% Jean-Pierre Demailly
% Universit\'e de Grenoble I, Institut Fourier
% \'Equations Diff\'erentielles et Analyse Num\'erique
% Collection Grenoble-Sciences, Presses Universitaires de Grenoble
% Chapitre X

\input equa-mac.tex

\blankline\blankline
\hfill\chtitle Chapitre X|%
\blankline\blankline
\hfill\title Stabilit\'e des solutions|
\vskip1.5pt
\hfill\title et points singuliers d'un champ de vecteurs|
\chaptitlerunning X -- Stabilit\'e des solutions et points singuliers|
\vskip3pc

\noindent On se propose ici d'\'etudier le comportement des solutions
d'une \'equation diff\'eren\-tielle et des lignes int\'egrales d'un
champ de vecteurs lorsque le temps $t$ tend vers l'infini. On
s'int\'eresse essentiellement au cas des \'equations lin\'eaires ou \og
voisines\fg\ de telles \'equations. Dans ce cas, le comportement des
solutions est gouvern\'e par le signe de la partie r\'eelle des valeurs
propres de la matrice associ\'ee \`a la partie lin\'eaire de
l'\'equation~: une solution est dite stable si les solutions associ\'ees
\`a des valeurs voisines de la donn\'ee initiale restent proches de la
solution consid\'er\'ee jusqu'\`a l'infini. Cette notion de stabilit\'e
(dite aussi stabilit\'e au sens de Lyapunov) ne devra pas \^etre
confondue avec la notion de stabilit\'e d'une m\'ethode num\'erique,
qui concerne la stabilit\'e de l'algorithme sur un intervalle de temps
fix\'e. On \'etudie finalement les diff\'erentes configurations
possibles des lignes int\'egrales au voisinages des points singuliers
non d\'eg\'en\'er\'es d'un champ de vecteurs plan.

\section 1. Stabilit\'e des solutions|

\subsection 1.1. D\'efinitions|

\noindent On consid\`ere le probl\`eme de Cauchy associ\'e \`a une \'equation
diff\'erentielle
$$
y'=f(t,y)\leqno{\rm (E)}
$$
avec condition initiale $y(t_0)=z_0$. On suppose que la solution de ce probl\`eme
existe sur $[t_0,+\infty[$.
\bigskip

\majdeb
\state D\'efinition -- | {\it Soit $y(t,z)$ la solution maximale de (E) tel
que $y(t_0,z)=z$. On dira que la solution $y(t,z_0)$ est {\rm stable} s'il existe une
boule $\overline B(z_0,r)$ et une constante $C\geq 0$ telles que
\vskip3pt
\parindent1.6pc
\item{\rm (i)} Pour tout $z\in\overline B(z_0,r)$, $t\mapsto y(t,z)$ est
d\'efinie sur $[t_0,+\infty[$~;
\medskip
\item{\rm (ii)} Pour tous $z\in\overline B(z_0,r)$ et $t\geq t_0$ on a
$$
\Vert y(t,z)-y(t,z_0)\Vert\leq C\Vert z-z_0\Vert.
$$}
\vskip-15pt\majfin

\majdeb {\it\parindent=1.6pc
\noindent La solution $y(t,z_0)$ est dite {\rm asymptotiquement stable} si elle est
stable et si la condition {\rm (ii')} plus forte que {\rm (ii)} est satisfaite~:
\vskip1pt
\item{\rm (ii')} Il existe une boule $\overline B(z_0,r)$ et une fonction
$\gamma:[t_0,+\infty[\rightarrow\bR_+$ continue avec
$\displaystyle\lim_{t\rightarrow+\infty}\gamma(t)=0$ telles que pour tous $z\in
\overline B(z_0,r)$ et $t\geq t_0$ on ait
$$
\Vert y(t,z)-y(t,z_0)\Vert\leq \gamma(t)\Vert z-z_0\Vert.
$$}
\vskip-\baselineskip
\majfin
\vskip\baselineskip

\noindent La signification g\'eom\'etrique de ces notions de stabilit\'e est
illustr\'ee par le sch\'ema suivant.

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\medskip\null

\subsection 1.2. Cas d'un syst\`eme lin\'eaire \`a coefficients constants|

\null
\vbox{\noindent Nous \'etudierons d'abord le cas le plus simple, \`a savoir
le cas d'un syst\`eme lin\'eaire sans second membre}
$$
Y'=AY,\qquad Y=\pmatrix{
y_1\cr
\vdots\cr
y_m\cr
},\qquad A=\pmatrix{
a_{11}&\dots&a_{1m}\cr
\vdots&&\vdots\cr
a_{m1}&\dots&a_{mm}\cr
}\leqno{\rm (E)}
$$
avec $y_j, a_{ij}\in\bC$~; le cas r\'eel peut bien entendu \^etre vu comme un cas
particulier du cas complexe. La solution du probl\`eme de Cauchy de condition
initiale $Y(t_0)=Z$ est donn\'ee par $Y(t,Z)=e^{(t-t_0)A}\cdot Z$. On a donc
$$
Y(t,Z)-Y(t,Z_0)=e^{(t-t_0)A}\cdot (Z-Z_0)
$$
et la stabilit\'e est li\'ee au comportement de $e^{(t-t_0)A}$ quand $t$ tend vers
$+\infty$, dont la norme $\VVert e^{(t-t_0)A}\VVert $ doit rester born\'ee. Distinguons
quelques cas.

\medskip
\noindent$\bullet$ $m=1$, $A=(a)$. On a alors
$$
|e^{(t-t_0)a}|=e^{(t-t_0)\Re(a)}.
$$
Les solutions sont stables si et seulement si cette quantit\'e reste born\'ee quand
$t$ tend vers $+\infty$, c'est-\`a-dire si $\Re(a)\leq 0$. De m\^eme, les solutions
sont asymptotiquement stables si et seulement si $\Re(a)<0$, et on peut alors prendre
$$
\gamma(t)=e^{(t-t_0)\Re(a)}~~
\build{\vlra15|}||{\textstyle t\rightarrow+\infty}|~~0.
$$

\medskip
\noindent$\bullet$ $m$ quelconque. Si $A$ est diagonalisable, on se ram\`ene apr\`es
un changement lin\'eaire de coordonn\'ees \`a
$$
\widetilde A=\pmatrix{
\lambda_1&&0\cr
&\ddots&\cr
0&&\lambda_m\cr
}
$$
o\`u $\lambda_1,\dots,\lambda_m$ d\'esignent les valeurs prores de $A$. Le syst\`eme
se ram\`ene aux \'equations ind\'ependantes $y'_j=\lambda_jy_j$ et admet pour
solution
$$
y_j(t,Z)=z_je^{\lambda_j(t-t_0)},\quad 1\leq j\leq m.
$$
Les solutions sont donc stables si et seulement si $\Re(\lambda_j)\leq 0$ pour tout
$j$ et asymptotiquement stables si et seulement si $\Re(\lambda_j)<0$ pour tout $j$.

