège Doctoral UJF\break «\?\?Du chaos à la complexité~: vers l’émergence d'une thématique pluridisciplinaire\?»\break (Maison des Sciences de l'Homme, du mardi 13 mai au jeudi 15 mai 2008). \vskip1cm (Version du 25 mai 2008) \vfill\eject \centerline{\fourteenbf Le concept de dimension} \bigskip La dimension d'un «\?espace\?» (ensemble de points dans lequel on se place) est classiquement le {\it nombre de coordonnées} nécessaires pour repérer un point de cet espace. C'est donc a priori un {\it nombre entier}. On va introduire ici une notion plus générale, qui conduit à des dimensions non nécessairement entières. \medskip {\it Objet de dimension 1} \InsertFig 5.000 10.000 { gsave 1 mm unit initcoordinates 0.600 setlinewidth 0.000 0.000 moveto 10.000 0.000 lineto stroke 40.000 0.000 moveto 70.000 0.000 lineto stroke 40.000 0.000 moveto 70.000 0.000 0.700 0.000 3 0 (I) sequence stroke grestore } \LabelTeX 22.000 -1.000 $\build\rightsquigarrow^{\textstyle\times 3}_{}$ \ELTX \LabelTeX 82.000 -1.000 $\times 3^1$ \ELTX \EndFig \medskip Par une homothétie de rapport 3, la mesure (longueur) est multipliée par~$3=3^1$, l'objet résultant contient 3 fois l'objet initial. La dimension d'un segment est $1$. \medskip {\it Objet de dimension 2} \InsertFig 5.000 35.000 { gsave 1 mm unit initcoordinates 0.600 setlinewidth 0.000 10.000 moveto 10.000 10.000 lineto 10.000 20.000 lineto 0.000 20.000 lineto closepath stroke 40.000 0.000 moveto 70.000 0.000 lineto stroke 40.000 0.000 translate 0.000 0.000 moveto 30.000 0.000 lineto 30.000 30.000 lineto 0.000 30.000 lineto closepath stroke 0.000 10.000 moveto 30.000 10.000 lineto 0.000 20.000 moveto 30.000 20.000 lineto 10.000 0.000 moveto 10.000 30.000 lineto 20.000 0.000 moveto 20.000 30.000 lineto stroke grestore } \LabelTeX 22.000 14.000 $\build\rightsquigarrow^{\textstyle\times 3}_{}$ \ELTX \LabelTeX 82.000 14.000 $\times 3^2$ \ELTX \EndFig \medskip Par une homothétie de rapport 3, la mesure (aire) de l'objet est multipliée par~$9=3^2$, l'objet résultant contient 9 fois l'objet initial. La dimension du carré est $2$. \medskip En généralisant, {\it pour un objet de dimension $d$}, l'effet d'une homothétie de rapport $3$ est de {\it multiplier la mesure par $3^d$}. \medskip {\bf Qu'en est-il d'un ensemble fractal ?}\\ \vbox{% {\it Prenons l'exemple\\ de la courbe de Koch}\\ (ou flocon de neige)\\ obtenue par itération\\ du procédé ci-contre $\longrightarrow$} \vskip -20mm \InsertImage 70 65 45 0 1.1 0 koch_curve.png \EndFig \vskip -38mm \InsertImage 5 35 35 0 1.1 0 flocon.gif \EndFig \InsertImage 0 25 135.5 0 1.1 0 koch_dilat.gif \LabelTeX 36 3 $\build\rightsquigarrow^{\textstyle\times 3}_{}$\ELTX \EndFig Par une homothétie de rapport $3$, l'objet devient un objet de nature identique, contenant $4$ morceaux de même taille que l'objet initial, donc de mesure 4 fois plus grande. Ceci conduit à poser $$ 3^d=4 ~~~\Longrightarrow~~~ d={\ln 4\over\ln 3}= 1.26185950714\ldots $$ Il nous faut admettre ici que la dimension n'est pas un entier, mais un nombre compris strictement entre $1$ et $2$~! Il est facile de voir d'autre part que la longueur de la courbe de Koch est infinie~: à chaque itération, la longueur est multipliée par $4/3$, donc si le segment initial est pris pour unité, la longueur de la $n$-ième itération est $(4/3)^n$, ce qui tend vers l'infini. D'autre part, l'aire est nulle. En effet, par récurrence sur le nombre d'itérations, on voit que la courbe de Koch est entièrement contenue dans le triangle isocèle ayant pour base le segment initial et une hauteur égale à $\sqrt{3}/6$ fois cette base. Comme la $n$-ième itération découpe dans la courbe de Koch $4^n$ parties dont la taille est homothétique à la courbe toute entière dans le rapport $3^{-n}$, celle-ci est contenue dans une réunion de $4^n$ petits triangles d'aire $\sqrt{3}/12\times(3^{-n})^2$, soit une aire totale $\sqrt{3}/12\times 4^n\times 9^{-n}$, qui tend bien vers~$0$ quand $n\to+\infty$. A titre d'exercice, on pourra chercher à évaluer la dimension du «\?triangle de Sierpinski\?» figuré ci-dessous. Ce fractal est obtenu en partant d'un triangle équilatéral, puis en enlevant le triangle central dans un découpage en 4 triangles semblables dans le rapport~$1/2$. On répète ensuite indéfiniment le procédé sur tous les triangles obtenus à chaque étape. \InsertImage 45 50 50 0 1.1 0 sierpinski_triangle.jpg \EndFig \medskip On montrera facilement que l'aire du triangle de Sierpinski est nulle. De manière analogue, on construit le «\?tapis de Sierpinski\?» en partant d'un carré, en le divisant en 9 carrés homothétiques dans le rapport $1/3$ et en enlevant le carré central. Dans l'espace de dimension~$3$, l'«\?éponge de Sierpinski\?» est obtenue à partir d'un cube divisé en 27 petits cubes, privé du cube central. On montrera que son aire est infinie, tandis que le volume est nul. \vskip1cm \centerline{\fourteenbf Mesures de Hausdorff} \medskip Les considérations qui précèdent se généralisent à l'aide des mesures $p$-dimensionnelles introduites par Felix Hausdorff (1868-1942) -- l'un des fondateurs de la topologie moderne. Si $(\cE,d)$ est un espace métrique quelconque, on définit la mesure de Hausdorff $p$-dimension­nelle d'une partie $A$ de $\cE$ par $$ \cH_p(A)=\lim_{\varepsilon\to 0} \cH_{p,\varepsilon}(A),\quad \cH_{p,\varepsilon}(A)=\inf_{\diam A_i\le\varepsilon}\sum_i(\diam A_i)^p $$ où $\cH_{p,\varepsilon}(A)$ est la borne inférieure des sommes $\sum_i(\diam A_i)^p$ étendue à toutes les partitions dénombrables $A=\bigcup A_i$ avec $\diam A_i\le\varepsilon$. Intuitivement, on charche à paver $A$ par des parties $A_i$ dont on évalue la $p$-mesure comme si elles étaient des «\?hyper-cubes\?» de dimension~$p$, puis on améliore la précision en