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{SECT 0 {EXCHG {PARA 267 "" 0 "" {TEXT 267 36 "Trajectoires de champs \+
de vecteurs I" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 268 "" 0 "" 
{TEXT 268 28 "Felice Ronga - 19 avril 2001" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 256 15 "I. Introduction" }}
{PARA 256 "" 0 "" {TEXT 257 145 "On vous propose d'etudier experimenta
lement l'allure des trajectoires de champs de vecteurs et des phenomen
es de bifurcation, a l'aide de Maple. " }}{PARA 257 "" 0 "" {TEXT -1 
0 "" }}{PARA 258 "" 0 "" {TEXT 258 27 "D'abord un peu de theorie. " }}
{PARA 258 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT 259 
56 "Systemes lineaires a coefficients constants dans le plan" }}{PARA 
259 "" 0 "" {TEXT 263 75 "Considerons un systeme de 2  equations linea
ires a coefficients constants :" }}{PARA 260 "" 0 "" {TEXT -1 0 "" }}
{PARA 261 "" 0 "" {TEXT 260 16 "                " }{TEXT 262 22 " x'(t
)=a x(t) + b y(t)" }}{PARA 0 "" 0 "" {TEXT -1 33 "            y'(t)=c \+
x(t) + d y(t)" }}{PARA 262 "" 0 "" {TEXT -1 0 "" }}{PARA 263 "" 0 "" 
{TEXT 261 97 "L'allure des trajectoires est determinee par les valeurs
 propres de la matrice des coefficients: " }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "A:=array(1..2,1..2,[[
a,b],[c,d]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7$7
$%\"aG%\"bG7$%\"cG%\"dG" }}}{PARA 263 "" 0 "" {TEXT -1 0 "" }}{PARA 0 
"" 0 "" {TEXT -1 33 "Voici la liste des cas possible :" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "1. A est diagonalisa
ble sur les reels." }}{PARA 0 "" 0 "" {TEXT -1 68 "   1a. les deux val
eurs propres sont positives ou les deux negatives" }}{PARA 0 "" 0 "" 
{TEXT -1 73 "   1b. une valeur propre est positive (resp. negative), l
'autre est nulle" }}{PARA 0 "" 0 "" {TEXT -1 59 "   1c. une valeur pro
pre est positive, l'autre est negative" }}{PARA 0 "" 0 "" {TEXT -1 
392 "2. A possede une valeur propre lambda non reelle, de la forme a+I
b; alors le conjugue a-Ib de lambda est distinct de lambda (i.e. b est
 different de 0), et est aussi valeur propre. Donc A est diagonalisabl
e sur les complexes. Soit v un vecteur propre complexe de A correspond
ant a la valeur propre a-Ib; en prenant pour base de R^2 la partie rel
le et imaginaire de v, A se met sous la forme :" }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 35 "A:=array(1..2,1..2,[[a,-b],[b,a]]);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7$7$%\"aG,$%\"bG!\"\"7$F,F
*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "Il \+
y a deux sous-cas :" }}{PARA 0 "" 0 "" {TEXT -1 44 "    2a. lambda est
 purement imaginaire (a=0)" }}{PARA 0 "" 0 "" {TEXT -1 90 "    2b. la \+
partie reelle a de lambda est strictement positive (resp. strictement \+
negative)" }}{PARA 0 "" 0 "" {TEXT -1 147 "3. A n'est pas diagonalisab
le. Alors elle possede necessairement une seule valeur propre, qui doi
t etre reelle. Elle peut se mettre sous la forme :" }}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 34 "A:=array(1..2,1..2,[[1,a],[0,1]]);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7$7$\"\"\"%\"aG7$\"\"!F*" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 
"avec a non nul. " }}}{SECT 1 {PARA 264 "" 0 "" {TEXT -1 23 "Allure de
s trajectoires" }}{PARA 0 "" 0 "" {TEXT -1 140 "On peut se faire une p
remi\217re id\216e de l'allure des trajectoires en dessinant le champ \+
de vecteurs; la commande \"fieldplot\" fait la besogne :" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}{PARA 7 "" 1 "" 
{TEXT -1 50 "Warning, the name changecoords has been redefined\n" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "fieldplot([-2*x,-3*y],x=-1..