Si $A$ n'est pas diagonalisable, il suffit de regarder ce qui se passe pour chaque
bloc d'une triangulation de $A$. Supposons donc
$$
A=\pmatrix{
\lambda&&\ast\cr
&\ddots&\cr
0&&\lambda\cr
}=\lambda I+N
$$
o\`u $N$ est une matrice nilpotente (triangulaire sup\'erieure) non nulle. Il vient
alors
$$
\eqalign{
e^{(t-t_0)A}&=e^{(t-t_0)\lambda I}\cdot e^{(t-t_0)}N\cr
&=e^{\lambda(t-t_0)}\sum_{k=0}^{m-1}\frac{(t-t_0)^k}{k!}\ N^k,\cr
}
$$
donc les coefficients de $e^{(t-t_0)A}$ sont des produits de $e^{\lambda(t-t_0)}$ par
des polyn\^omes de degr\'e $\leq m-1$ non tous constants (car $N\not=0$, donc le
degr\'e est au moins 1). Si $\Re(\lambda)<0$, les coefficients tendent vers 0, et si
$\Re(\lambda)>0$ leur module tend vers $+\infty$ car la croissance de l'exponentielle
l'emporte sur celle des polyn\^omes. Si $\Re(\lambda)=0$, on a
$|e^{\lambda(t-t_0)}|=1$ et par suite $e^{(t-t_0)A}$ est non born\'ee. On voit donc
que les solutions sont asympotiquement stables si et seulement si $\Re(\lambda)<0$ et
sinon elle sont instables. En r\'esum\'e, on peut \'enoncer~:

\bigskip
\majdeb
\state Th\'eor\`eme -- |  \it Soient $\lambda_1,\dots,\lambda_m$ les valeurs
propres complexes de la matrice $A$. Alors les solutions du syst\`eme lin\'eaire
$Y'=AY$ sont
\vskip3pt
\parindent1pc
\item{$\bullet$} asymptotiquement stables si et seulement si $\Re(\lambda_j)<0$ pour
tout $j=1,\dots,m$. 
\vskip1pt
\item{$\bullet$} stables si et seulement si pour tout $j$, ou bien
$\Re(\lambda_j)<0$, ou bien $\Re(\lambda_j)=0$ et le bloc correspondant est
diagonalisable. 
\par\majfin\par

\subsection 1.3. Petite perturbation d'un syst\`eme lin\'eaire|

\noindent On consid\`ere dans $\bK^m=\bR^m$ ou $\bC^m$ un syst\`eme de la forme
$$
Y'=AY+g(t,Y)\leqno{\rm (E)}
$$
o\`u $g:[t_0,+\infty[\ \times\ \bK^m\rightarrow\bK^m$ est une fonction continue. On se
propose de montrer que si la partie lin\'eaire est asymptotiquement stable et si la
\og perturbation\fg\ $g$ est suffisamment petite, en un sens \`a pr\'eciser, alors
les solutions de (E) sont encore asymptotiquement stables.


\bigskip
\majdeb
\state Th\'eor\`eme -- |  \it On suppose que les valeurs propres complexes
$\lambda_j$ de $A$ sont de partie r\'eelle $\Re\lambda_j<0$.
\vskip3pt
\parindent1.6pc
\item{\rm (a)} S'il existe une fonction $k:[t_0,+\infty[\ \rightarrow\bR_+$ continue
telle que $\displaystyle\lim_{t\rightarrow+\infty}k(t)=0$ et
$$
\forall t\in[t_0,+\infty[,\quad\forall Y_1,Y_2\in\bK^m,\quad \Vert
g(t,Y_1)-g(t,Y_2)\Vert\leq k(t)\Vert Y_1-Y_2\Vert,
$$
\item{} alors toute solution de (E) est asymptotiquement stable.
\vskip1pt
\item{\rm (b)} Si $g(t,0)=0$ et s'il existe $r_0>0$ et une fonction continue
$k:[0,r_0]\rightarrow\bR_+$ telle que $\displaystyle\lim_{r\rightarrow 0}k(r)=0$ et
$$
\forall t\in[t_0,+\infty[,\quad\forall Y_1,Y_2\in\overline B(0,r),\quad \Vert
g(t,Y_1)-g(t,Y_2)\Vert\leq k(r)\Vert Y_1-Y_2\Vert 
$$
\item{} pour $r\leq r_0$, alors il existe une boule $\overline B(0,r_1)\subset
\overline B(0,r_0)$ telle que toute solution $Y(t,Z_0)$ de valeur initiale
$Z_0\in\overline B(0,r_1)$ soit asymptotiquement stable.
\par\majfin\par

\bigskip
\noindent{\prem D\'emonstration.*} Si $\bK=\bR$, on peut toujours
\'etendre le syst\`eme \`a $\bC^m$ en posant par exemple $\widetilde
g(t,Y)=g(t,\Re(Y))$ pour $Y\in\bC^m$. On se placera donc dans $\bC^m$.
Il existe alors une base $(e_1,\dots,e_m)$ dans laquelle $A$ se met
sous forme triangulaire
$$
A=\pmatrix{
\lambda_1&a_{12}&\dots&a_{1m}\cr
&\lambda_2&&\vdots\cr
\vdots&&\ddots&a_{m-1m}\cr
\noalign{\vskip-6pt}
&\dots&\dots&\cr
0&\dots&\dots&\lambda_m\cr
}
$$
Posons $\widetilde e_j=\varepsilon^je_j$ avec $\varepsilon>0$ petit. Il vient
$$
\eqalign{
A\widetilde e_j&=\varepsilon^j(a_{1j}e_1+\dots+a_{j-1j}e_{j-1}+\lambda_je_j)\cr
&=\varepsilon^{j-1}a_{1j}\widetilde e_1+\dots+\varepsilon a_{j-1j}\widetilde
e_{j-1}+\lambda_j\widetilde e_j\cr 
}
$$
de sorte que dans la base $(\widetilde e_j)$ les coefficients non diagonaux peuvent
\^etre rendus arbitrairement petits. On supposera donc qu'on a $|a_{ij}|\leq
\varepsilon$, et on pourra choisir $\varepsilon$ aussi petit qu'on veut.
Consid\'erons deux solutions $Y(t,Z)$ et $Y(t,Z_0)$~:
$$
\eqalign{
Y'(t,Z)\kern4pt&=AY(t,Z)+g(t,Y(t,Z)),\cr
Y'(t,Z_0)&=AY(t,Z_0)+g(t,Y(t,Z_0))\cr
}
$$
et cherchons \`a \'evaluer la diff\'erence $\Delta(t)=Y(t,Z)-Y(t,Z_0)$ en distinguant
les deux cas (a) et (b).