demandant que le diamètre des $A_i$ tende vers~zéro. On montre que $\cH_p$ définit une «\?mesure borélienne dénombrablement additive\?», c'est-à-dire qu'elle est définie sans ambiguité sur tous les ouverts, fermés et leurs unions, intersections finies ou dénombrables répétées autant de fois que l'on veut, avec la propriété que $\cH_p(\bigcup A_i)=\sum_i\cH_p(A_i)$ pour de parties boréliennes disjointes -- sur les parties non boréliennes, on obtient seulement une «\?mesure extérieure\?» qui vérifie $\cH_p(\bigcup A_i)\le\sum_i\cH_p(A_i)$. Dans $\cE=\bR^n$ avec sa structure euclidienne habituelle, cette mesure coïncide pour $p$ entier avec la mesure d'aide euclidienne $p$-dimensionnelle, à un facteur de proportionnalité près égal au volume de la boule euclidienne de dimension~$p$ et de diamètre $1$. Ces mesures permettent de donner une définition précise de la «\?dimension de Hausdorff\?»~: \medskip {\bf Définition.} {\it Si $(\cE,d)$ est un espace métrique et $A$ une partie de $\cE$, ont dit que $A$ est de dimension de Hausdorff égale à $p_0$ si on a $$ \hbox{$\cH_p(A)=+\infty$~ pour $pp_0$.} $$} Il est facile de voir que dans $\bR^n$, la dimension d'une partie quelconque est toujours de dimension${}\le n$ (c'est-à-dire que $\cH_p(A)=0$ pour $p>n$)$\,$: on pourra pour cela commencer par montrer que la $\cH_p$-mesure d'un $n$-cube est nulle pour $p>n$, puis utiliser la sous-additivité dénombrable à l'aide d'un recouvrement de $A$ par des cubes. Par ailleurs, en général, on ne peut rien dire sur $\cH_{p_0}(A)$, cette valeur peut très bien être nulle, finie ou infinie. \vfill\eject \centerline{\fourteenbf Systèmes dynamiques discrets} \medskip L'une des façons les plus usuelles d'obtenir des objets fractal est de considérer un «\?système dynamique\?». D'un point de vue mathématique, il s'agit simplement de considérer une certaine transformation continue $f:X\to X$ d'un espace $X$, puis de regarder l'évolution des points par itération de~$f$. Autrement dit, étant donné un point initial $x_0$, on regarde la suite définie par la relation de récurrence $x_{n+1}=f(x_n)$, c'est-à-dire encore ce qu'on appelle l'orbite de $x_0$ sous l'action des composées successives $$ f^{[n]}=f\circ f\circ\cdots\circ f:X\to X,\qquad x_n=f^{[n]}(x_0). $$ En mathématiques, on cherche toujours à étudier d'abord les situations non triviales les plus simples possibles. L'une de celles-ci consiste à regarder dans le plan complexe $\bC$ la transformation quadratique $$ P_c:\bC\to\bC,\qquad P_c(z)=z^2+c~é\hbox{où $c$ est un paramètre.} $$ On considère donc {\it l'orbite} $z_n=P_c^{[n]}(z_0)$ d'un point $z_0\in\bC$ donné. On notera que $P_c^{[2]}(z)=(z^2+c)^2+c$ est un polynôme de degré $4\;$; de manière générale, $P_c^{[n]}$ est un polynôme de degré~$2^n$. On introduit par définition~: {\bf Ensemble de Julia de $P_c$.} {\it On appelle ensemble de Julia rempli $K_c$ l'ensemble des points initiaux $z_0$ tels que la suite $(z_n)$ reste bornée, et ensemble de Julia le bord $J_c=\partial K_c$ de l'ensemble de Julia rempli~$K_c$.} Si $c=0$, on a simplement $P_0^{[n]}(z)=z^{2^n}$ et on voit aussitôt que $K_0$ consiste en le disque unité fermé $|z|\le 1$ (donc $J_0$ est le cercle unité). Pour tout autre valeur $c\ne 0$ on obtient un ensemble fractal. Voici par exemple une image de $J_c$ et $K_c$ pour la valeur $c=0,328075517+0,022051744\,i$ du paramètre~: \InsertImage 45 65 52 0 1.1 0 julia1.png \LabelTeX -6 30 $K_c$\ELTX \EndFig \medskip On notera que si on remplace $z_0$ par $z_0'=z_1=P_c(z_0)$, on obtient la même suite avec simplement un décalage d'indice, donc $z_0\in K_c$ si et seulement si $z_0'=P_c(z_0)\in K_c$. Par conséquent, ceci montre que $P_c(K_c)=K_c$ et $P_c^{-1}(K_c)=K_c$, autrement dit $J_c$ et $K_c$ sont {\bf auto-similaires} sous l'action de la transformation $P_c$. La propriété d'autosimilarité est une caractéristique assez générale des objets fractals - notamment tous ceux obtenus par le procédé qui vient d'être décrit. Il existe cependant des objets fractals plus compliqués, comme l'ensemble de Mandelbrot (du nom du mathématicien franco-américain Benoît Mandelbrot, né à Varsovie en 1924). \medskip {\bf Ensemble de Mandelbrot} {\it C'est l'ensemble $M$ des valeurs complexes $c$ du paramètre telles que l'ensemble de Julia $K_c$ associé à $P_c$ soit connexe.} \InsertImage 8 75 120 0 1.1 0 mandelbrot.png \LabelTeX 81 27 $M$\ELTX \LabelTeX 13 64 $K_c$\ELTX \LabelTeX 13 31 $K_{c'}$\ELTX \LabelTeX 90 34.5 $\times$\ELTX \LabelTeX 87 38 $c=0$\ELTX \EndFig \smallskip L'ensemble de Mandelbrot $M$ consiste en le domaine intérieur blanc de la partie droite de l'image (avec ses infinies ramifications~$\ldots$). Sur cette partie droite figurent aussi la position de deux valeurs complexes $c$ et $c'$ (points rouge et violet), et, à gauche, les ensembles de Julia associés $K_c$ et $K_{c'}$. On voit que $K_c$ est connexe, donc $c$ est dans l'ensemble de Mandelbrot, tandis que $K_{c'}$ n'est pas connexe, donc $c'$ n'est pas dans $M$. Si on note $M_{z_0}$ l'ensemble des valeurs de $c$ telles que la suite $z_n$ de point initial $z_0=0$ soit bornée, on peut démontrer que l'on a en fait $M=M_0$. L'ensemble de Mandelbrot $M$ est seulement partiellement auto-similaire$\,$; il comporte des parties qui sont des répliques approximatives de l'ensemble complet, mais aussi d'autres qui sont dissemblables~: \vbox{% \InsertImage 28 60 80 0 1.1 0 mandelbrot2.png \EndFig \vskip-10mm } La structure de $M$ est beaucoup plus compliquée que celle des ensembles de Julia~$J_c$, comme il est déjà apparent sur les images ci-dessus. On sait par exemple que $M$ est connexe, mais on ne sait pas à ce jour si $M$ est localement connexe$\,$; rappelons qu'un espace est dit localement connexe si tout