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G-%*AXESSTYLEG6#%%NONEG" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 
45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 67 "ici pa
r contre on soupsonne que les trajectoires sont des ellipses." }}
{PARA 0 "" 0 "" {TEXT -1 122 "On retrouve ces champ de vecteurs ci-des
sous, ou on va  calculer explicitement les trajectoires et confirmer n
os soupsons." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 120 "Maple met a dispo
sition la procedure \"dsolve\" pour calculer des solutions d'equations
 differentielles. Voici son usage :" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 24 "on donne les equations :" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "e1:=diff(x(t),t)=-2*x(t);e2:=diff(y
(t),t)=-3*y(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#e1G/-%%diffG6$-%
\"xG6#%\"tGF,,$F)!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#e2G/-%%dif
fG6$-%\"yG6#%\"tGF,,$F)!\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "on
 donne les conditions initiales :" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 20 "ini1:=x(0)=1,y(0)=1;" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>%%ini1G6$/-%\"xG6#\"\"!\"\"\"/-%\"yGF)F+" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 114 "on demande de resoudre; les solutions seront sous form
e d'une liste, qu'on assignera a la nouvelle variable \"s1\" :" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "s1:=dsolve(\{e1,e2,ini1\},\{
x(t),y(t)\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#s1G<$/-%\"xG6#%\"t
G-%$expG6#,$F*!\"#/-%\"yGF)-F,6#,$F*!\"$" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 21 "ini2:=x(0)=1,y(0)=-1;" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%%ini2G6$/-%\"xG6#\"\"!\"\"\"/-%\"yGF)!\"\"" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 37 "s2:=dsolve(\{e1,e2,ini2\},\{x(t),y(t)\});" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#s2G<$/-%\"xG6#%\"tG-%$expG6#,$F*!
\"#/-%\"yGF),$-F,6#,$F*!\"$!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
107 "Maple a donc pu nous calculer les solutions explicitement. On va \+
calculer une solution dans chaque cadran :" }}{PARA 0 "" 0 "" {TEXT 
-1 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 117 "ini3:=x(0)=-1,y
(0)=1;ini4:=x(0)=-1,y(0)=-1;s3:=dsolve(\{e1,e2,ini3\},\{x(t),y(t)\});s
4:=dsolve(\{e1,e2,ini4\},\{x(t),y(t)\});" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%%ini3G6$/-%\"xG6#\"\"!!\"\"/-%\"yGF)\"\"\"" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%%ini4G6$/-%\"xG6#\"\"!!\"\"/-%\"yGF)F+" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>%#s3G<$/-%\"xG6#%\"tG,$-%$expG6#,$F*!\"#!\"\"/
-%\"yGF)-F-6#,$F*!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#s4G<$/-%\"
xG6#%\"tG,$-%$expG6#,$F*!\"#!\"\"/-%\"yGF),$-F-6#,$F*!\"$F1" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT 266 270 "ATTENTION : Maple renverse parfo
is  l'ordre de x et y dans le resultat de l'appel a \"dsolve\", ce qui
 a pour consequence de fausser les graphiques. Ce phenomene est aleato
ire et n'est pas reproductible sur demande. Il faut en tenir compter l