\bigskip
(a) Observons dans ce cas que $f(t,Y)=AY+g(t,Y)$ est lipschitzienne en $Y$ avec
constante de Lipschitz $\VVert A\VVert +k(t)$. Le crit\`ere V 3.4 montre donc d\'ej\`a que
toutes les solutions sont globalement d\'efinies sur $[t_0,+\infty[$. Nous avons
$$
\eqalign{
&\Delta'(t)=A\Delta(t)+g(t,Y(t,Z))-g(t,Y(t,Z_0)),\cr
&\Vert g(t,Y(t,Z))-g(t,Y(t,Z_0))\Vert\leq k(t)\Vert \Delta(t)\Vert\cr
}
$$
Notons $(\delta_j(t))_{1\leq j\leq m}$ les composantes de $\Delta(t)$ et
$$
\rho(t)=\Vert \Delta(t)\Vert^2=\sum_{j=1}^m\delta_j(t)\overline{\delta_j(t)}.
$$
Par diff\'erentiation on obtient
$$
\eqalign{
\rho'(t)&=\sum_{j=1}^m\delta'_j(t)\overline{\delta_j(t)}+\delta_j(t)
\overline{\delta'_j(t)}=2\Re\sum_{i=1}^m\delta'_j(t)\overline{\delta_j(t)},\cr
&=2\Re(^t\overline{\Delta(t)}\Delta'(t))\cr
&=2\Re(^t\overline{\Delta(t)}A
\Delta(t))+2\Re\Big({}^t\overline{\Delta(t)}(g(t,Y(t,Z))-g(t,Y(t,Z_0)))\Big).\cr
}$$
La deuxi\`eme partie r\'eelle est major\'ee par
$$
2\Vert\Delta(t)\Vert\cdot\Vert g(t,Y(t,Z))-g(t,Y(t,Z_0))\Vert\leq
2k(t)\Vert\Delta(t)\Vert^2=2k(t)\rho(t). 
$$
On a par ailleurs
$$
^t\overline{\Delta(t)}A\Delta(t)=\sum_{j=1}^m\lambda_j|\delta_j(t)|^2+\sum_{i<j}a_{ij}
\overline{\delta_i(t)}\delta_j(t),
$$
de sorte que
$$
\Re(^t\overline{\Delta(t)}A\Delta(t))\leq\sum_{j=1}^m(\Re\lambda_j)|\delta_j(t)|^2+
\Big(\sum_{i<j}|a_{ij}|\Big)\Vert\Delta(t)\Vert^2.
$$
Comme $\Re(\lambda_j)<0$ par hypoth\`ese et $|a_{ij}|\leq\varepsilon$, il y a un
choix de $\varepsilon$ tel que
$$
\Re(^t\overline{\Delta(t)}A\Delta(t))\leq-\alpha\sum_{j=1}^m|\delta_j(t)|^2=
-\alpha\rho(t)
$$
avec $\alpha>0$. On obtient alors
$$
\eqalign{
\rho'(t)&\leq-2\alpha\rho(t)+2k(t)\rho(t),\cr
\frac{\rho'(t)}{\rho(t)}&\leq-2\alpha+2k(t),\cr
\ln\,\frac{\rho(t)}{\rho(t_0)}&\leq -2\int^t_{t_0}(\alpha-k(u))du,\cr
\rho(t)&\leq\Vert Z-Z_0\Vert^2\exp\Big(-2\int^t_{t_0}(\alpha-k(u))du\Big)\cr
}
$$
car $\rho(t_0)=\Vert Z-Z_0\Vert^2$. On notera que $\rho(t)=\Vert\Delta(t)\Vert^2$ ne
peut s'annuler que si les deux solutions co\"{\i}ncident identiquement. En 
prenant la racine carr\'ee, on obtient
$$
\Vert Y(t,Z)-Y(t,Z_0)\Vert\leq \gamma(t)\Vert Z-Z_0\Vert
$$
avec
$$
\gamma(t)=\exp\,\Big(-\int^t_{t_0}(\alpha-k(u))du\Big).
$$
Comme
$\displaystyle\lim_{u\rightarrow+\infty}(\alpha-k(u))=\alpha>0$,
l'int\'egrale diverge vers $+\infty$ et
$\displaystyle\lim_{t\rightarrow+\infty}\gamma(t)=0$. Les solutions
sont donc bien asymptotiquement stables.

\bigskip
\noindent(b) Ce cas est un peu plus d\'elicat car on ne sait pas {\it a
priori} si toutes les solutions sont globales~; elles ne le seront
d'ailleurs pas en g\'en\'eral si $Z_0\not\in\overline B(0,r_0)$, vu
que les hypoth\`eses ne concernent que ce qui se passe pour
$Y\in\overline B(0,r_0)$. Comme $g(t,0)=0$, on a toutefois la solution
globale $Y(t)=0$, c'est-\`a-dire que $Y(t,0)=0$ pour
$t\in[t_0,+\infty[$. De plus on a
$$
\Vert g(t,Y(t,Z))-g(t,Y(t,Z_0))\Vert\leq k(r)\Vert\Delta(t)\Vert,
$$
\`a condition de supposer que $t\mapsto Y(t,Z)$ et $t\mapsto
Y(t,Z_0)$ prennent toutes leurs valeurs dans $\overline B(0,r)\subset
\overline B(0,r_0)$. Sous cette hypoth\`ese, les m\^emes calculs que
pr\'ec\'edemment donnent
$$
\eqalignno{
\rho(t)\leq\Vert Z-Z_0\Vert^2~&\exp\Big(-2\int^t_{t_0}(\alpha-k(r)du\Big),\cr
\Vert Y(t,Z)-Y(t,Z_0)\Vert\leq{}&\exp\Big(-(t-t_0)(\alpha-k(r)\Big)\Vert
Z-Z_0\Vert,&(*)\cr 
}$$
et en particulier pour $Z_0=0$~:
$$
\Vert Y(t,Z)\Vert\leq \exp\,\Big(-(t-t_0)(\alpha-kr)\Big)\Vert Z\Vert.
$$
Comme $\displaystyle\lim_{r\rightarrow 0}k(r)=0$, on peut choisir $r_1<r_0$ tel que
$k(r_1)<\alpha$, c'est-\`a-dire $\alpha-k(r_1)>0$. L'in\'egalit\'e pr\'ec\'edente
montre alors que pour $Z\in B(0,r_1)$ la solution maximale $Y(t,Z)$ contenue dans la
boule ouverte $B(0,r_1)$ v\'erifie les in\'egalit\'es\break $\Vert Y(t,Z)\Vert\leq\Vert
Z\Vert<r_1$. Cette solution maximale est n\'ecessairement d\'efinie globalement sur
$[t_0,+\infty[$. Sinon l'intervalle maximal serait un intervalle born\'e $[t_0,t_1[$,
n\'ecessairement ouvert \`a droite d'apr\`es les r\'esultats de V 2.4. Comme la
d\'eriv\'ee de $t\mapsto Y(t,Z)$ est major\'ee par
$$
\Vert AY+g(t,Y)\Vert\leq (\VVert A\VVert +k(r_1))\Vert Y\Vert\leq M
$$
avec $M=(\VVert A\VVert +k(r_1))r_1$, la fonction $Y(t,Z)$ v\'erifierait le crit\`ere de
Cauchy
$$
\lim_{t,t'\rightarrow t_1-0}\Vert Y(t,Z)-Y(t',Z)\Vert=0.
$$
Elle aurait donc une limite $\displaystyle Y_1=\lim_{t\rightarrow t_1-0}Y(t,Z)$ avec
$\Vert Y_1\Vert\leq\Vert Z\Vert<r_1$ et se prolongerait sur un voisinage \`a droite
de $t_1$ en une solution enti\`erement contenue dans $B(0,r_1)$, contradiction.
Quitte \`a diminuer encore un peu $r_1$, on voit que toute solution $Y(t,Z)$ avec
$Z\in\overline B(0,r_1)$ est globale et enti\`erement contenue dans $\overline
B(0,r_1)$. Par cons\'equent $(*)$ est satisfaite pour tous $t\in[t_0,+\infty[$ et
$Z,Z_0\in\overline B(0,r_1)$ avec la constante $\alpha-k(r_1)>0$, ce qui d\'emontre le
th\'eor\`eme.\hfill\blacksquare

