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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 263 37 "Trajectoires de champs \+
de vecteurs II" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" 
{TEXT 264 28 "Felice Ronga - 19 avril 2001" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 
0 "" {TEXT 256 23 "II. Bifurcation de Hopf" }}{PARA 3 "" 0 "" {TEXT 
262 68 "Nous allons etudier l'allure des trajectoires du champs de vec
teurs " }}{PARA 0 "" 0 "" {TEXT -1 23 "    x'=-y+x(e-x^2-y^2);" }}
{PARA 0 "" 0 "" {TEXT -1 22 "    y'=x+y(e-x^2-y^2);" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "lorsque e varie de -0.1
 a 0.1 en passant par la valeur 0." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 107 "Tout d'abord, une procedure utile (deja \+
proposee dans la correction de l'exercice 2); les parametres sont :" }
}{PARA 0 "" 0 "" {TEXT -1 78 "eq1,eq2 : partie de droite de la premier
e, respectivement la deuxieme equation" }}{PARA 0 "" 0 "" {TEXT -1 63 
"x0,y0 : coordonnee x, respectivement y de la condition initiale" }}
{PARA 0 "" 0 "" {TEXT -1 49 "a,b : temps de depart et d'arrivee respec
tivement" }}{PARA 0 "" 0 "" {TEXT -1 14 "coul : couleur" }}{PARA 0 "" 
0 "" {TEXT -1 176 "npoints : nombre de points de la trajectoire qui se
ront calcules. Plus (b-a) est grand, plus il faudra calculer de points
 pour avoir une bonne approximation de la trajectoire. " }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "with(plots):with(plottools):#si ce \+
n'est deja fait" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 247 "palin:=
proc(eq1,eq2,x0,y0,a,b,coul,npoints)\nlocal e1,e2,ini;\ne1:=diff(x(t),
t)=eq1;\ne2:=diff(y(t),t)=eq2;\nini:=x(0)=x0,y(0)=y0;\nodeplot(dsolve(
\{e1,e2,ini\},\{x(t),y(t)\},numeric),[x(t),y(t)],a..b,scaling=constrai
ned,color=coul,numpoints=npoints);\nend:" }}}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT 257 13 "Exercice 3.  " }{TEXT -1 95 "Mont
rer que les valeurs propres de la partie lineaire de ce champs en (0,0
)  valent e+I et e-I." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 64 "champ:=-y(t)+x(t)*(e-x(t)^2-y(t)^2),x(t)+
y(t)*(e-x(t)^2-y(t)^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&champG6$
,&-%\"yG6#%\"tG!\"\"*&-%\"xGF)\"\"\",(%\"eGF/*$)F-\"\"#F/F+*$)F'F4F/F+
F/F/,&F-F/*&F'F/F0F/F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "f
1:=subs(e=0.1,champ[1]),subs(e=0.1,champ[2]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#f1G6$,&-%\"yG6#%\"tG!\"\"*&-%\"xGF)\"\"\",($F/F+F/*$
)F-\"\"#F/F+*$)F'F4F/F+F/F/,&F-F/*&F'F/F0F/F/" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 123 "p1:=palin(f1,0.2,0,0,10,blue,50):p2:=palin(f1,s
qrt(0.1),0,0,10,red,50):p3:=palin(f1,0.4,0,0,10,green,50):display(p1,p
2,p3);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6'-%'CUR
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2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve \+
3" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 223 "On voit que la trajectoire
 par (sqrt(e),0) est un cercle attracteur, car les trajectoires proche
s du cercle (en rouge), a l'interieur(en bleu) ou a l'exterieur (en ve
rt), sont des spirales qui s'enroulent autour du cercle. " }}{PARA 0 "