point admet des voisinages connexes arbitrairement petits. L'espace appelé «\?peigne dénombrable\?» est un exemple typique d'espace connexe non localement connexe~: \InsertFig 20 35 { gsave 1 mm unit 0 0 moveto 100 0 lineto stroke /k 1 def 1 1 15 { pop 100 k div dup 0 moveto 25 lineto stroke /k k 1.414213 mul def } for 0.3 setlinewidth 0 0 moveto 0 25 lineto stroke 1 0 0 setrgbcolor 0 13 moveto 0.7 disk 0.1 setlinewidth 0 13 moveto 3 circle grestore } \EndFig \medskip Le mathématicien japonais Shishikura a pu démontrer en 1992 que la frontière de l'en\-semble de Mandelbrot est de dimension de Hausdorff $2$ (donc maximale dans le plan), mais on ne sait pas si sa mesure $2$-dimensionnelle est nulle ou non (les experts pensent que cette mesure est bien nulle, avis aux amateurs~!). Pour obtenir l'image de $M=M_0$ à l'aide d'un programme d'ordinateur, on procède comme suit. On convient d'une valeur assez grande, disons $10^9$, qui teste le caractère non borné de la suite, et un nombre d'itérations assez grand, disons $N=150$. Étant donné une valeur complexe $c$ fixée représentée par un pixel de l'écran d'ordinateur, on calcule la suite $z_n$ telle que $z_0=0$ et $z_{n+1}=P_c(z_0)$, pour $n\le N=150$. Si toutes ces valeurs restent inférieures à $10^9$, il est probable que la suite $(z_n)$ est bornée, donc on décide que $c\in M$ (ou du moins dans son approximation calculée par l'ordinateur), et on affecte la couleur blanche au pixel. Sinon il existe une valeur $n_010^9$ pour $n=n_0+1$. On arrête alors le calcul en estimant que la suite $(z_n)$ va tendre vers l'infini, et on affecte au pixel une couleur (brunâtre ou violette dans les deux images de la page précédente) qui dépend de la valeur de $n_0$~: sur la schéma ci-dessus, la couleur est d'autant plus foncée que $n_0$ est plus grand. Ceci donne les zones colorées qui «\?entourent\?» l'ensemble de Mandelbrot. En réalité, il ne s'agit pas d'un simple artifice de représentation graphique, ces zones colorées ont une signification mathématique et physique. On peut en effet regarder les fonctions $$ G_n(z,c)=2^{-n}\log_+|P_c^{[n]}(z)| $$ où $\log_+t=\max(0,\log t)$ est la partie positive du logarithme. Du fait que $P_c^{[n]}$ est de degré~$2^n$, on peut démontrer assez facilement que $G_n(z,c)$ converge vers une fonction $G(z,c)\ge 0$ qui est nulle sur le graphe $\Gamma=\{(z,c)\in\bC^2\,;\;z\in K_c\}$ (là où la suite $P_c^{[n]}(z)$ est bornée), et strictement positive en dehors de $\Gamma$. La valeur de $G(z,c)$ mesure la vitesse à laquelle la suite tend vers l'infini. Comme les fonctions holomorphes sont harmoniques, on vérifie facilement que $\Delta_z G(z,c)=0$ pour $z\notin J_c$. La fonction $z\mapsto G(z,c)$ s'interprète comme le potentiel électrostatique créé par un conducteur métallique cylindrique qui aurait la forme de $K_c$, et dont le bord $J_c=\partial K_c$ serait chargé électriquement. Les zones colorées évoquées plus haut sont délimitées par les lignes équipotentielles~$\ldots$ L'ensemble $\Gamma$ est quand à lui un ensemble fractal de complexité monstrueuse dans $\bC^2$ (donc dans un espace de dimension réelle $4$), dont les tranches $c=\hbox{Cte}$ sont les ensembles de Julia remplis $K_c$, tandis que les tranches $z=\hbox{Cte}$ sont les ensembles de Mandelbrot~$M_z$. \vbox{% \InsertImage 10 75 100 0 1.1 0 gnofract4d.png \EndFig \vskip-65.5mm \InsertFig 20.000 40.000 { gsave 1 mm unit initcoordinates 0.200 setlinewidth 0 25 moveto 90 0 2.4 vector 30 -5 moveto 60 90 2.4 vector 1 0 0 setrgbcolor 56 20 moveto 74.9 20 lineto stroke 0 0 1 setrgbcolor 60 8 moveto 60 42 lineto stroke grestore } \LabelTeX 88 21 $z\in\bC$\ELTX \LabelTeX 26.3 53 $c$\ELTX \LabelTeX 60 4 $M_z$\ELTX \LabelTeX 76 19 $K_c$\ELTX \LabelTeX 46 49 $\Gamma$\ELTX \EndFig } \vskip7mm Le programme {\tt gnofract4d} permet d'explorer ce fractal quadridimensionnel. Pour les experts, signa­lons que le $(1,1)$-courant positif $T=i\partial \overline\partial G(z,c)$ dans $\bC^2$ a précisément comme support l'intérieur de $\Gamma$, ses tranches sont donc les ensembles $M_z$ et $J_c$. La théorie des courants positifs est un outil essentiel de l'analyse complexe contemporaine, particulièrement pour l'étude des systèmes dynamiques holomorphes~$\ldots$ \medskip {\bf Logiciels} Voici 3 logiciels libres assez agréables à utiliser~: -- Fractint : {\tt http://www.fractint.org} (Linux/MacOS X/Windows)\\ -- Xaos : {\tt http://wmi.math.u-szeged.hu/xaos/doku.php} (Linux/MacOS X/Windows)\\ -- Gnofract4d : {\tt http://gnofract4d.sourceforge.net/} (Linux/MacOS X) Sous Windows, un logiciel réputé (mais non libre) est~: -- Ultrafractal : {\tt http://www.ultrafractal.com } \vfill\eject \centerline{\fourteenbf L'entropie topologique} \bigskip On considère de nouveau une transformation $f:X\to X$, et on cherche à mesurer «\?combien $f$ mélange\?» l'espace~$X$. Voici une illustration de la transformation dite du boulanger (ou de la «\?pâte feuilletée\?»), du carré unité dans lui-même, itérée 17 fois~: \InsertImage 8 154 120 0 1.1 0 joconde.png \EndFig \smallskip La transformation du carré $X=[0,1]^2$ figurée ci-dessus est donnée par $$ f(x,y)=\big(2x,(1+y)/2\big)~~\hbox{si $x\le 1/2$},\qquad f(x,y)=\big(2x-1,(1-y)/2\big)~~\hbox{si $x\ge 1/2$}. $$ La périodicité d'ordre 17 observée ici est un artefact dû à la discrétisation de l'image en carrés de $256\times 256$ pixels. En réalité la transformation du boulanger est apériodique et conduirait à un mélange parfait de l'image si celle-ci était non discrétisée. D'un point de vue mathématique, le mélange plus ou moins grand provoqué par la transformation $f$ est décrit par un nombre appelé «\?entropie topologique\?». Voici une définition précise, due à Bowen et Dinaburg. {\bf Définition.