\section 2. Points singuliers d'un champ de vecteurs|

\subsection 2.1. Position du probl\`eme|

\noindent On suppose donn\'e un champ de vecteurs de classe $C^1$ dans un ouvert
$\Omega\subset \bR^2$, c'est-\`a-dire une application
$$
\Omega\rightarrow\bR^2,\qquad M=\pmatrix{x\cr y\cr}\mapsto\overrightarrow
V(M)=\pmatrix{
f(x,y)\cr
g(x,y)\cr
} 
$$
o\`u $f,g$ sont de classe $C^1$ sur $\Omega$. On consid\`ere le syst\`eme
diff\'erentiel associ\'e
$$
\frac{\overrightarrow{dM}}{dt}=\overrightarrow V(M)\Longleftrightarrow\cases{
x'(t)=f(x(t),y(t))\hfill\cr
\noalign{\vskip3pt}
y'(t)=g(x(t),y(t))\hfill\cr
}.
$$
Gr\^ace au th\'eor\`eme de Cauchy-Lipschitz, on sait que par tout point il passe une
courbe int\'egrale unique. Un probl\`eme g\'eom\'etrique int\'eressant est de
d\'ecrire l'allure de la famille des courbes int\'egrales passant au voisinage d'un
point  $M_0$ donn\'e.

\bigskip
\state Premier cas~:| $\overrightarrow V(M_0)\not=\overrightarrow 0$. 
Dans ce cas, l'angle entre $\overrightarrow V(M)$ et $\overrightarrow
V(M_0)$ tend vers 0 quand $M$ tend vers 0. Par cons\'equent, les
tangentes aux lignes int\'egrables sont sensiblement parall\`eles les
unes aux autres dans un petit voisinage de $M_0$. Un tel point $M_0$
est dit r\'egulier~:

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\EndFig
\vskip-42mm

\state Deuxi\`eme cas~:| $\overrightarrow V(M_0)=\overrightarrow 0$. On voit
alors facilement sur des exemples qu'il y a plusieurs configurations possibles
pour le champ des tangentes~:

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\vskip-5mm

$$
\overrightarrow V\pmatrix{x\cr y\cr}=
\pmatrix{x\cr y\cr}\qquad\qquad
\overrightarrow V\pmatrix{x\cr y\cr}=\pmatrix{x\cr-y\cr}\qquad\qquad
\overrightarrow V\pmatrix{x\cr y\cr}=\pmatrix{-y\cr x\cr} 
$$
Si $\overrightarrow V(M_0)=\overrightarrow 0$, on dit que $M_0$ est
un {\it point singulier} (ou {\it point critique}) du champ de
vecteurs. Un tel point donne \'evidemment une solution constante
$M(t)=M_0$ de (E). Pour \'etudier les solutions voisines, on supposera
apr\`es translation \'eventuelle de l'origine des coordonn\'ees que
$M_0=0$. On a alors $f(0,0)=g(0,0)=0$, de sorte que le syst\`eme
diff\'erentiel peut s'\'ecrire
$$
\left\{\eqalign{
\dfrac{dx}{dt}&=f(x,y)=ax+by+o(|x|+|y|)\hfill\cr
\noalign{\vskip3pt}
\dfrac{dy}{dt}&=g(x,y)=cx+dy+o(|x|+|y|).\hfill\cr}
\right.
$$
Introduisons la matrice
$$
A=\pmatrix{
a&b\cr
\noalign{\vskip3pt}
c&d\cr
}=\pmatrix{
f'_x(0,0)&f'_y(0,0)\cr
\noalign{\vskip3pt}
g'_x(0,0)&g'_y(0,0)\cr
}.
$$
Le syst\`eme diff\'erentiel consid\'er\'e s'\'ecrit maintenant
$$
\frac{dM}{dt}=AM+G(M)
$$
avec $G(0,0)=G'_x(0,0)=G'_y(0,0)=0$. La fonction continue
$$
k(r)=\sup_{M\in \overline{B}(0,r)}\VVert G'(M)\VVert
$$ 
tend vers 0 quand $r$ tend vers 0 et
le th\'eor\`eme des accroissements finis donne
$$
\Vert G(M_1)-G(M_2)\Vert\leq k(r)\Vert\overrightarrow{M_1M_2}\Vert
$$
pour tous $M_1,M_2\in\overline B(0,r)$. L'hypoth\`ese (b) du
th\'eor\`eme du \S\ 1.3 est donc satisfaite. Dire que le point $M_0$
est asymptotiquement stable signifie que les lignes int\'egrales
issues d'un point $M_1$ voisin de $M_0$ convergent toutes vers $M_0$
(\`a peu pr\`es uniform\'ement \`a la m\^eme vitesse) quand le temps
tend vers $+\infty$. On peut donc \'enoncer~:

\bigskip
\majdeb
\state Proposition -- |  {\it Pour qu'un point singulier $M_0=(x_0,y_0)$ soit
asymptotiquement stable, il suffit que les valeurs propres de la matrice 
jacobienne
$$
A=\pmatrix{
f'_x(x_0,y_0)&f'_y(x_0,y_0)\cr
\noalign{\vskip3pt}
g'_x(x_0,y_0)&g'_y(x_0,y_0)\cr
}
$$
soient de partie r\'eelle $<0$.
}
\par\majfin\par

\bigskip
\state Remarque -- |  Contrairement au cas d'un syst\`eme lin\'eaire, on ne
peut pas d\'ecider de la nature du point critique si la matrice jacobienne 
a une valeur propre de partie r\'eelle nulle. Consid\'erons par exemple le 
syst\`eme
$$
\left\{
\eqalign{
\dfrac{dx}{dt}&=\alpha x^3\cr
\noalign{\vskip2pt}
\dfrac{dy}{dt}&=\beta y^3\cr
}\right.\qquad t\in[t_0,+\infty[{}=[0,+\infty[,
$$
qui admet l'origine comme point critique avec matrice jacobienne $A=0$. On voit
facilement que la solution du probl\`eme de Cauchy est
$$
x(t)=x_0(1-2\alpha x^2_0 t)^{-1/2},\quad
y(t)=y_0(1-2\,\beta y_0^2 t)^{-1/2}. 
$$
Par cons\'equent l'origine est un point asymptotiquement stable si
$\alpha<0$ et $\beta<0$, instable d\`es que $\alpha>0$ ou $\beta>0$.
Dans ce dernier cas, les solutions ne sont en fait m\^eme pas
globalement d\'efinies~: lorsque $\alpha>0$, $\beta\leq 0$ et
$x_0\not=0$, la solution maximale n'est d\'efinie que pour
$t\in[0,1/(2\,\alpha x_0^2)[$.