" 0 "" {TEXT -1 86 "En realite, les trajectoire exterieures au cercle \+
ressemblent a des etoiles filantes :" }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
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c:OGK,6.uF\\v7$$!31(4Uer#)G-$F0$!3_,bD=Q8*4\"F37$$!3)RFLeL!G#*HF0$!3gE
!p2\"H87HF37$$!3H!oym<#o^HF0$!3\"=OW<m6jm%F37$$!34bZ+Hf)4!HF0$!3n,B`c;
s#Q'F37$$!3UQ:T^V\\SGF0$!3??HhW(ec0)F37$$!3Qw.+f6aqFF0$!3#H\"*o=^a'z'*
F37$$!3')4B*RH!\\\"p#F0$!3RNZKk![\\7\"F07$$!3IL$o$*3NPg#F0$!3!45,-([,w
7F07$$!3APi5+Xp2DF0$!3y%p?E#[p?9F07$$!3&**>JjR6QS#F0$!3I_LP'QY&e:F0-Fd
en6&FfenFgenF+F+-%(SCALINGG6#%,CONSTRAINEDG-%+AXESLABELSG6%Q\"x6\"Q\"y
Fj]r-%%FONTG6#%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 1 1.000000 45.000000 
45.000000 0 0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
115 "Les trajectoires sont des spirales qui tendent rapidement vers l'
origine, comme prevu par le theoreme de Liapounov." }}{PARA 0 "" 0 "" 
{TEXT -1 182 "On voit donc que lorsque e varie de -0.1 a +0.1, le poin
t critique attracteur (l'origine) donne naissance, lorsque e=0, a un p
etit cercle attracteur. Ce type de phenomene est appele " }{TEXT 265 
19 "bifurcation de Hopf" }{TEXT -1 58 " (Eberhardt), du nom de celui q
ui les a etudie en premier." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT 266 18 "Premiere variation" }{TEXT -1 311 " sur
 ce theme : l'equation suivante correspond a un champs de vecteurs qui
 est aussi une deformation de (-y,x), mais c'est une deformation diffe
rente. Neanmoins, qualitativement cet exemple ressemble au precedent, \+
 comme vous etes invites a vous en rendre compte en etudiant les traje
ctoires pres de l'origine :" }}{PARA 0 "" 0 "" {TEXT -1 17 "  x'=-y+x(
e-x^2);" }}{PARA 0 "" 0 "" {TEXT -1 16 "  y'=x+y(e-y^2);" }}{PARA 0 "
" 0 "" {TEXT -1 107 "En particulier, pour e>0 on devrait avoir un cycl
e attracteur, qui ressemble a une patate de rayon sqrt(e)." }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 258 8 "Reponse \+
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 50 "champ:=-y(t)+x(t)*(e-x(t)^2),x(t)+y(t)*(e-y(t)^2);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%&champG6$,&-%\"yG6#%\"tG!\"\"*&-%\"xGF)\"
\"\",&%\"eGF/*$)F-\"\"#F/F+F/F/,&F-F/*&F'F/,&F1F/*$)F'F4F/F+F/F/" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "f1:=subs(e=0.4,champ[1]),sub
s(e=0.1,champ[2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f1G6$,&-%\"yG
6#%\"tG!\"\"*&-%\"xGF)\"\"\",&$\"\"%F+F/*$)F-\"\"#F/F+F/F/,&F-F/*&F'F/
,&$F/F+F/*$)F'F5F/F+F/F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 115 "Par \+
tatonnement a partir du point (sqrt(0.4),0), on voit que le point (sqr
t(0.4)-0.06,0) semble etre sur un cycle :" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 40 "palin(f1,sqrt(0.4)-0.05,0,0,10,red,100);" }}{PARA 
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{PARA 0 "" 0 "" {TEXT -1 59 "Voyons le comportement des trajectoires v
oisines du cycle :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 147 "p1:=
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0,10,red,100):p3:=palin(f1,sqrt(0.4),0,0,10,green,100):display(p1,p2,p