orsqu'on trace les graphiques." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 193 "On va calculer les graphiques des 4 traj
ectoires pour t=-1 a t=0. Pour cela, on applique la commande \"rhs\" (
=\"right hand side\") a s1,..,s4  pour extraire les parametrisations d
es trajectoires :" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 22 "rhs(s1[1]);rhs(s1[2]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%$expG6#,$%\"tG!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#-%$expG6#,$%\"tG!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 294 "w
ith(plots):p1:=plot([rhs(s1[1]),rhs(s1[2]),t=0..1],color=blue):p2:=plo
t([rhs(s2[1]),rhs(s2[2]),t=0..1],color=red):p3:=plot([rhs(s3[1]),rhs(s
3[2]),t=0..1],color=green):p4:=plot([rhs(s4[2]),rhs(s4[1]),t=0..1],col
or=yellow):ini:=pointplot([[1,1],[-1,1],[-1,-1],[1,-1]],symbol=circle,
color=black):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 308 "Il est preferab
le de terminer ces commandes par \":\" plutot que \";\", car la repons
e est un texte interminable decrivant le graphique (essayez). Maintena
nt on peut afficher les graphiques calcules; les conditions initiales \+
sont representes par de petits cercles et permettent d'identifier le s
ens de parcours :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "displa
y(p1,p2,p3,p4,ini,title=`2 valeurs propres reelles negatives (noeud)`,
scaling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 307 307 307 
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F[il7$Fh_lF^il7$F[`lFail7$F^`lFdil7$Fa`lFgil7$Fd`lFjil7$Fg`lF]jl7$Fj`l
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CIRCLEG-%+AXESLABELSG6%Q!6\"F_bm-%%FONTG6#%(DEFAULTG-%&TITLEG6#%L2~val
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IEWG6$FdbmFdbm" 1 2 0 1 10 0 2 9 1 4 1 1.000000 45.000000 45.000000 0 
0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" }}}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 251 "Si t augmente, on s'approche de l'origine. On \+
dit qu'un tel point singulier est un attracteur; on parle de repulseur
 lorsque, au contraire, toutes les trajectoires s'eloignent de l'origi
ne; c'est le cas si les 2 valeurs propres sont reelles positives." }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
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0)=-1/10;ini4:=x(0)=1/10,y(0)=-1/10;" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%%ini1G6$/-%\"xG6#\"\"!#\"\"\"\"#5/-%\"yGF)F+" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%%ini2G6$/-%\"xG6#\"\"!#!\"\"\"#5/-%\"yGF)#\"\"\"F-" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%ini3G6$/-%\"xG6#\"\"!#!\"\"\"#5/-%
\"yGF)F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%ini4G6$/-%\"xG6#\"\"!#
\"\"\"\"#5/-%\"yGF)#!\"\"F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
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=black,symbol=circle):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 148 "
s1:=dsolve(\{e1,e2,ini1\},\{x(t),y(t)\});s2:=dsolve(\{e1,e2,ini2\},\{x
(t),y(t)\});s3:=dsolve(\{e1,e2,ini3\},\{x(t),y(t)\});s4:=dsolve(\{e1,e
2,ini4\},\{x(t),y(t)\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#s1G<$/-
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1]),rhs(s3[2]),t=0..1],color=green):p4:=plot([rhs(s4[2]),rhs(s4[1]),t=