\bigskip
Si la matrice jacobienne est inversible (valeurs propres $\not=0$), le
th\'eor\`eme d'inversion locale montre que la fonction
$M\mapsto\overrightarrow V(M)$ d\'efinit une bijection d'un voisinage
de $M_0$ sur un voisinage de $0$~; en particulier, pour $M$ assez
voisin et distinct de $M_0$ on aura $\overrightarrow
V(M)\not=\overrightarrow 0$, de sorte que $M_0$ est un point singulier
{\it isol\'e}. Ce n'est pas toujours le cas si la matrice est
d\'eg\'en\'er\'ee~: le champ $\overrightarrow V(x,y)=(x,0)$ admet par
exemple toute une droite $x=0$ de points singuliers. On exclura en
g\'en\'eral ces situations qui peuvent \^etre extr\^emement
compliqu\'ees.

\bigskip
\majdeb
\state D\'efinition -- |  {\it On dira qu'un point singulier $M_0$ 
est non d\'eg\'en\'er\'e si
$$
\det\,\pmatrix{
f'_x(x_0,y_0)&f'_y(x_0,y_0)\cr
\noalign{\vskip3pt}
g'_x(x_0,y_0)&g'_y(x_0,y_0)\cr
}\not=0.
$$} 
\vskip-12pt
\par\majfin\par

\bigskip
Nous nous proposons maintenant d'\'etudier les diff\'erentes
configurations possibles pour un point singulier non d\'eg\'en\'er\'e.
On verra au chapitre XI, \S\ 2.3 que les courbes int\'egrales ont
tendance \`a ressembler \`a celles du syst\`eme lin\'eaire $dM/dt=AM$
lorsqu'on se rapproche du point critique, tout au moins sur un
intervalle de temps $[t_0,t_1]$ fix\'e~; ceci n'est pas
n\'ecessairement vrai sur tout l'intervalle $[t_0,+\infty[$ (voir 
\S~2.3 pour des exemples). On se restreindra dans un premier temps au cas
lin\'eaire.

\subsection 2.2. Cas d'un champ lin\'eaire de vecteurs|

\noindent Consid\'erons le syst\`eme
$$
\dfrac{dM}{dt}=AM,\qquad
\left\{
\eqalign{
\dfrac{dx}{dt}&=ax+by\cr
\noalign{\vskip1pt}
\dfrac{dy}{dt}&=cx+dy\cr
}\right.
\quad\hbox{\rm o\`u}\quad A=\pmatrix{a&b\cr c&d\cr}.
$$
On supposera $\det\,A\not=0$, de sorte que le champ de vecteurs
$\overrightarrow V(M)=AM$ admet l'origine pour seul point critique.
Comme le champ des tangentes est invariant par les homoth\'eties de
centre O, les courbes int\'egrales se d\'eduisent les unes des autres
par homoth\'eties. Distinguons maintenant plusieurs cas en fonction
des valeurs propres de $A$.

\bigskip
\noindent {\tenpointbf (a) Les valeurs propres $\lambda_1,\lambda_2$ de $A$ 
sont r\'eelles.}
\medskip
\noindent$\bullet$ Supposons de plus $\lambda_1\not=\lambda_2$. Dans ce cas la
matrice $A$ est diagonalisable. Apr\`es changement de base on peut supposer
$$
A=\pmatrix{
\lambda_1&0\cr
\noalign{\vskip3pt}
0&\lambda_2\cr
}
$$
et le syst\`eme se r\'eduit \`a
$$
\left\{
\eqalign{
\dfrac{dx}{dt}&=\lambda_1 x\cr
\noalign{\vskip3pt}
\dfrac{dy}{dt}&=\lambda_2 y.\cr
}\right.
$$
\vbox{
La solution du probl\`eme de Cauchy avec $M(0)=(x_0,y_0)$ est donc
$$
\cases{
x(t)=x_0e^{\lambda_1t}\hfill\cr
\noalign{\vskip3pt}
y(t)=y_0e^{\lambda_2t}\hfill\cr
}
$$
de sorte que les courbes int\'egrales sont les courbes
$$
y=C|x|^{\lambda_2/\lambda_1},\quad C\in\bR
$$
et la droite d'\'equation $x=0$. Distinguons deux sous-cas~:}

\medskip $\ast$ $\lambda_1,\lambda_2$ de m\^eme signe et, disons,
$|\lambda_1|<|\lambda_2|$.  On a alors $\lambda_2/\lambda_1>1$. On
dit qu'on a affaire \`a un {\it n{\oe}ud impropre}~:
\vskip12mm

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$\matrix{
0<\lambda_1<\lambda_2\cr
\noalign{\vskip3pt}
\hbox{n{\oe}ud impropre instable}
}\qquad\qquad\qquad\qquad\qquad
\matrix{
\lambda_2<\lambda_1<0\cr
\noalign{\vskip3pt}
\hbox{n{\oe}ud impropre stable}
}$\ELTX
\EndFig
\vskip-23mm}

$\ast$ $\lambda_1,\lambda_2$ de signes oppos\'es, par exemple
$\lambda_1<0<\lambda_2$. Il s'agit d'un {\it col} (toujours instable)~:

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\EndFig
\vskip-40mm}

\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
int\'egrales sont donn\'ees 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
\`a un {\it n{\oe}ud propre}~:

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\vskip23mm

$$
\matrix{
\lambda>0\cr
\noalign{\vskip5pt}
\hbox{N{\oe}ud propre instable}\cr
}\qquad\qquad\qquad\qquad\qquad\matrix{
\lambda<0\cr
\noalign{\vskip5pt}
\hbox{N{\oe}ud propre stable}\cr
}
$$
\vskip7.1mm

$\ast$ $A$ est non diagonalisable. Alors il existe une base dans
laquelle la matrice $A$ et le syst\`eme s'\'ecrivent
$$
A=\pmatrix{
\lambda&0\cr
1&\lambda\cr
},\qquad\qquad\left\{\eqalign{
\dfrac{dx}{dt}&=\lambda x\cr
\noalign{\vskip3pt}
\dfrac{dy}{dt}&=x+\lambda y.\cr}
\right.
$$
Le courbes int\'egrales sont donn\'ees 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 int\'egrale avec $x_0\not=0$ passe par un point
tel que $|x(t)|=1$, on obtient toutes les courbes int\'egrales autres
que $x=0$ en prenant $x_0=\pm 1$, d'o\`u
$$
\left\{\eqalign{
t&=\dfrac{1}{\lambda}\ \ln\,|x|\cr
\noalign{\vskip3pt}
y&=y_0|x|+\dfrac{x}{\lambda}\ \ln\,|x|\cr}
\right.
$$
On dit qu'il s'agit d'un {\it n{\oe}ud 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 d\'eduisent par homoth\'eties.