} {\it Soit $(X,d)$ un espace métrique compact et $f:X\to X$ une application $($supposée en général continue$)$. On définit une nouvelle distance $d_n$ en posant $$ d_n(x,y)=\max\{d(f^{[i]}(x),f^{[i]}(y)): 0\leq i0$, toute petit écart initial $X_{i,0}$ par rapport à la postion d'équilibre $X_i=0$ finira par entraîner une très grande déviation (le cas $\Re \lambda_i=0$ est indéterminé, comme on le verra plus loin). La stabilité est donc essentiellement liée au signe des valeurs propres du système linéarisé près d'un point critique. On peut démontrer le théorème important suivant~: \medskip {\bf Théorème.} {\it Pour qu'un système différentiel $(S): \overrightarrow{dM}/dt=\overrightarrow{V}(M)$ possède un état d'équi\-libre stable $M_0$ $($en un point singulier $M_0$ tel que $\overrightarrow{V}(M_0)=\overrightarrow{0})$, il suffit que la matrice $A=\big(\partial V_i/\partial x_j(M_0)\big)$ associée au système linéarisé ait toutes ses valeurs propres complexes $\lambda_i$ de partie réelle $\Re\lambda_i$ strictement négative.} \medskip {\bf Cas d'un champ linéaire de vecteurs en dimension $2$} Pour fixer les idées nous allons donner la représentation graphique des solutions de tous les systèmes linéaires non dégénérés de dimension $2$, donc des systèmes du type $$ \dfrac{dM}{dt}=AM,\qquad \left\{ \eqalign{ \dfrac{dx}{dt}&=ax+by\cr \noalign{\vskip1pt} \dfrac{dy}{dt}&=cx+dy\cr }\right. \quad\hbox{\rm où}\quad A=\pmatrix{a&b\cr c&d\cr},~~ \det(A)=ad-bc\ne 0. \leqno(S) $$ La discussion dépend de manière essentielle de ce que sont les valeurs propres $\lambda_1$, $\lambda_2$. \medskip {\bf (a) Les valeurs propres $\lambda_1,\lambda_2$ de $A$ sont réelles.} \medskip \noindent$\bullet$ Supposons de plus $\lambda_1\not=\lambda_2$. Dans ce cas la matrice $A$ est diagonalisable. Après changement de base on peut supposer $$ A=\pmatrix{ \lambda_1&0\cr \noalign{\vskip3pt} 0&\lambda_2\cr } $$ et le système se réduit à $$ \left\{ \eqalign{ \dfrac{dx}{dt}&=\lambda_1 x\cr \noalign{\vskip3pt} \dfrac{dy}{dt}&=\lambda_2 y\cr }\right.\qquad \hbox{de solution}\qquad \cases{ x(t)=x_0e^{\lambda_1t}\hfill\cr \noalign{\vskip3pt} y(t)=y_0e^{\lambda_2t},\hfill\cr } $$ de sorte que les courbes intégrales sont les courbes $$ y=C|x|^{\lambda_2/\lambda_1},\quad C\in\bR $$ et la droite d'équation $x=0$. Distinguons deux sous-cas~: \medskip $\ast$ $\lambda_1,\lambda_2$ de même signe et, disons, $|\lambda_1|<|\lambda_2|$. On a alors $\lambda_2/\lambda_1>1$. 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85.697 59.758 85.045 59.906 84.398 60.059 83.757 60.218 83.121 60.383 82.491 60.554 81.868 60.731 81.252 60.916 80.643 61.108 80.042 61.308 79.449 61.516 78.865 61.734 78.290 61.961 77.725 62.198 77.171 62.447 76.629 62.708 76.100 62.983 75.585 63.273 75.086 63.579 74.605 63.905 74.145 64.252 73.707 64.624 73.296 65.025 72.918 65.463 72.579 65.948 72.293 66.495 72.080 67.147 72.009 67.897 72.182 68.508 72.425 69.064 72.705 69.587 73.011 70.085 73.337 70.564 73.678 71.027 74.033 71.477 74.398 71.917 74.772 72.346 75.155 72.768 75.546 73.182 75.943 73.589 76.346 73.989 76.755 74.385 77.168 74.775 77.587 75.160 78.009 75.541 78.436 75.917 78.866 76.290 79.300 76.659 79.738 77.025 80.178 77.388 80.622 77.748 81.068 78.104 81.517 78.458 81.968 78.810 82.422 79.158 82.878 79.505 83.336 79.849 83.797 80.191 84.259 80.531 84.724 80.869 85.190 81.206 85.658 81.540 86.128 81.872 86.599 82.203 87.072 82.532 87.547 82.860 88.023 83.186 88.501 83.510 88.980 83.833 89.460 84.155 89.942 84.475 90.425 84.794 90.909 85.112 91.395 85.428 91.881 85.744 92.369 86.058 92.858 86.370 93.348 86.682 93.839 86.993 94.331 87.303 94.824 87.611 95.319 ] curve stroke 47.189 49.957 moveto [ 47.189 49.957 47.498 50.445 47.808 50.932 48.120 51.418 48.432 51.903 48.745 52.386 49.060 52.869 49.376 53.350 49.693 53.829 50.011 54.308 50.331 54.785 50.652 55.260 50.974 55.734 51.298 56.207 51.624 56.678 51.950 57.148 52.279 57.615 52.609 58.081 52.940 58.546 53.274 59.008 53.609 59.469 53.946 59.927 54.285 60.384 54.626 60.838 54.969 61.290 55.315 61.739 55.663 62.187 56.013 62.631 56.365 63.073 56.721 63.512 57.079 63.948 57.440 64.381 57.804 64.810 58.172 65.236 58.543 65.658 58.917 66.076 59.296 66.490 59.679 66.899 60.066 67.303 60.459 67.701 60.856 68.094 61.260 68.480 61.670 68.859 62.087 69.231 62.511 69.593 62.945 69.945 63.388 70.286 63.843 70.613 64.312 70.925 64.796 71.217 65.300 71.484 65.830 71.719 66.395 71.905 67.012 72.000 67.760 71.836 68.325 71.557 68.813 71.216 69.251 70.829 69.650 70.408 70.018 69.958 70.360 69.485 70.680 68.991 70.981 68.480 71.265 67.953 71.534 67.411 71.789 66.857 72.033 66.291 72.265 65.715 72.486 65.128 72.698 64.533 72.901 63.929 73.096 63.317 73.283 62.697 73.463 62.071 73.636 61.437 73.803 60.798 73.963 60.153 74.118 59.502 74.267 58.846 74.410 58.184 74.549 57.518 74.683 56.847 74.812 56.172 74.937 55.492 75.057 54.809 75.174 54.121 75.286 53.429 75.395 52.734 75.500 52.035 75.602 51.333 75.700 50.628 75.795 49.919 75.886 49.207 ] curve stroke 40.000 72.000 moveto 95.000 72.000 lineto stroke 61.500 50.000 moveto 72.500 95.000 lineto stroke 69.800 84.000 moveto 0.000 255.000 1.8 vector 60.100 83.500 moveto 0.000 285.000 1.8 vector 53.500 80.400 moveto 0.000 315.000 1.8 vector 49.000 76.800 moveto 0.000 340.000 1.8 vector 47.000 72.000 moveto 0.000 0.000 1.8 vector 48.500 67.800 moveto 0.000 20.000 1.8 vector 51.000 64.600 moveto 0.000 40.000 1.8 vector 55.600 62.000 moveto 0.000 55.000 1.8 vector 64.000 60.000 moveto 0.000 75.000 1.8 vector 74.000 60.100 moveto 0.000 102.000 1.8 vector 80.800 63.100 moveto 0.000 130.000 1.8 vector 85.300 67.000 moveto 0.000 153.000 1.8 vector 87.000 72.000 moveto 0.000 180.000 1.8 vector 85.800 76.200 moveto 0.000 202.000 1.8 vector 