0..1],color=yellow):display(p1,p2,p3,p4,ini,title=`2 valeurs propres r
eelles de signe oppose (point selle)`,scaling=constrained);" }}{PARA 
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l7$$\"3(\\')*R0>#[d)Fgcl$!328FG1Z_\\FF\\cl7$$\"3T%\\U-KN(z)*Fgcl$\"3;A
'fd5Ydk*Fgcl7$$!3**yDmO)p4Y#Fgcl$\"36nNGFy$zv\"Fgcl7$$!3f!>LAUR:N&Fddl
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4K<kDJ?F^gl7$$!35NHlc$\\(Q5F^gl$!3G*3lp)o,[7F^gl7$$\"3%fb&p\"z<_g$Fahl
$!3yooeaWzy`F[hl7$$\"3hMR'pE<-I$F[hl$\"3;Qq\\;Rc08Fahl7$$!3jO\")Q(**G'
=UF[hl$!3oh0!yi;%*3#F[hl7$$\"3OUNV-j%='RFdil$!3h^o\"ya'>25F[hl7$$\"3EZ
cA`(zh*HFdil$\"3co(zp!)G&QPFdil7$$!3&z\\\"\\yZU&3\"Fdil$\"3/(*>n&y1,%=
Fgil7$$!39hr.l]5I**Fejl$!3%*\\'*ze1bLRFgil7$$\"3\"[*=a>W\")p8Fgil$\"3j
gL\">2@\\F&Fejl7$$!3s))\\_zi!p6\"Fejl$\"3Gj&\\jS!eXJFejl7$$!35J2r!*y\"
HB\"Fejl$!3n!32x$\\,Z6Fejl7$$\"32.&)Qoe\"=;$Fb[m$!3_:o!fTR9J\"Fb[m7$$
\"3KU:LLY9[oF]\\m$\"3[i;7$R$**=9Fb[m7$$!3#f6\"\\^]v%H%F]\\m$!3![`h*p7$
*[8F]\\m7$$\"3vI**oS3g.=F[]m$!3y)e\")*pNB[6F]\\m7$$\"3d[gxNL;()QF[]m$
\"3e!eS(*o\\6W$F[]m7$$!3Bgd*3ki\"z$*F^]m$\"3`&*fa<7$Hd%F^]m7$$!3sWD,6M
\\fGF^]m$!3WYj$o>qYh%F^]m7$$\"3l7?p\"4poM\"F^]m$\"3?J#y%4C\"p-$F\\^m7$
$\"3J0'))GN\\uE#Fg^m$\"3rR@Kp,.pUF\\^m7$$!3\"4^Bo(*[*49F\\^m$!3CL$o(\\
lc!z*Fg^m7$$\"3Ux%o9LbIJ#Fg^m$!33_@cP?&Hs#Fg^m7$$\"3a9qpU7?n$*Fd_m$\"3
9,O5pT3;9Fg^m-Ff_m6&Fh_mFi_mF(F(-%'POINTSG6&F'F_`m-Ff_m6&Fh_mF)F)F)-%'
SYMBOLG6#%'CIRCLEG-%+AXESLABELSG6%Q!6\"Fhdo-%%FONTG6#%(DEFAULTG-%&TITL
EG6$%eo2~valeurs~propres~complexes~conjuguees~de~partie~reelle~negativ
e~(spirale)G-F[eo6%%*HELVETICAG%%BOLDG\"#5-%%VIEWG6$F]eoF]eo" 1 2 0 1 
10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "C
urve 3" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 129 "et on obtient des spi
rales, qui tendent vers l'origine lorsque t augmente. Dans ce dernier \+
cas, la matrice des coefficients est :" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 40 "A:=array(1..2,1..2,[[-1/10,-1],[1,-1]]);" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7$7$#!\"\"\"#5F+7$\"\"\"F+" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "Verifions que l'on est dans le c
as 2 b:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }
}{PARA 7 "" 1 "" {TEXT -1 80 "Warning, the protected names norm and tr
ace have been redefined and unprotected\n" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 15 "eigenvalues(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$
,&#!#6\"#?\"\"\"*&^##F'F&F'-%%sqrtG6#\"$>$F'F',&F$F'*&^##!\"\"F&F'F+F'
F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 91 "Si les deux valeurs propres
 sont purement imaginaires, les trajectoires sont des ellipses :" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 58 "e1:=diff(x(t),t)=x(t)-5*y(t);e2:=diff(y(t),t)=2*x(t)-y(t);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#e1G/-%%diffG6$-%\"xG6#%\"tGF,,&F)\"
\"\"*&\"\"&F.-%\"yGF+F.!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#e2G
/-%%diffG6$-%\"yG6#%\"tGF,,&-%\"xGF+\"\"#F)!\"\"" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 94 "s1:=dsolve(\{e1,e2,x(0)=2/3,y(0)=0\},\{x(t),y