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FAULTG" 1 2 0 1 10 0 2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "Curve
 1" "Curve 2" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 104 "On voit donc qu
e cette famille de champs de vecteurs presente un comportement semblab
le a la precedente." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT 267 18 "Deuxieme variation" }{TEXT -1 5 " : l'" }
{TEXT 260 24 "equation de Van der Pol." }}{PARA 0 "" 0 "" {TEXT -1 91 
"Il s'agit de l'equation suivante, qui provient de l'etude d'un certai
n circuit electrique :" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 20 "  x''+e(x^2-1)x'+x=0" }}{PARA 0 "" 0 "" {TEXT -1 37 
"Elle se ramene au systeme d'ordre 1 :" }}{PARA 0 "" 0 "" {TEXT -1 7 "
  x'=y;" }}{PARA 0 "" 0 "" {TEXT -1 17 " y'=-x-e(x^2-1)y;" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "Si on prend e=0, o
n trouve le systeme lineaire " }}{PARA 0 "" 0 "" {TEXT -1 10 "   x'=-y
; " }}{PARA 0 "" 0 "" {TEXT -1 8 "   y'=x;" }}{PARA 0 "" 0 "" {TEXT 
-1 159 " dont les trajectoires sont des cercles centres a l'origine. D
e nouveau, si on donne a e des valeurs positives un cycle limite attra
cteur apparait. Montrez-le." }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 261 7 "R
eponse" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "champ:=y(t),-x(t)-
e*(x(t)^2-1)*y(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&champG6$-%\"y
G6#%\"tG,&-%\"xGF(!\"\"*(%\"eG\"\"\",&*$)F+\"\"#F0F0F0F-F0F&F0F-" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "f1:=subs(e=1,champ[1]),subs(
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G,&-%\"xGF(!\"\"*&,&*$)F+\"\"#\"\"\"F3F3F-F3F&F3F-" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 82 "p1:=palin(f1,0.1,0,0,20,blue,200):p2:=palin
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431e,ss>F`]m$\"3Zhl$**>J(=XF*7$$\"3!*)Qio30'z>F`]m$\"3]$fVR5'4**RF*7$$
\"32Qd*o&om&)>F`]m$\"3fy[x:px+NF*7$$\"3'G?YLFCu)>F`]m$\"3lJg<wi-ZLF*7$
$\"35:I(*4L5*)>F`]m$\"3)ROY47#Q&>$F*7$$\"33epsW]q!*>F`]m$\"3E5T:5@$e/$
F*7$$\"3oT_D\\0B#*>F`]m$\"3od&f)yIO)*GF*-Fg[y6&Fi[yFj[yF+F+-%(SCALINGG
6#%,CONSTRAINEDG-%+AXESLABELSG6%Q\"x6\"Q\"yFf^il-%%FONTG6#%(DEFAULTG" 
1 2 0 1 10 0 2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Cur
ve 2" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 5 "" 
0 "" {TEXT -1 8 "Exercice" }}{PARA 0 "" 0 "" {TEXT -1 289 "Calculer le
s valeurs propres de la partie lineaire du champs de vecteurs preceden
t. Montrer que leur partie reelle passe de valeurs negatives a des val
eurs positives lorsque e traverse la valeur 0. Montrer (experimentalem
ent) que pour e=-0.1 le champs possede un cycle repulseur par (2,0)." 
}}{SECT 1 {PARA 20 "" 0 "" {TEXT -1 7 "Reponse" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 48 "f2:=subs(e=-0.1,champ[1]),subs(e=-0.1,champ[2]);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f2G6$-%\"yG6#%\"tG,&-%\"xGF(!\"
\"*($\"\"\"F-F0,&*$)F+\"\"#F0F0F0F-F0F&F0F0" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 116 "p1:=palin(f2,1,0,0,10,blue,100):p2:=palin(f2,2,0,
0,10,red,100):p3:=palin(f2,2.1,0,0,10,green,100):display(p1,p2,p3);" }
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