82.800 79.100 moveto 0.000 217.000 1.8 vector 78.400 82.000 moveto 0.000 238.000 1.8 vector grestore grestore } \LabelTeX 92.500 68.500 $x$\ELTX \LabelTeX 69.000 92.500 $y$\ELTX \LabelTeX 162.500 68.500 $x$\ELTX \LabelTeX 139.000 92.500 $y$\ELTX \LabelTeX 47.000 36.000 $\matrix{ 0<\lambda_1<\lambda_2\cr \noalign{\vskip3pt} \hbox{nœud impropre instable} }\qquad\qquad\qquad\qquad\qquad \matrix{ \lambda_2<\lambda_1<0\cr \noalign{\vskip3pt} \hbox{nœud impropre stable} }$\ELTX \EndFig \vskip-23mm} $\ast$ $\lambda_1,\lambda_2$ de signes opposés, par exemple $\lambda_1<0<\lambda_2$. Il s'agit d'un {\it col} (toujours instable)~: \vbox{ \InsertFig -18.000 100.000 { 1 mm unit gsave 50.000 50.000 moveto 110.000 94.000 rectangle clip newpath 80.000 72.000 moveto 0.700 disk 50.251 73.004 moveto [ 50.251 73.004 51.009 73.048 51.764 73.095 52.516 73.145 53.265 73.198 54.010 73.255 54.752 73.317 55.490 73.382 56.223 73.452 56.951 73.528 57.674 73.609 58.391 73.696 59.102 73.790 59.806 73.892 60.503 74.001 61.192 74.118 61.873 74.244 62.544 74.381 63.206 74.528 63.857 74.686 64.497 74.856 65.125 75.040 65.740 75.236 66.342 75.447 66.930 75.673 67.504 75.914 68.063 76.171 68.606 76.444 69.135 76.733 69.647 77.040 70.144 77.362 70.625 77.701 71.091 78.056 71.542 78.427 71.978 78.813 72.399 79.213 72.807 79.628 73.202 80.056 73.584 80.497 73.954 80.950 74.312 81.414 74.660 81.888 74.998 82.373 75.327 82.867 75.646 83.369 75.958 83.880 76.261 84.398 76.557 84.923 76.847 85.455 77.129 85.992 77.406 86.536 77.678 87.084 77.944 87.638 78.205 88.195 78.462 88.757 78.714 89.324 78.962 89.893 79.207 90.466 79.448 91.043 79.685 91.622 79.920 92.204 80.152 92.789 80.380 93.376 80.606 93.965 80.830 94.557 ] curve stroke 90.402 94.181 moveto [ 90.402 94.181 90.320 93.485 90.243 92.786 90.170 92.083 90.103 91.375 90.042 90.663 89.988 89.945 89.942 89.221 89.905 88.490 89.878 87.751 89.863 87.002 89.863 86.242 89.880 85.484 89.915 84.744 89.969 84.024 90.044 83.324 90.141 82.647 90.262 81.993 90.407 81.363 90.579 80.759 90.778 80.182 91.005 79.632 91.261 79.111 91.547 78.619 91.863 78.155 92.209 77.720 92.584 77.314 92.989 76.936 93.423 76.585 93.883 76.261 94.370 75.961 94.882 75.685 95.417 75.431 95.974 75.198 96.551 74.984 97.147 74.788 97.760 74.608 98.388 74.443 99.031 74.292 99.687 74.153 100.355 74.026 101.034 73.909 101.722 73.801 102.419 73.702 103.125 73.610 103.837 73.525 104.556 73.447 105.281 73.374 106.012 73.307 106.748 73.245 107.488 73.186 108.232 73.132 108.980 73.082 109.732 73.035 110.251 73.004 ] curve stroke 49.749 70.996 moveto [ 49.749 70.996 50.505 70.951 51.258 70.903 52.007 70.851 52.753 70.795 53.495 70.735 54.233 70.671 54.965 70.602 55.693 70.527 56.414 70.446 57.129 70.358 57.837 70.263 58.538 70.160 59.229 70.047 59.911 69.925 60.582 69.791 61.242 69.645 61.889 69.486 62.521 69.312 63.138 69.121 63.738 68.912 64.318 68.683 64.878 68.433 65.416 68.159 65.930 67.859 66.417 67.533 66.878 67.177 67.309 66.791 67.710 66.373 68.079 65.923 68.417 65.439 68.722 64.921 68.994 64.370 69.235 63.785 69.444 63.168 69.622 62.520 69.771 61.842 69.893 61.136 69.988 60.404 70.059 59.646 70.106 58.866 70.132 58.065 70.138 57.243 70.126 56.411 70.099 55.598 70.059 54.801 70.009 54.017 69.951 53.244 69.884 52.481 69.811 51.726 69.732 50.979 69.648 50.238 69.559 49.503 ] curve stroke 79.342 49.899 moveto [ 79.342 49.899 79.559 50.468 79.779 51.033 80.000 51.596 80.225 52.156 80.452 52.714 80.681 53.268 80.914 53.819 81.150 54.366 81.389 54.910 81.632 55.450 81.879 55.985 82.130 56.517 82.385 57.043 82.644 57.565 82.909 58.082 83.179 58.593 83.454 59.098 83.736 59.596 84.024 60.089 84.318 60.574 84.620 61.051 84.930 61.521 85.248 61.982 85.574 62.435 85.910 62.877 86.256 63.310 86.613 63.733 86.980 64.144 87.359 64.544 87.750 64.932 88.154 65.307 88.572 65.669 89.003 66.018 89.449 66.353 89.909 66.674 90.384 66.980 90.875 67.272 91.380 67.549 91.901 67.812 92.438 68.060 92.989 68.295 93.555 68.515 94.135 68.722 94.729 68.915 95.337 69.097 95.957 69.266 96.590 69.424 97.234 69.571 97.889 69.708 98.554 69.836 99.228 69.955 99.911 70.065 100.603 70.168 101.302 70.264 102.008 70.353 102.721 70.436 103.440 70.513 104.164 70.585 104.893 70.653 105.627 70.715 106.366 70.774 107.109 70.829 107.855 70.880 108.604 70.929 109.357 70.974 109.749 70.996 ] curve stroke 50.040 76.712 moveto [ 50.040 76.712 50.686 76.871 51.325 77.038 51.957 77.211 52.583 77.391 53.202 77.579 53.814 77.773 54.418 77.975 55.016 78.186 55.605 78.403 56.187 78.629 56.760 78.864 57.326 79.106 57.883 79.358 58.432 79.617 58.973 79.886 59.505 80.163 60.028 80.449 60.543 80.744 61.050 81.048 61.548 81.361 62.037 81.682 62.517 82.012 62.990 82.351 63.453 82.699 63.909 83.054 64.356 83.419 64.795 83.791 65.227 84.172 65.650 84.560 66.066 84.957 66.474 85.361 66.875 85.772 67.269 86.190 67.656 86.616 68.036 87.048 68.409 87.487 68.776 87.932 69.137 88.384 69.492 88.842 69.841 89.305 70.184 89.774 70.522 90.248 70.855 90.728 71.182 91.213 71.504 91.702 71.822 92.196 72.135 92.695 72.444 93.198 72.749 93.706 73.049 94.217 73.345 94.732 ] curve stroke 98.059 94.222 moveto [ 98.059 94.222 98.124 93.501 98.201 92.792 98.290 92.095 98.392 91.411 98.507 90.741 98.636 90.084 98.779 89.441 98.937 88.812 99.109 88.199 99.297 87.600 99.500 87.017 99.719 86.449 99.954 85.897 100.206 85.361 100.474 84.842 100.759 84.339 101.060 83.852 101.379 83.382 101.714 82.929 102.066 82.491 102.435 82.071 102.820 81.666 103.222 81.277 103.639 80.904 104.073 80.546 104.522 80.204 104.986 79.875 105.464 79.562 105.957 79.262 106.464 78.976 106.984 78.702 107.516 78.441 108.062 78.192 108.619 77.955 109.187 77.729 109.767 77.513 110.357 77.308 ] curve stroke 50.036 66.557 moveto [ 50.036 66.557 50.626 66.342 51.205 66.117 