\InsertFig -49.000 102.000 {
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100.569  75.007   100.995  75.385   101.419  75.764   101.840  76.145   
102.260  76.528   102.678  76.912   103.095  77.299   103.510  77.686   
103.923  78.076   104.334  78.467   ] scurve 
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100.734  67.469   101.211  67.796   101.685  68.125   102.156  68.458   
102.625  68.793   103.091  69.131   103.554  69.471   104.014  69.813   
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\vskip-50mm

$$
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\lambda>0\cr
\noalign{\vskip3pt}
\hbox{N{\oe}ud exceptionnel instable}\cr
}\qquad\qquad\qquad\qquad\matrix{
\lambda<0\cr
\noalign{\vskip3pt}
\hbox{N{\oe}ud exceptionnel stable}\cr
}
$$

\bigskip
\noindent {\tenpointbf (b) Les valeurs propres de $A$ sont non r\'eelles.}
\medskip

On a des valeurs propres complexes conjugu\'ees $\alpha+i\beta$,
$\alpha-i\beta$ avec disons $\beta>0$, et il existe une base dans
laquelle la matrice $A$ et le syst\`eme s'\'ecrivent
$$
A=\pmatrix{
\alpha&-\beta\cr
\noalign{\vskip3pt}
\beta&\alpha\cr
},\qquad\left\{\eqalign{
\dfrac{dx}{dt}&=\alpha x-\beta y\hfill\cr
\noalign{\vskip3pt}
\dfrac{dy}{dt}&=\beta x+\alpha y.\hfill\cr}
\right.
$$
La mani\`ere la plus rapide de r\'esoudre un tel syst\`eme 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 g\'en\'erale est
$$
z(t)=z_0e^{(\alpha+i\beta)t}=z_0e^{\alpha t}e^{i\beta t}.
$$
En coordonn\'ees polaires $z=re^{i\theta}$, l'\'equation 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 g\'en\'eral
graphiquement une ellipse car la base utilis\'ee ci-dessus n'est pas
n\'ecessairement orthonorm\'ee). On dit alors que le point singulier
est un {\it foyer}, respectivement un {\it centre}~:

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116.468  75.756   115.807  75.888   115.130  76.002   114.434  76.098   
113.721  76.174   112.987  76.229   112.232  76.262   111.454  76.269   
110.662  76.251   109.897  76.206   109.156  76.139   108.439  76.050   
107.742  75.941   107.066  75.812   106.409  75.665   105.771  75.501   
105.150  75.319   104.547  75.121   103.961  74.906   103.392  74.676   
102.839  74.429   102.303  74.167   101.783  73.889   101.280  73.595   
100.794  73.285   100.325  72.959    99.874  72.616    99.441  72.255   
 99.026  71.876    98.632  71.479    98.259  71.062    97.908  70.624   
 97.581  70.164    97.280  69.679    97.007  69.168    96.767  68.627   
 96.563  68.053    96.401  67.441    96.290  66.783    96.241  66.070   
 96.274  65.286    96.386  64.564    96.556  63.909    96.774  63.306   
 97.031  62.747    97.323  62.225    97.645  61.735    97.995  61.275   
 98.369  60.842    98.767  60.434    99.187  60.049    99.628  59.686   
100.089  59.345   100.570  59.024   101.070  58.724   101.588  58.443   
102.126  58.183   102.682  57.942   103.257  57.721   103.851  57.520   
104.465  57.341   105.099  57.183   105.755  57.048   106.433  56.937   
107.135  56.852   107.863  56.796   108.619  56.770   109.407  56.778   
110.176  56.823   110.911  56.898   111.614  57.003   112.290  57.134   
112.939  57.290   113.562  57.469   114.162  57.671   114.738  57.896   
115.291  58.141   115.822  58.409   116.329  58.697   116.813  59.008   
117.273  59.341   117.709  59.697   118.118  60.078   118.500  60.484   
118.851  60.918   119.170  61.381   119.452  61.879   119.691  62.414   
119.879  62.995   120.005  63.629   120.050  64.334   119.981  65.121   
119.819  65.798   119.591  66.398   119.312  66.939   118.989  67.433   
118.630  67.886   118.237  68.303   117.815  68.687   117.364  69.041   
116.887  69.365   116.383  69.662   115.855  69.931   115.301  70.173   
114.722  70.388   114.116  70.574   113.483  70.731   112.822  70.856   
112.130  70.947   111.404  71.000   110.640  71.010   109.859  70.972   
109.128  70.889   108.438  70.769   107.787  70.614   107.170  70.426   
106.587  70.209   106.035  69.962   105.516  69.685   105.028  69.379   
104.573  69.043   104.154  68.676   103.771  68.275   103.430  67.836   
103.136  67.357   102.899  66.828   102.732  66.239   102.660  65.573   
102.732  64.793   102.926  64.140   103.200  63.582   103.532  63.092   
103.913  62.658   104.335  62.271   104.795  61.926   105.291  61.621   
105.821  61.353   106.384  61.123   106.982  60.932   107.616  60.781   
108.289  60.674   109.004  60.616   109.769  60.617   110.526  60.683   
111.220  60.803   111.861  60.970   112.455  61.180   113.003  61.430   
113.507  61.719   113.965  62.048   114.376  62.420   114.733  62.837   
115.029  63.308   115.246  63.843   115.357  64.462   115.297  65.207   
115.071  65.842   114.752  66.358   114.365  66.791   113.923  67.160   
113.433  67.472   112.897  67.732   112.316  67.939   111.688  68.092   
111.010  68.185   110.273  68.205   109.508  68.138   108.831  67.998   
108.226  67.798   107.684  67.543   107.205  67.234   106.791  66.871   
106.450  66.447   106.202  65.949   ] scurve 
[ 106.202  65.949   106.086  65.354   106.203  64.602   106.504  64.043   
106.900  63.611   107.367  63.268   107.895  63.000   108.483  62.805   
109.132  62.687   109.851  62.661   110.601  62.751   111.233  62.932   
111.772  63.189   112.223  63.517   112.577  63.920   112.808  64.415   
112.839  65.042   112.563  65.685   112.150  66.109   111.649  66.412   
111.070  66.613   110.414  66.707   109.666  66.663   109.026  66.481   
108.520  66.200   108.142  65.824   107.928  65.336   108.021  64.679   
108.417  64.199   108.932  63.911   109.542  63.765   110.253  63.787   
110.870  63.992   111.299  64.324   111.518  64.787   111.308  65.441   
110.816  65.781   110.208  65.911   109.490  65.794   109.041  65.481   
108.890  65.009   109.371  64.444   109.976  64.335   110.613  64.560   
110.812  64.984   110.229  65.474   109.604  65.359   109.418  64.957   
110.032  64.649   110.434  64.979   109.743  65.155   109.937  64.812   
110.165  65.097   109.878  65.096   ] scurve 
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\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'homoth\'etie de deux spires
cons\'ecutives de la \hbox{spirale est $e^{2\pi\alpha/\beta}$}\kern-1.8pt.