(t)\});s2:=dsolve(\{e1,e2,x(0)=1,y(0)=0\},\{x(t),y(t)\});" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>%#s1G<$/-%\"yG6#%\"tG,$-%$sinG6#,$F*\"\"$#\"\"
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{MPLTEXT 1 0 206 "p1:=plot([rhs(s1[2]),rhs(s1[1]),t=0..2*Pi/3],color=b
lue):p2:=plot([rhs(s2[2]),rhs(s2[1]),t=0..2*Pi/3],color=red):display(p
1,p2,title=`2 valeurs propres imaginaires conjuguees (centre)`,scaling
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ires~conjuguees~(centre)G-%(SCALINGG6#%,CONSTRAINEDG-%%VIEWG6$Fc[mFc[m
" 1 2 0 1 10 0 2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "C
urve 2" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "Ici on dit que le poin
t singulier est un centre." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{SECT 1 {PARA 265 "" 0 "" {TEXT -1 10 "Exercice 1" }}{PARA 0 "" 0 "" 
{TEXT -1 267 "Tracer les trajectoires des systemes suivants; determine
r la nature du point critique (0,0). Controlez vos reponses en calcula
nt les valeurs propres de la matrice du systeme. De preference, ecrive
z une procedure pour dessiner des trajectoires a partir des  equations
." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 13 "a) \+
  x'=-x-2y" }}{PARA 0 "" 0 "" {TEXT -1 14 "      y'=4x.5y" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "b)  x'=-3x+4y" }}
{PARA 0 "" 0 "" {TEXT -1 14 "     y'=-2x+3y" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "c)  x'=5x+2y" }}{PARA 0 "" 0 "
" {TEXT -1 14 "    y'=-17x-5y" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 264 
21 "Sytemes non lineaires" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 202 "Lors
qu'on a une equation non lineaire, Maple n'est pas capable de trouver \+
explicitement des solution. Il faut lui demander  de calculer les solu
tions numeriquement par l'option \"numeric\" dans \"dsolve\" :" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 110 "nonlin:=dsolve(\{diff(x(t),t)=-x(t)+y(t)^2,diff(y(t),t)=-2*y(t)
+x(t)^2,x(0)=0.1,y(0)=0.1\},\{x(t),y(t)\},numeric);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%'nonlinGf*6#%(rkf45_xG6'%\"iG%(rkf45_sG%)outpointG%
#r1G%#r2G6#%aoCopyright~(c)~1993~by~the~University~of~Waterloo.~All~ri
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G!\")\"\"&F\\oFNFG\"\"($FG!\"*\"\")\"&++$\"\"*\"%+5\"#5FR\"#6FR\"#7FR
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numeric_solnall_rkf45G6,%&loc_FG-%$varG6#FD-Fes6#F_q-Fes6#Fgq-Fes6#%'l
oc_F1G-Fes6#%'loc_F2G-Fes6#%'loc_F3G-Fes6#%'loc_F4G-Fes6#%'loc_F5G-Fes
6#%)loc_workG@$/Fjr%*lasterrorGC%>8'-%+searchtextG6$.Ffr-%(convertG6$-
%#opG6$FG7#Fjr%%nameG>8(-Fdu6$.%)hardwareGFgu@%50FbuFR0F`vFR-Fas6,FcsF
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6#,&8$FGFGFG&F_q6#Fiw/FiwFdqF06%FDF_qFgqF0" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 159 "La reponse est une procedure capable de calculer des val
eurs numeriques de la solution ayant les conditions initiales donnees,
 pour des valeurs de t choisies :" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 10 "nonlin(0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%/%\"t
G\"\"!/-%\"xG6#F%$\"\"\"!\"\"/-%\"yGF*F+" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 68 "Pour tracer la solution, on utiliser la procedure ad hoc \+