51.774 65.879 52.331 65.630 52.875 65.369 53.407 65.094 53.926 64.806 54.431 64.504 54.922 64.187 55.398 63.855 55.859 63.508 56.304 63.144 56.734 62.764 57.147 62.367 57.543 61.953 57.922 61.522 58.284 61.073 58.629 60.606 58.956 60.122 59.266 59.619 59.558 59.099 59.832 58.560 60.090 58.004 60.329 57.430 60.552 56.839 60.758 56.231 60.948 55.607 61.121 54.966 61.279 54.309 61.421 53.636 61.548 52.949 61.661 52.247 61.760 51.530 61.845 50.801 61.917 50.058 61.977 49.302 ] curve stroke 87.019 49.899 moveto [ 87.019 49.899 87.311 50.395 87.608 50.887 87.908 51.375 88.213 51.859 88.522 52.339 88.836 52.814 89.154 53.284 89.476 53.749 89.804 54.210 90.137 54.665 90.475 55.115 90.819 55.560 91.168 55.999 91.523 56.432 91.884 56.859 92.251 57.280 92.625 57.694 93.005 58.102 93.392 58.504 93.786 58.898 94.187 59.286 94.595 59.666 95.010 60.039 95.433 60.405 95.864 60.763 96.303 61.113 96.749 61.455 97.204 61.790 97.667 62.116 98.138 62.434 98.618 62.745 99.106 63.047 99.602 63.340 100.107 63.626 100.620 63.903 101.142 64.172 101.672 64.433 102.211 64.686 102.758 64.930 103.313 65.167 103.877 65.396 104.448 65.617 105.028 65.830 105.615 66.036 106.210 66.234 106.812 66.426 107.421 66.610 108.038 66.788 108.661 66.958 109.291 67.123 109.928 67.281 110.571 67.433 ] curve stroke 80.000 72.000 moveto 50.000 72.000 moveto 110.000 72.000 lineto stroke 74.500 50.000 moveto 85.500 94.000 lineto stroke 90.000 72.000 moveto 0.000 180.000 1.8 vector 70.000 72.000 moveto 0.000 0.000 1.8 vector 82.500 82.000 moveto 0.000 76.000 1.8 vector 77.500 62.000 moveto 0.000 256.000 1.8 vector 91.900 78.000 moveto 0.000 135.000 1.8 vector 100.900 84.000 moveto 0.000 130.000 1.8 vector 65.000 84.000 moveto 0.000 45.000 1.8 vector 71.000 78.000 moveto 0.000 36.000 1.8 vector 68.000 66.000 moveto 0.000 -45.000 1.8 vector 89.000 66.000 moveto 0.000 215.000 1.8 vector 97.500 62.000 moveto 0.000 215.000 1.8 vector 57.500 62.000 moveto 0.000 -44.000 1.8 vector grestore} \LabelTeX 107.500 68.000 $x$\ELTX \LabelTeX 82.000 92.000 $y$\ELTX \EndFig \vskip-45mm} \noindent$\bullet$ Les valeurs propres sont confondues~: $\lambda_1=\lambda_2=\lambda$. Deux cas sont possibles~: \medskip $\ast$ $A$ est diagonalisable. Alors $A$ est en fait diagonale et les courbes intégrales sont données par $$ \cases{ x(t)=x_0e^{\lambda t}\hfill\cr \noalign{\vskip3pt} y(t)=y_0e^{\lambda t},\hfill\cr } $$ ce sont les droites $y=\alpha x$ et $x=0$. On dit qu'on a affaire à un {\it nœud propre}~: \InsertFig 33.000 28.000 { 1 mm unit -25.000 0.000 moveto 25.000 0.000 lineto stroke 0.000 22.000 moveto 0.000 -22.000 lineto stroke 17.000 17.000 moveto 16.000 16.000 moveto -16.000 -16.000 lineto stroke -16.000 16.000 moveto 16.000 -16.000 lineto stroke 0.000 0.000 moveto 0.700 disk 0.000 12.000 moveto 0.000 90.000 1.8 vector -8.500 8.500 moveto 0.000 135.000 1.8 vector -12.000 0.000 moveto 0.000 180.000 1.8 vector -8.500 -8.500 moveto 0.000 225.000 1.8 vector 0.000 -12.000 moveto 0.000 -90.000 1.8 vector 8.500 -8.500 moveto 0.000 -45.000 1.8 vector 12.000 0.000 moveto 0.000 0.000 1.8 vector 8.500 8.500 moveto 0.000 45.000 1.8 vector 70.000 0.000 translate -25.000 0.000 moveto 25.000 0.000 lineto stroke 0.000 22.000 moveto 0.000 -22.000 lineto stroke 17.000 17.000 moveto 16.000 16.000 moveto -16.000 -16.000 lineto stroke -16.000 16.000 moveto 16.000 -16.000 lineto stroke 0.000 0.000 moveto 0.700 disk 0.000 12.000 moveto 0.000 -90.000 1.8 vector -8.500 8.500 moveto 0.000 -45.000 1.8 vector -12.000 0.000 moveto 0.000 0.000 1.8 vector -8.500 -8.500 moveto 0.000 45.000 1.8 vector 0.000 -12.000 moveto 0.000 90.000 1.8 vector 8.500 -8.500 moveto 0.000 135.000 1.8 vector 12.000 0.000 moveto 0.000 180.000 1.8 vector 8.500 8.500 moveto 0.000 225.000 1.8 vector} \LabelTeX 23.000 -3.500 $x$\ELTX \LabelTeX -3.000 20.600 $y$\ELTX \LabelTeX 93.000 -3.500 $x$\ELTX \LabelTeX 67.000 20.600 $y$\ELTX \EndFig \vskip20mm $$ \matrix{ \lambda>0\cr \noalign{\vskip5pt} \hbox{Nœud propre instable}\cr }\qquad\qquad\qquad\qquad\qquad\matrix{ \lambda<0\cr \noalign{\vskip5pt} \hbox{Nœud propre stable}\cr } $$ $\ast$ $A$ est non diagonalisable. Alors il existe une base dans laquelle la matrice $A$ et le système s'écrivent $$ A=\pmatrix{ \lambda&0\cr 1&\lambda\cr },\qquad\qquad\left\{\eqalign{ \dfrac{dx}{dt}&=\lambda x\cr \noalign{\vskip3pt} \dfrac{dy}{dt}&=x+\lambda y.\cr} \right. $$ Le courbes intégrales sont données par $$ \left\{ \eqalign{ x(t)&=x_0e^{\lambda t}\cr \noalign{\vskip3pt} y(t)&=(y_0+x_0t)e^{\lambda t}.\cr} \right. $$ Comme toute courbe intégrale avec $x_0\not=0$ passe par un point tel que $|x(t)|=1$, on obtient toutes les courbes intégrales autres que $x=0$ en prenant $x_0=\pm 1$, d'où $$ \left\{\eqalign{ t&=\dfrac{1}{\lambda}\ \ln\,|x|\cr \noalign{\vskip3pt} y&=y_0|x|+\dfrac{x}{\lambda}\ \ln\,|x|\cr} \right. $$ On dit qu'il s'agit d'un {\it nœud exceptionnel}. Pour construire les courbes, on tracera par exemple d'abord la courbe $y=\dfrac{x}{\lambda}\ \ln\,|x|$ passant par $(x_0,y_0)=(\pm 1,0)$. Toutes les autres s'en déduisent par homothéties. \InsertFig -49.000 102.000 { 1 mm unit gsave 50.000 50.000 moveto 104.000 94.000 rectangle clip newpath 50.023 89.282 moveto [ 50.023 89.282 50.593 89.508 51.167 89.728 51.748 89.941 52.334 90.149 52.928 90.349 53.528 90.542 54.135 90.728 54.751 90.905 55.375 91.072 56.009 91.230 56.652 91.378 57.307 91.513 57.974 91.636 58.654 91.744 59.350 91.837 60.062 91.911 60.794 91.965 61.547 91.995 62.326 91.997 63.109 91.968 63.861 91.908 64.583 91.819 65.277 91.702 65.945 91.558 66.586 91.388 67.202 91.194 67.795 90.975 68.364 90.734 68.911 90.470 69.437 90.185 69.942 89.879 70.426 89.553 70.891 89.208 71.337 88.843 71.765 88.460 72.174 88.058 72.565 87.639 72.939 87.202 73.296 86.749 