\subsection 2.3. Singularit\'es de champs de vecteurs non lin\'eaires|

\noindent L'objet de ce paragraphe est essentiellement de mettre en
garde le lecteur contre un certain nombre d'id\'ees fausses
fr\'equemment rencontr\'ees, en particulier l'id\'ee que les courbes
int\'egrales d'un champ de vecteurs quelconque au voisinage d'un point
singulier ressemblent toujours \`a celles du syst\`eme lin\'eaire
associ\'e. En fait, ce n'est en g\'en\'eral pas le cas lorsque le
syst\`eme lin\'earis\'e pr\'esente {\it un centre}, et cela peut de
m\^eme ne pas \^etre le cas lorsque celui-ci pr\'esente {\it un n{\oe}ud}.
Les deux exemples ci-dessous illustrent le ph\'enom\`ene.

\bigskip
\state Exemple 1 -- |  On consid\`ere le syst\`eme diff\'erentiel
$$
{\rm (S)}\qquad\left\{\eqalign{
\dfrac{dx}{dt}&=-y-x(x^2+y^2)\cr
\noalign{\vskip2pt}
\dfrac{dy}{dt}&=\kern8pt x-y(x^2+y^2).\cr}
\right.
$$
L'origine est un point critique non d\'eg\'en\'er\'e, et le
syst\`eme lin\'eaire associ\'e\break $dx/dt=-y$, $dy/dt=x$ pr\'esente
un centre d'apr\`es le \S 2.2. Si l'on passe en coordonn\'ees polaires
$(r,\theta)$ le syst\`eme (S) devient
$$
\left\{\eqalign{
r\,\dfrac{dr}{dt}&=x\,\dfrac{dx}{dt}+y\,\dfrac{dy}{dt}=-(x^2+y^2)^2\cr
\noalign{\vskip2pt}
\dfrac{d\theta}{dt}&=\dfrac{x\,\dfrac{dy}{dt}
-y\,\dfrac{dx}{dt}}{x^2+y^2}=1\hfill\cr}
\right.
\qquad\Longleftrightarrow\qquad
\left\{\eqalign{
\dfrac{dr}{r^3}&=-dt\cr
\noalign{\vskip2pt}
d\theta&=dt\phantom{\raise4pt\hbox{$\bigg|$}}\cr}
\right.
$$
car $rdr=xdx+ydy$ et $xdy-ydx=r^2d\theta$ (exercice~!). Les courbes
int\'egrales de l'\'equation $-dr/r^3=d\theta$ sont donn\'ees par
$1/2r^2=\theta-\theta_0$, soit $r=(2(\theta-\theta_0))^{-1/2}$ pour
$\theta>\theta_0$. On a ici $\theta=t+C$,
$\lim_{\theta\rightarrow+\infty}r(\theta)=0$. On voit que
les courbes int\'egrales sont des spirales convergeant vers 0 quand
$t\rightarrow+\infty$, l'origine est donc {\it un foyer} stable.

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 71.581  82.367    71.934  82.819    72.339  83.212    72.796  83.549   
 73.306  83.825    73.873  84.036    74.506  84.168    75.224  84.199   
 75.976  84.099    76.609  83.902    77.159  83.635    77.642  83.309   
 78.065  82.930    78.432  82.501    78.743  82.021    78.992  81.488   
 79.172  80.895    79.269  80.230    79.251  79.474    79.112  78.771   
 78.890  78.169    78.603  77.642    78.261  77.177    77.868  76.770   
 77.425  76.418    76.933  76.122    76.386  75.887    75.779  75.723   
 75.098  75.646    74.321  75.688    73.640  75.844    73.050  76.077   
 72.532  76.372    72.075  76.720    71.673  77.118    71.325  77.565   
 71.033  78.061    70.801  78.610    70.639  79.219    70.562  79.900   
 70.603  80.677    70.756  81.362    70.985  81.954    71.276  82.477   
 71.620  82.940    72.012  83.347    72.452  83.702    72.941  84.003   
 73.481  84.245    74.078  84.422    74.744  84.518    75.497  84.508   
 76.211  84.380    76.827  84.170    77.369  83.897    77.850  83.570   
 78.276  83.194    78.650  82.771    78.972  82.301    79.239  81.783   
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 79.372  78.433    79.129  77.856    78.829  77.344    78.480  76.888   
 78.083  76.485    77.641  76.132    77.153  75.831    76.615  75.586   
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 73.252  75.585    72.691  75.841    72.191  76.151    71.745  76.509   
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 70.291  78.995    70.188  79.656    70.181  80.397    70.288  81.134   
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 71.790  83.690    72.232  84.043    72.719  84.346    73.253  84.596   
 73.839  84.788    74.485  84.909    75.205  84.943    75.973  84.863   
 76.634  84.694    77.218  84.458    77.740  84.169    78.209  83.831   
 78.629  83.450    79.001  83.025    79.325  82.557    79.600  82.045   
 79.820  81.484    79.978  80.869    80.061  80.189    80.044  79.428   
 79.922  78.711    79.725  78.083    79.469  77.524    79.162  77.021   
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 74.317  74.833    73.612  74.964    72.991  75.166    72.434  75.425   
 71.933  75.732    71.480  76.084    71.073  76.477    70.711  76.911   
 70.395  77.386    70.126  77.904    69.909  78.468    69.752  79.084   
 69.666  79.762    69.672  80.518    69.781  81.248    69.964  81.890   
 70.206  82.464    70.498  82.983    70.834  83.453    71.211  83.878   
 71.627  84.258    72.084  84.595    72.581  84.887    73.122  85.131   
 73.710  85.322    74.352  85.450    75.060  85.501    75.853  85.450   
 76.545  85.310    77.160  85.104    77.716  84.845    78.219  84.540   
 78.677  84.193    79.091  83.807    79.462  83.382    79.791  82.918   
 80.076  82.415    80.314  81.869    80.500  81.276    80.624  80.629   
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 75.175  86.302    75.966  86.250    76.664  86.119    77.295  85.930   
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 65.654  80.031    65.646  80.790    65.700  81.561    65.807  82.274   
 65.957  82.939    66.145  83.563    66.365  84.152    66.615  84.708   
 66.893  85.236    67.196  85.736    67.524  86.210    67.875  86.659   
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154.087 110.925   154.779 111.033   155.470 111.142   156.160 111.252   
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\vskip-60mm