\"odeplot\" :" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 44 "odeplot(nonlin,[x(t),y(t)],0..1,color=blue);" }}
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" 1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 130 "En fait, l'equation qu'on a pri
s est une perturbation de l'equation lineaire x'(t)=-x(t),'(t)=-2y(t) \+
par des termes quadratiques. " }}{PARA 0 "" 0 "" {TEXT -1 385 "On appe
lle point critique ou singulier d'un champs de vecteurs (f(x,y),g(x,y)
) un points (a,b) ou' il est nul : f(a,b)=g(a,b)=0; dans l'exemple pre
cedent, (0,0) est un point critique du champs (-x+y^2,-2y+x^2). La par
tie lineaire du champs (f(x,y),g(x,y)) en un point critique est le ter
me d'ordre 1 du developpement de Taylor, c'est-a-dire l'application li
neaire dont la matrice est :" }}{PARA 0 "" 0 "" {TEXT -1 13 "         \+
    " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 63 "array(1..2,1..2,[[Diff(f,x),Diff(f,y)],[Diff(g,x),Dif
f(g,y)]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7$7$-%%DiffG
6$%\"fG%\"xG-F)6$F+%\"yG7$-F)6$%\"gGF,-F)6$F3F/" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 42 "les derivees etant prises au point (a,b). " }}{PARA 
0 "" 0 "" {TEXT -1 3 "Le " }{TEXT 265 21 "theoreme de Liapunov " }
{TEXT -1 320 "affirme que lorsque les parties reelles des valeurs prop
res de la partie lineaire d'un champ de vecteur en un point critique s
ont non nulles, les trajectoires du champs de vecteur ressemblent a ce
lles de sa partie lineaire. C'est ce que montre l'exemple precedent, q
ue l'on reprend maintenant avec plus de trajectoires :" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 162 "e1:=d
iff(x(t),t)=-x(t)+y(t)^2;e2:=diff(y(t),t)=-2*y(t)+x(t)^2;ini1:=x(0)=0.
1,y(0)=0.1;ini2:=x(0)=0.1,y(0)=-0.1;ini3:=x(0)=-0.1,y(0)=-0.1;ini4:=x(
0)=-0.1,y(0)=0.1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#e1G/-%%diffG6$
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{XPPMATH 20 "6#>%#e2G/-%%diffG6$-%\"yG6#%\"tGF,,&F)!\"#*$)-%\"xGF+\"\"
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20 "6#>%%ini3G6$/-%\"xG6#\"\"!$!\"\"F,/-%\"yGF)F+" }}{PARA 11 "" 1 "" 
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{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 409 "p1:=odeplot(dsolve(\{e1,e2,
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(dsolve(\{e1,e2,ini2\},\{x(t),y(t)\},numeric),[x(t),y(t)],0..10,color=
blue):p3:=odeplot(dsolve(\{e1,e2,ini3\},\{x(t),y(t)\},numeric),[x(t),y
(t)],0..10,color=green):p4:=odeplot(dsolve(\{e1,e2,ini4\},\{x(t),y(t)
\},numeric),[x(t),y(t)],0..10,color=brown):ini:=pointplot([[0.1,0.1],[
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4" "Curve 5" }}}}{SECT 1 {PARA 266 "" 0 "" {TEXT -1 11 "Exercice 2." }
}{PARA 0 "" 0 "" {TEXT -1 209 "Calculer les points critiques des syste
mes suivants et determiner leur nature, en tracant des trajectoires et
 en calculant les valeurs propres des parties lineaires des champs de \+
vecteurs aux points critiques." }}{PARA 0 "" 0 "" {TEXT -1 13 "a) x'=-
x+y^2;" }}{PARA 0 "" 0 "" {TEXT -1 14 "    y'=-x+y^2;" }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "b) x'=x^2-y^2-x;" }}
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