73.637 86.278 73.961 85.791 74.269 85.288 74.561 84.769 74.837 84.234 75.098 83.683 75.343 83.116 75.573 82.534 75.788 81.936 75.988 81.323 76.172 80.694 76.341 80.049 76.494 79.387 76.632 78.709 76.753 78.013 76.858 77.300 76.945 76.567 77.014 75.813 77.062 75.036 77.088 74.231 77.087 73.388 77.050 72.517 77.020 72.158 ] curve stroke 50.016 81.313 moveto [ 50.016 81.313 50.521 81.605 51.029 81.892 51.540 82.176 52.056 82.456 52.575 82.732 53.099 83.004 53.627 83.271 54.160 83.533 54.698 83.790 55.241 84.041 55.789 84.286 56.343 84.526 56.904 84.758 57.472 84.984 58.047 85.201 58.629 85.411 59.220 85.611 59.821 85.801 60.432 85.981 61.054 86.148 61.688 86.302 62.337 86.441 63.001 86.563 63.683 86.665 64.387 86.744 65.114 86.797 65.872 86.816 66.665 86.795 67.418 86.732 68.131 86.629 68.807 86.490 69.447 86.315 70.054 86.106 70.629 85.866 71.173 85.596 71.689 85.298 72.176 84.971 72.638 84.618 73.073 84.239 73.484 83.835 73.870 83.407 74.233 82.956 74.574 82.482 74.892 81.985 75.189 81.466 75.464 80.926 75.718 80.364 75.952 79.781 76.164 79.176 76.356 78.550 76.527 77.901 76.676 77.230 76.804 76.536 76.909 75.816 76.990 75.069 77.044 74.289 77.067 73.469 77.049 72.579 77.018 72.152 ] curve stroke 50.037 74.225 moveto [ 50.037 74.225 50.494 74.565 50.954 74.902 51.416 75.236 51.880 75.568 52.348 75.898 52.818 76.224 53.290 76.547 53.766 76.868 54.245 77.184 54.728 77.498 55.213 77.808 55.703 78.114 56.196 78.416 56.693 78.713 57.195 79.007 57.701 79.295 58.212 79.579 58.727 79.857 59.249 80.129 59.776 80.395 60.310 80.654 60.850 80.906 61.398 81.151 61.953 81.386 62.518 81.612 63.092 81.828 63.676 82.033 64.273 82.224 64.883 82.401 65.508 82.561 66.151 82.701 66.815 82.818 67.504 82.908 68.223 82.963 68.980 82.974 69.757 82.929 70.478 82.829 71.149 82.678 71.772 82.481 72.351 82.240 72.888 81.958 73.387 81.637 73.850 81.280 74.277 80.888 74.671 80.462 75.034 80.005 75.365 79.516 75.667 78.997 75.939 78.448 76.183 77.870 76.399 77.262 76.586 76.624 76.744 75.955 76.872 75.253 76.969 74.514 77.031 73.732 77.050 72.888 77.017 72.146 ] curve stroke 50.033 65.882 moveto [ 50.033 65.882 50.443 66.270 50.855 66.655 51.269 67.039 51.684 67.421 52.101 67.802 52.519 68.180 52.940 68.557 53.362 68.932 53.786 69.305 54.212 69.676 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70.070 76.209 70.598 76.467 71.115 76.728 71.608 76.980 71.998 ] curve stroke 49.992 57.731 moveto [ 49.992 57.731 50.364 58.157 50.737 58.581 51.112 59.005 51.487 59.428 51.863 59.850 52.241 60.270 52.619 60.689 52.999 61.107 53.380 61.524 53.762 61.939 54.146 62.353 54.531 62.766 54.917 63.177 55.305 63.587 55.694 63.995 56.084 64.402 56.477 64.807 56.870 65.210 57.266 65.612 57.663 66.011 58.062 66.409 58.463 66.805 58.866 67.199 59.271 67.590 59.679 67.979 60.088 68.366 60.500 68.750 60.914 69.132 61.331 69.511 61.751 69.887 62.173 70.260 62.599 70.629 63.027 70.995 63.460 71.358 63.896 71.716 64.336 72.070 64.780 72.420 65.228 72.765 65.682 73.104 66.140 73.437 66.605 73.765 67.076 74.085 67.554 74.397 68.040 74.701 68.534 74.994 69.039 75.277 69.555 75.546 70.084 75.801 70.629 76.037 71.193 76.250 71.781 76.434 72.400 76.581 73.061 76.674 73.787 76.681 74.517 76.559 75.128 76.320 75.636 75.981 76.058 75.554 76.400 75.046 76.669 74.462 76.867 73.799 76.990 73.045 77.012 72.129 ] 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103.665 94.240 104.005 94.702 ] curve stroke 76.995 71.902 moveto [ 76.995 71.902 77.084 71.442 77.375 71.095 77.937 71.021 78.663 71.290 79.198 71.620 79.679 71.978 80.131 72.355 80.562 72.746 80.979 73.147 81.385 73.556 81.781 73.973 82.170 74.395 82.552 74.823 82.929 75.256 83.301 75.692 83.668 76.132 84.032 76.576 84.392 77.022 84.748 77.472 85.102 77.923 85.453 78.378 85.802 78.834 86.148 79.292 86.492 79.753 86.834 80.215 87.174 80.679 87.512 81.144 87.849 81.612 88.184 82.080 88.517 82.550 88.849 83.021 89.180 83.494 89.509 83.967 89.837 84.442 90.164 84.918 90.490 85.395 90.814 85.873 91.138 86.352 91.460 86.832 91.782 87.313 92.102 87.794 92.422 88.277 92.741 88.760 93.059 89.245 93.376 89.730 93.692 90.215 94.008 90.702 94.322 91.189 94.636 91.677 94.950 92.165 95.262 92.654 95.574 93.144 95.886 93.634 96.196 94.125 96.506 94.617 ] curve stroke 77.010 71.962 moveto [ 77.010 71.962 77.660 72.650 77.994 73.163 78.308 73.677 78.611 74.194 78.907 74.714 79.197 75.236 79.483 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0.000 185.000 1.8 vector 92.200 62.600 moveto 0.000 205.000 1.8 vector 95.200 70.500 moveto 0.000 218.000 1.8 vector 94.400 75.500 moveto 0.000 222.000 1.8 vector 92.600 80.200 moveto 0.000 228.000 1.8 vector 89.900 84.700 moveto 0.000 236.000 1.8 vector 85.500 87.700 moveto 0.000 246.000 1.8 vector [ 2.000 1.000 ] 0 setdash 104.000 72.000 moveto 50.000 72.000 lineto stroke [ ] 0 setdash grestore} \LabelTeX 101.000 69.500 $x$\ELTX \LabelTeX 79.000 92.000 $y$\ELTX \LabelTeX 171.000 69.500 $x$\ELTX \LabelTeX 148.800 92.000 $y$\ELTX \EndFig \vskip-50mm $$ \matrix{ \lambda>0\cr \noalign{\vskip3pt} \hbox{Nœud exceptionnel instable}\cr }\qquad\qquad\qquad\qquad\matrix{ \lambda<0\cr \noalign{\vskip3pt} \hbox{Nœud exceptionnel stable}\cr } $$ {\bf (b) Les valeurs propres de $A$ sont non réelles.} \medskip On a des valeurs propres complexes conjuguées $\alpha+i\beta$, $\alpha-i\beta$ avec disons $\beta>0$, et il existe une base dans laquelle la matrice $A$ et le système s'écrivent $$ A=\pmatrix{ \alpha&-\beta\cr \noalign{\vskip3pt} \beta&\alpha\cr },\qquad\left\{\eqalign{ \dfrac{dx}{dt}&=\alpha x-\beta y\hfill\cr \noalign{\vskip3pt} \dfrac{dy}{dt}&=\beta x+\alpha y.