\state Exemple 2 -- |  On consid\`ere maintenant le syst\`eme diff\'erentiel
$$
{\rm (S)}\qquad\left\{\eqalign{
\dfrac{dx}{dt}&=-x-\dfrac{2y}{\ln\,(x^2+y^2)}\cr
\noalign{\vskip2pt}
\dfrac{dy}{dt}&=-y+\dfrac{2x}{\ln\,(x^2+y^2)}\cr}
\right.
$$
sur le disque unit\'e ouvert $x^2+y^2<1$. Observons que $2y/\ln\,(x^2+y^2)$ se
prolonge en une fonction de classe $C^1$ au voisinage de $(0,0)$~: elle admet en effet
une limite \'egale \`a 0 \`a l'origine, ainsi que ses d\'eriv\'ees partielles
$$
\dfrac{-4xy}{(x^2+y^2)(\ln\,(x^2+y^2))^2},\quad
\dfrac{2}{\ln\,(x^2+y^2)}-\dfrac{4y^2}{(x^2+y^2)(\ln\,(x^2+y^2))^2}. 
$$
Il en est de m\^eme pour le terme $2x/\ln\,(x^2+y^2)$. L'origine est donc un point
singulier, et le syst\`eme lin\'eaire associ\'e $dx/dt=-x$, $dy/dt=-y$ pr\'esente
{\it un n{\oe}ud propre}. Pour r\'esoudre (S), on utilise de nouveau les coordonn\'ees
polaires $(r,\theta)$. Il vient
$$
\left\{\eqalign{
\dfrac{dr}{dt}&=-\dfrac{x^2+y^2}{r}=-r\cr
\noalign{\vskip2pt}
\dfrac{d\theta}{dt}&=\dfrac{1}{x^2+y^2}\
\dfrac{2x^2+2y^2}{\ln\,(x^2+y^2)}=\dfrac{1}{\ln\,r}.\cr}
\right.
$$
La solution du probl\`eme de Cauchy est donn\'ee par
$$
\eqalign{
r&=r_0e^{-t}\quad{\rm avec}\quad r_0<1,\hfill\cr
d\theta&=\dfrac{dt}{\ln\,r_0-t},\quad \theta=\theta_0-\ln\,(1-t/\ln\,r_0)\hfill\cr
}
$$
pour une donn\'ee initiale $(r_0,\theta_0)$ en $t=0$. La solution
est d\'efinie sur $[\ln\,r_0,+\infty[$ et on a $\lim_{t\rightarrow
+\infty}r(t)=0$, $\lim_{t\rightarrow+\infty}\theta(t)=-\infty$. On a
ici encore une spirale convergeant vers 0 (c'est peu visible sur
le sch\'ema ci-dessous car $\theta$ tend vers $-\infty$ tr\`es
lentement). L'origine est donc un {\it foyer stable}. Comme
$\lim_{t\rightarrow \ln\,r_0+0}r(t)=1_-$ et
$\lim_{t\rightarrow\ln\,r_0+0}\theta(t)=+\infty$, la courbe s'enroule
en spiralant \`a l'int\'erieur du cercle $r=1$ quand
$t\rightarrow\ln\,r_0+0$.

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\section 3. Probl\`emes|

{\parindent1.4pc
\noindent{\bf 3.1.} On consid\`ere sur $\bR^2$ le champ de vecteurs
$$
\overrightarrow V\pmatrix{
x\cr y\cr
}=\pmatrix{
x^2-y^2\cr
2xy\cr
}.
$$

\item{(a)} D\'eterminer les points critiques.

\medskip
\item{(b)} En posant $z=x+iy$, calculer la solution correspondant \`a la donn\'ee
initiale $z_0$ au temps $t=0$.

\medskip
\item{(c)} En d\'eduire que les courbes int\'egrales sont les deux demi-axes $0x,0x'$
et les cercles passant par l'origine, centr\'es sur l'axe $y'0y$.

\medskip
\item{(d)} Montrer que les solutions telles que $z_0\in\bC\setminus[0,+\infty[$ sont
asymptotiquement stables. Qu'en est-il si $z_0\in[0,+\infty[$~?

\bigskip
\noindent{\bf 3.2.} On \'etudie dans $\bR^2$ le syst\`eme diff\'erentiel
$$
\frac{d\overrightarrow M}{dt}=\overrightarrow V(M)\leqno{\rm (S)}
$$
o\`u $\overrightarrow V$ d\'esigne le champ de vecteurs qui \`a tout point 
$M(x,y)\in\bR^2$ associe le vecteur
$$
\overrightarrow V(M)=(-x^2-y,-x+y^2).
$$
D\'eterminer les points critiques du champ de vecteurs $\overrightarrow V$.
Calculer les solutions $t\mapsto\widetilde M(t)=(\widetilde x(t),\widetilde y(t))$ du
syst\`eme diff\'erentiel obtenu en lin\'earisant $\overrightarrow V$ au voisinage de
chacun des points critiques. Faire un sch\'ema indiquant l'allure des solutions au
voisinage des points critiques. Ces points sont-ils stables~?

\bigskip
\noindent{\bf 3.3.} M\^emes questions pour le champ de vecteurs $\overrightarrow
V(x,y)=(-1+x^2+y^2,-x)$.

\bigskip
\noindent{\bf 3.4.} On consid\`ere le champ de vecteurs d\'efini par
$$
\overrightarrow V\pmatrix{x\cr y\cr}=\pmatrix{
-y+x\,\sin\,\Big(\dfrac{\pi}{x^2+y^2}\Big)\,\exp\,\Big(-\dfrac{1}{x^2+y^2}\Big)\cr
\noalign{\vskip2pt}
\hfill
x+y\,\sin\,\Big(\dfrac{\pi}{x^2+y^2}\Big)\,\exp\,\Big(-\dfrac{1}{x^2+y^2}\Big)\cr }
$$
pour $(x,y)\not=(0,0)$, et par $\overrightarrow V(0,0)=\overrightarrow 0$.

\medskip

\item{(a)} Montrer que le champ $\overrightarrow V$ est de classe $C^{\infty}$ sur
$\bR^2$~; on pourra commencer par montrer que la fonction
$$
t\mapsto\sin\,(\pi/t)\,\exp\,(-1/t),\quad t>0,
$$
\item{} se prolonge en une fonction de classe $C^{\infty}$ sur $[0,+\infty[$.

\medskip
\item{(b)} Montrer que le syst\`eme diff\'erentiel $dM/dt=\overrightarrow V(M)$ se
ram\`ene \`a une \'equation de la forme $dr/d\theta=f(r)$~; on ne cherchera pas \`a
r\'esoudre explicitement cette \'equation. En d\'eduire qu'il y a une infinit\'e de
cercles concentriques $(C_k)_{k\geq 1}$ de rayons $R_k$ d\'ecroissant vers 0, qui
sont des courbes int\'egrales du champ.

\medskip
\item{(c)} \'Etudier la convergence des int\'egrales
$\int^{R_k}_{R_{k+1}}dr/f(r)$ en chacune des bornes. Montrer que la
courbe int\'egrale issue d'un point de coordonn\'ees polaires
$(r_0,\theta_0)$ telles que $R_{k+1}<r_0<R_k$ est une spirale
admettant les cercles $r=R_k$ et $r=R_{k+1}$ comme asymptotes.
\'Etudier de m\^eme le comportement \`a l'infini des courbes
int\'egrales.
\par}

\bye