\hfill\cr} \right. $$ La manière la plus rapide de résoudre un tel système est de poser $z=x+iy$. On trouve alors $$ \frac{dz}{dt}=(\alpha+i\beta)x+(-\beta+\alpha i)y=(\alpha+i\beta)(x+iy)=(\alpha+i\beta)z, $$ de sorte que la solution générale est $$ z(t)=z_0e^{(\alpha+i\beta)t}=z_0e^{\alpha t}e^{i\beta t}. $$ En coordonnées polaires $z=re^{i\theta}$, l'équation devient $$ \cases{ r=r_0e^{\alpha t}\hfill\cr \noalign{\vskip3pt} \theta=\theta_0+\beta t\hfill\cr },\quad{\rm soit}\quad r=r_0e^{\frac{\scriptstyle\alpha}{\scriptstyle\beta}\,(\theta-\theta_0)}. $$ Il s'agit d'une spirale logarithmique si $\alpha\not=0$ et d'un cercle si $\alpha=0$ (noter que ce cercle donne en général graphiquement une ellipse car la base utilisée ci-dessus n'est pas nécessairement orthonormée). On dit alors que le point singulier est un {\it foyer}, respectivement un {\it centre}~: \InsertFig -42.000 83.000 { gsave 1 mm unit gsave 50.000 50.000 moveto 80.000 80.000 rectangle clip newpath 64.878 64.904 moveto [ 64.878 64.904 65.231 65.033 64.727 65.091 65.131 64.748 65.378 65.182 64.690 65.307 64.413 64.890 65.160 64.517 65.750 64.836 65.677 65.379 65.079 65.656 64.364 65.553 63.952 65.187 63.971 64.619 64.495 64.204 65.142 64.087 65.910 64.261 66.350 64.610 66.526 65.117 66.102 65.861 65.551 66.140 64.894 66.249 64.117 66.151 63.546 65.893 63.141 65.519 62.924 65.017 63.052 64.319 63.468 63.839 63.991 63.533 64.601 63.349 65.299 63.289 66.083 63.390 66.695 63.610 67.187 63.920 67.565 64.317 67.808 64.817 67.836 65.462 67.549 66.121 67.133 66.560 66.635 66.887 66.068 67.126 65.435 67.280 64.729 67.342 63.935 67.287 63.255 67.130 62.670 66.898 62.165 66.600 61.740 66.236 61.400 65.802 61.166 65.284 61.085 64.650 61.270 63.883 61.599 63.347 62.011 62.921 62.485 62.574 63.014 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169.000 65.000 moveto 0.000 80.000 1.8 vector 165.000 65.000 moveto 0.000 76.000 1.8 vector 161.000 65.000 moveto 0.000 72.000 1.8 vector 155.000 65.000 moveto 0.700 disk grestore} \EndFig \vskip-57mm $$ \matrix{ \alpha>0\cr \noalign{\vskip3pt} \hbox{Foyer instable}\cr }\qquad\qquad\qquad\quad\matrix{ \alpha<0\cr \noalign{\vskip3pt} \hbox{Foyer stable}\cr }\qquad\qquad\qquad\qquad\matrix{ \alpha=0\quad\cr \noalign{\vskip3pt} \hbox{Centre}\quad\cr } $$ \noindent Si $\alpha\not=0$, le rapport d'homothétie de deux spires consécutives de la \hbox{spirale est $e^{2\pi\alpha/\beta}$}. Comme on le voit, la description de la situation en dimension $2$ est toujours très simple. C'est là un fait général, qui résulte du théorème dit de Poincaré-Bendixson~: \medskip {\bf Th\'eor\`eme de Poincar\'e-Bendixson.} {\it Soit $M\mapsto \overrightarrow V(M)$ un champ de vecteurs de classe $C^1$ dans un ouvert $\Omega$ du plan. On suppose qu'il existe une partie fermée bornée $K\subset\Omega$ ne contenant aucun z\'ero du champ de vecteurs $\overrightarrow V$ et stable par l'équation d'évolution en temps $t\ge 0$ $[$hypothèse qui implique en particulier que les courbes intégrales sont d\'efini pour tout point initial $(x_0,y_0)\in K$ et tout $t\ge 0\,]$. Alors $K$ est constitu\'e d'une réunion d'orbites p\'eriodiques du flot.} \medskip {\bf Bibliographie mathématique élémentaire~:} Jean-Pierre demailly, {\it Analyse numérique et équations différentielles}, Presses universitaires de Grenoble, 3${}^{\rm e}$ édition, 2006, ISBN 2-7061-0715-4. \vfill\eject \centerline{\fourteenbf Le système différentiel de Lorenz (1963)} \bigskip Contrairement à ce qui se passe en dimension~$2$, les systèmes différentiels d'évolution peuvent dès la dimension $3$ donner lieu à des situations géométriques très compliquées et à des comportements essentiellement «\?chaotiques\?». Nous allons illustrer ceci par un système différentiel célèbre, appelé {\it système de Lorenz}. \medskip Il s'agit dans l'espace $\Omega=\bR^3$ du système différentiel non linéaire $$ \cases{ \displaystyle{dx\over dt}=10 (y - x)\cr \noalign{\vskip5pt} \displaystyle{dy\over dt}=28x-y- xz\cr \noalign{\vskip4pt} \displaystyle{dz\over dt}=xy- {8\over 3}z.\cr}\leqno(S) $$ Voici quelques représentations graphiques des trajectoires (nous avons extrait les images de l'exposé d'\'Etienne Ghys au Congrès International des mathématiciens de Madrid (2006))~: on obtient ce qu'on appelle l'attracteur de Lorenz, qui a une structure fractale. \InsertImage 5 52 60 0 1.1 0 ghys002.png \EndFig \vskip-52mm \InsertImage 70 52 60 0 1.1 0 ghys003.png \EndFig \InsertImage 37.5 52 60 0 1.1 0 ghys004.png \EndFig \medskip La complexité de ces trajectoires est très grande. En particulier, les trajectoires pério\-diques forment des nœuds, et il apparaît des nœuds de différents types. Voici quelques illustrations de nœuds apparaissant au sein des trajectoires~: \InsertImage 5 52 60 0 1.1 0 ghys005.png \EndFig \vskip-52mm \InsertImage 70 52 60 0 1.1 0 ghys006.png \EndFig \medskip Un problème particulièrement intéressant étudié par Ghys est de classifier tous les types de nœuds pouvant apparaître dans l'attracteur de Lorenz, et ensuite, de déterminer la façon dont ils peuvent s'enlacer~: \InsertImage 5 52 60 0 1.1 0 ghys044.png \EndFig \vskip-52mm \InsertImage 70 52 60 0 1.1 0 ghys045.png \EndFig \medskip Il est surprenant de constater que du chaos émerge tout de même certaines formes d'ordre extrêmement complexes, correspondant à des situations mathématiques très riches~$\ldots$ \bigskip {\bf Références.} \'Etienne Ghys, {\tt http://www.umpa.ens-lyon.fr/$\,\tilde{\kern3pt}\,$ghys/} et en particulier, sur {\tt http://www.umpa.ens-lyon.fr/$\,\tilde{\kern3pt}\,$ghys/Publis.html} l'exposé {\it Knots and Dynamics}, Proceedings du Congrès International de Madrid 2006, {\tt http://www.umpa.ens-lyon.fr/$\,\tilde{\kern3pt}\,$ghys/articles/icm.pdf}\\ {\tt http://www.umpa.ens-lyon.fr/$\,\tilde{\kern3pt}\,$ghys/articles/ghys-icm.pdf} \end % Local Variables: % TeX-command-default: "uTeX" % End: