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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 256 20 "ATTRACTEURS ETRANGES" }
}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 29 "with(plots): with(plottools):" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT 259 21 "L'attracteur de H\351n
on" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
257 13 "Visualisation" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 473 "h
\351non:=proc(x0,y0,n,a,b) # n repr\351sente le nombre de points de l'
orbite de point initial [x0;y0]; a et b sont des param\350tres \nlocal
 point,s,i,f,g;\nf:=unapply(y+1-a*x^2,x,y); g:=unapply(b*x,x,y); # fon
ctions coordonn\351es de la fonction d'it\351ration\npoint:=[x0,y0]; s
:=NULL; # initialisations\nfor i to n do s:=s,point; point:=[f(point[1
],point[2]),g(point[1],point[2])] od; # construction de la s\351quence
 des points de l'orbite\nplot([s],style=POINT,symbol=POINT, color=RED)
\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "h\351non(0,0,2000
,1.4,0.3);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT 260 27 "Etude d'une zone de capture" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 122 "display(polygonplot([[-1.33,0.42],[1.32,0.133],[1.
245,-0.14],[-1.06,-0.5]],style=LINE,color=RED),h\351non(0,0,1000,1.4,0
.3));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "h\351non(1.292,0,2
0,1.4,0.3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1104 "bassin_h
\351non:=proc(point) local phi,psi,x,y,i,z,appartenance,r\351ponse;\np
hi:=(x,y)->y+1-1.4*x^2; psi:=x->.3*x;\nx:=op(1,point); y:=op(2,point);
 r\351ponse:=false; # initialisations\nappartenance:=(x,y)->(x+1.33)*(
0.133-0.42)-(y-.42)*(1.32+1.33)>=0 and (x-1.32)*(-.14-.133)-(y-.133)*(
1.245-1.32)>=0 and (x-1.245)*(-.5+.14)*(y+.14)*(-1.06-1.245)<=0 and (x
+1.06)*(0.42+0.5)-(y+0.5)*(-1.33+1.06)>=0;\n# le point est-il dans la \+
zone de capture?\nfor i to 200 do \nif traperror(appartenance(x,y))=la
sterror then r\351ponse:=false; break\nelse if appartenance(x,y)=true \+
then r\351ponse:=true; break\n      else z:=x; x:=traperror(phi(x,y));
  if x=lasterror then r\351ponse:=false; break\n                      \+
                                else  y:=traperror(psi(z)); if y=laste
rror then r\351ponse:=false; break\n                                  \+
                                                         fi\n         \+
                                             fi\n      fi\nfi\nod;\n# \+
tous ces traperror sont destin\351s \340 \351viter un message d'erreur
 pour d\351passement dans la valeur absolue des donn\351es calcul\351e
s\nr\351ponse # bool\351en final\nend:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 24 "bassin_h\351non([1.292,0]);" }}}{EXCHG {PARA 0 "> " 
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{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "bassin_h\351non([-2,6]);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "bassin_h\351non([1.2,0.1]);
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7]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "bassin_h\351non([1.
2,1]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "bassin_h\351non([
1.2,1.3]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "bassin_h\351n
on([1.2,2]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "bassin_h
\351non([1.2,-.1]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "bass
in_h\351non([1.2,-0.7]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT 261 51 "Agrandissement de certaines parties de l
'attracteur" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 672 "zoom_h\351n
on:=proc(xmin,xmax,ymin,ymax) # les arguments repr\351sentent les limi
tes du trac\351\nlocal f,g,point,s,compteur;\nf:=unapply(y+1-1.4*x^2,x
,y); g:=unapply(.3*x,x,y); # fonctions coordonn\351es de la fonction d
'it\351ration\npoint:=[0,0]; s:=NULL; compteur:=0; # initialisations\n
while compteur<=200 do point:=[f(point[1],point[2]),g(point[1],point[2
])];\n                                if point[1]>=xmin and point[1]<=
xmax and point[2]>=ymin and point[2]<=ymax\n                          \+
      then s:=s,point; compteur:= compteur+1\n                        \+
        fi\n                                od;\nplot([s],x=xmin..xmax
,y=ymin..ymax,style=POINT,symbol=POINT, color=RED)\nend:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "zoom_h\351non(0.9,1,0.1,0.15);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "zoom_h\351non(0.94,0.96,0.13
,0.14);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT 262 48 "Etude de la sensibilit\351 aux conditions initiales" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 585 "sensibilit\351_h\351non:=pr
oc(point1,point2,n) # n repr\351sente le nombre de points calcul\351s
\nlocal point,punkt,punto,s,i,f,g;\nf:=unapply(y+1-1.4*x^2,x,y); g:=un
apply(0.3*x,x,y); # fonctions coordonn\351es de la fonction d'it\351ra
tion\npunkt:=point1; punto:=point2; point:=[point1[1],point2[1]]; s:=N
ULL; # initialisations\nfor i to n do\n             s:=s,point;\n     \+
        punkt:=[f(punkt[1],punkt[2]),g(punkt[1],punkt[2])];\n         \+
    punto:=[f(punto[1],punto[2]),g(punto[1],punto[2])];\n             \+
point:=[punkt[1],punto[1]]\n             od; \nplot([s],style=POINT,sy
mbol=POINT, color=RED)\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 54 "sensibilit\351_h\351non([-1,0.1],[-0.999999,0.100001],3000);" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "sensibilit\351_h\351non([0,
0],[0.00001,0.000001],3000);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT 263 68 "Essai d'autres param\350tres, construc
tion d'un diagramme de Feigenbaum" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 24 "h\351non(0,0,1000,0.3,0.3);" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 24 "h\351non(0,0,1000,0.3,0.4);" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 22 "h\351non(0,0,1000,1,0.3);" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 1219 "it\351rateur_h\351non:=proc(d\351but,fi
n,pas)\nlocal a,s,f,g,point,i,k,u,v;\na:=d\351but; s:=\{\}; \nwhile a<
=fin do\n                  f:=unapply(y+1-a*x^2,x,y); g:=unapply(0.3*x
,x,y);\n                  for i from 0 to 20\n                        \+
  do\n                          point:=[1,-1+i/10];                # p
ourvu que ce point soit dans un bassin d'attraction!\n                \+
          for k to 50 do u:=traperror(f(point[1],point[2]));\n        \+
                                      if u=lasterror then break\n     \+
                                         else v:=traperror(g(point[1],
point[2]));\n                                                     if v
=lasterror then break fi\n                                            \+
  fi;\n                                              point:=[u,v]   \n
                                          od\n                        \+
  od;\n                  for k to 100 do point:=[f(point[1],point[2]),
g(point[1],point[2])];\n                                        s:=s u
nion \{[a,evalf(point[1],4)]\}     # 4 d\351cimales pour ne pas surcha
rger la structure s\n                                   od;\n         \+
         a:=a+pas\n                  od;\nplot([op(s)],'a'=d\351but..f
in,style=POINT,symbol=POINT)\nend:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 29 "it\351rateur_h\351non(0,1.42,0.01);" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 264 83 "Confirmation d
e la pr\351sence simultan\351e de deux attracteurs pour certains param
\350tres" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1158 "h\351non1:=pr
oc(x0,y0,n,a,b) # n repr\351sente le nombre de points de l'orbite de p
oint initial [x0;y0]; a et b sont des param\350tres \nlocal point,s,i,
f,g;\nf:=unapply(y+1-a*x^2,x,y); g:=unapply(b*x,x,y); # fonctions coor
donn\351es de la fonction d'it\351ration\npoint:=[x0,y0]; s:=NULL; # i
nitialisations\nfor i to 50 do point:=[f(point[1],point[2]),g(point[1]
,point[2])] od; # \351limination de premiers it\351r\351s\nfor i to n \+
do s:=s,point; point:=[f(point[1],point[2]),g(point[1],point[2])] od; \+
# construction de la s\351quence des points de l'orbite\nplot([s],styl
e=POINT,symbol=POINT, color=RED)\nend:\n\nh\351non2:=proc(x0,y0,n,a,b)
 # n repr\351sente le nombre de points de l'orbite de point initial [x
0;y0]; a et b sont des param\350tres \nlocal point,s,i,f,g;\nf:=unappl
y(y+1-a*x^2,x,y); g:=unapply(b*x,x,y); # fonctions coordonn\351es de l
a fonction d'it\351ration\npoint:=[x0,y0]; s:=NULL; # initialisations
\nfor i to 50 do point:=[f(point[1],point[2]),g(point[1],point[2])] od
;  # \351limination de premiers it\351r\351s\nfor i to n do s:=s,point
; point:=[f(point[1],point[2]),g(point[1],point[2])] od; # constructio
n de la s\351quence des points de l'orbite\nplot([s],style=POINT,symbo
l=POINT, color=BLUE)\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
74 "display(h\351non1(.5,0,1000,1.08,.3),h\351non2(-0.1111,-0.105555,1
000,1.08,.3));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "
" 0 "" {TEXT 258 23 "L'attracteur de R\366ssler" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(DEtool
s):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
265 61 "Diverses visualisations de l'attracteur de R\366ssler pour c=5
.7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 305 "# a)L'attracteur de \+
R\366ssler pour c=5.7, approximation par une solution du syst\350me di
ff\351rentiel de R\366ssler\nDEplot3d(\{D(x)(t)=-(y(t)+z(t)),D(y)(t)=x
(t)+0.2*y(t),D(z)(t)=0.2+x(t)*z(t)-5.7*z(t)\},[x(t),y(t),z(t)],t=0..10
0,[[x(0)=-1,y(0)=0,z(0)=0]],stepsize=0.05,axes=FRAME,linecolor=RED,ori
entation=[-130,60]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 246 "# \+
b)La m\352me chose, mais entre t=100 et t=200\nDEplot3d(\{D(x)(t)=-(y(
t)+z(t)),D(y)(t)=x(t)+0.2*y(t),D(z)(t)=0.2+x(t)*z(t)-5.7*z(t)\},[x(t),
y(t),z(t)],t=100..200,[[x(0)=-1,y(0)=0,z(0)=0]],stepsize=0.05,axes=FRA
ME,linecolor=RED,orientation=[-130,60]);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 246 "# c)La m\352me chose, mais entre t=100 et t=300\nDEp
lot3d(\{D(x)(t)=-(y(t)+z(t)),D(y)(t)=x(t)+0.2*y(t),D(z)(t)=0.2+x(t)*z(
t)-5.7*z(t)\},[x(t),y(t),z(t)],t=100..300,[[x(0)=-1,y(0)=0,z(0)=0]],st
epsize=0.05,axes=FRAME,linecolor=RED,orientation=[-130,60]);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 306 "# d)Le m\352me attracteur, \+
la solution \351tant trac\351e point par point ; \nDEplot3d(\{D(x)(t)=
-(y(t)+z(t)),D(y)(t)=x(t)+0.2*y(t),D(z)(t)=0.2+x(t)*z(t)-5.7*z(t)\},[x
(t),y(t),z(t)],t=100..300,[[x(0)=-1,y(0)=0,z(0)=0]],stepsize=0.05,axes
=FRAME,linecolor=RED,style=POINT,symbol=POINT,thickness=0,orientation=
[-130,60]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 307 "# e)Encore \+
cet attracteur, la couleur de la courbe variant avec le param\350tre t
 pour cr\351er une animation\nDEplot3d(\{D(x)(t)=-(y(t)+z(t)),D(y)(t)=
x(t)+0.2*y(t),D(z)(t)=0.2+x(t)*z(t)-5.7*z(t)\},[x(t),y(t),z(t)],t=100.
.300,[[x(0)=-1,y(0)=0,z(0)=0]],stepsize=0.05,axes=FRAME,linecolor=t/30
0,orientation=[-130,60]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
282 "# f)Toujours l'attracteur de R\366ssler, les conditions initiales
 \351tant modifi\351es\nDEplot3d(\{D(x)(t)=-(y(t)+z(t)),D(y)(t)=x(t)+0
.2*y(t),D(z)(t)=0.2+x(t)*z(t)-5.7*z(t)\},[x(t),y(t),z(t)],t=100..300,[
[x(0)=0,y(0)=-8,z(0)=8]],stepsize=0.05,axes=FRAME,linecolor=t/300,orie
ntation=[-130,60]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT 266 48 "Etude de la sensibilit\351 aux conditions init
iales" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 740 "sensibilit\351_ro
ssler:=proc(point1,point2,n)\n# point1 et point2 correspondent \340 de
ux vecteurs initiaux, n au nombre de points [x1(t),x2(t)] construits\n
local solution,s,i,solution1,solution2;\nsolution:=(x0,y0,z0)->dsolve(
\{D(x)(t)=-(y(t)+z(t)),D(y)(t)=x(t)+0.2*y(t),D(z)(t)=0.2+x(t)*z(t)-5.7
*z(t),y(0)=y0,x(0)=x0,z(0)=z0\},\{x(t),y(t),z(t)\},type=numeric,method
=rkf45,maxfun=60000,relerr=Float(1,4-Digits));\ns:=NULL; solution1:=so
lution(op(point1)); solution2:=solution(op(point2));\nfor i from 100 t
o n+100 do s:=s,[subs(solution1(i),x(t)),subs(solution2(i),x(t))] od;
\n# (on commence \340 100 pour \351liminer les 100 premiers termes, qu
i correspondront certainement \340 des valeurs encore tr\350s voisines
)\nplot([s],style=POINT,symbol=CROSS, color=RED)\nend:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "sensibilit\351_rossler([-1,0,0],[-1
.001,0.001,-0.001],500);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT 267 74 "L'attracteur de R\366ssler pour quelques
 valeurs bien choisies du param\350tre c" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 282 "# c=2.5   l'attracteur est une solution p\351riodiqu
e, un seul minimum de l'abscisse\nDEplot3d(\{D(x)(t)=-(y(t)+z(t)),D(y)
(t)=x(t)+0.2*y(t),D(z)(t)=0.2+x(t)*z(t)-2.5*z(t)\},[x(t),y(t),z(t)],t=
100..200,[[x(0)=-1,y(0)=0,z(0)=0]],stepsize=0.1,axes=FRAME,linecolor=R
ED,orientation=[-130,60]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
319 "# c=4   l'attracteur est une solution p\351riodique, quatre minim
a de l'abscisse, on est au-del\340 de la deuxi\350me bifurcation\nDEpl
ot3d(\{D(x)(t)=-(y(t)+z(t)),D(y)(t)=x(t)+0.2*y(t),D(z)(t)=0.2+x(t)*z(t
)-4*z(t)\},[x(t),y(t),z(t)],t=100..200,[[x(0)=-1,y(0)=0,z(0)=0]],steps
ize=0.1,axes=FRAME,linecolor=RED,orientation=[-130,60]);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 333 "# c=5.32   solution p\351riodique,
 trois minima de l'abscisse, fen\352tre de p\351riode 3 dans la deuxi
\350me partie du diagramme de Feigenbaum\nDEplot3d(\{D(x)(t)=-(y(t)+z(
t)),D(y)(t)=x(t)+0.2*y(t),D(z)(t)=0.2+x(t)*z(t)-5.32*z(t)\},[x(t),y(t)
,z(t)],t=100..200,[[x(0)=-1,y(0)=0,z(0)=0]],stepsize=0.1,axes=FRAME,li
necolor=RED,orientation=[-130,60]);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 253 "# c=7   attracteur chaotique \351trange, comme pour \+
c=5.7\nDEplot3d(\{D(x)(t)=-(y(t)+z(t)),D(y)(t)=x(t)+0.2*y(t),D(z)(t)=0
.2+x(t)*z(t)-7*z(t)\},[x(t),y(t),z(t)],t=100..200,[[x(0)=-1,y(0)=0,z(0
)=0]],stepsize=0.1,axes=FRAME,linecolor=RED,orientation=[-130,60]);" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 296 "# c=8   l'attracteur est \+
une solution p\351riodique, cinq minima de l'abscisse, fen\352tre de p
\351riode 5\nDEplot3d(\{D(x)(t)=-(y(t)+z(t)),D(y)(t)=x(t)+0.2*y(t),D(z
)(t)=0.2+x(t)*z(t)-8*z(t)\},[x(t),y(t),z(t)],t=100..200,[[x(0)=-1,y(0)
=0,z(0)=0]],stepsize=0.1,axes=FRAME,linecolor=RED,orientation=[-130,60
]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" 
{TEXT 268 22 "L'attracteur de Lorenz" }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(DEtools):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 266 "# vue globale de l'attracte
ur de Lorenz pour R=28\nDEplot3d(\{D(x)(t)=10*(-x(t)+y(t)),D(y)(t)=28*
x(t)-y(t)-x(t)*z(t),D(z)(t)=-8/3*z(t)+x(t)*y(t)\},[x(t),y(t),z(t)],t=1
80..220,[[x(0)=8.4,y(0)=8.4,z(0)=27]],stepsize=0.02,axes=FRAME,linecol
or=ORANGE,orientation=[-50,65]);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 316 "# vue sur un intervalle de temps plus court pour mie
ux visualiser la dynamique du syst\350me\nDEplot3d(\{D(x)(t)=10*(-x(t)
+y(t)),D(y)(t)=28*x(t)-y(t)-x(t)*z(t),D(z)(t)=-8/3*z(t)+x(t)*y(t)\},[x
(t),y(t),z(t)],t=184.4..191.5,[[x(0)=8.4,y(0)=8.4,z(0)=27]],stepsize=0
.015,axes=FRAME,linecolor=0.16*t-29.5,orientation=[-50,65]);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 270 "# une solution p\351riodiqu
e attractive pour R=100.5\nDEplot3d(\{D(x)(t)=10*(-x(t)+y(t)),D(y)(t)=
100.5*x(t)-y(t)-x(t)*z(t),D(z)(t)=-8/3*z(t)+x(t)*y(t)\},[x(t),y(t),z(t
)],t=100..120,[[x(0)=8.4,y(0)=8.4,z(0)=27]],stepsize=0.015,axes=FRAME,
linecolor=ORANGE,orientation=[-50,65]);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT 269 39 "Reconstruction des att
racteurs \351tranges" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT 270 21 "Une premi\350re approche" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 404 "cr\351e_liste:=proc(fonction_d_it\351ration,
traitement,z0) local z,s,j;\n# cr\351atrice d'une liste de donn\351es \+
obtenue artificiellement en appliquant un certain traitement (pour se \+
ramener \340 une seule variable) aux it\351r\351s par une certaine fon
ction d'it\351ration \340 partir d'une donn\351e initiale z0.\nz:=(fon
ction_d_it\351ration@@50)(z0); s:=NULL;\nfor j to 100 do s:=s,traiteme
nt(z); z:=fonction_d_it\351ration(z) od;\n[s]\nend:" }}{PARA 0 "> " 0 
"" {MPLTEXT 1 0 250 "approche1:=proc(liste) local s,k;\n# construit le
 graphe des points [donn\351e num\351ro k;donn\351e num\351ro (k+1)] \+
\340 partir d'une liste de donn\351es \ns:=NULL;\nfor k from 1 to nops
(liste)-1 do s:=s,[liste[k],liste[k+1]] od;\nplot([s],style=POINT,symb
ol=POINT)\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "(approch
e1@cr\351e_liste)(x->rand(),x->x/10^10,1996);" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 46 "(approche1@cr\351e_liste)(x->4*x*(1-x),x->x,0.1)
;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "(approche1@cr\351e_lis
te)(liste->[liste[2]+1-1.4*liste[1]^2,.3*liste[1]],liste->liste[1],[0,
0]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 281 "# essai de la prem
i\350re approche pour reconstituer l'attracteur de R\366ssler\nwith(DE
tools):\nf:=dsolve(\{D(x)(t)=-(y(t)+z(t)),D(y)(t)=x(t)+0.2*y(t),D(z)(t
)=0.2+x(t)*z(t)-5.7*z(t),y(0)=0,x(0)=-1,z(0)=0\},\{x(t),y(t),z(t)\},ty
pe=numeric):\n(approche1@cr\351e_liste)(t->t+1,T->subs(f(T),z(t)),0);
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 271 
51 "Une deuxi\350me approche pour les attracteurs spatiaux" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "with(DEtools): with(plots): with(pl
ottools):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 415 "# une reconst
ruction de l'attracteur de R\366ssler par un la m\351thode des phases
\nrossler:=dsolve(\{D(x)(t)=-(y(t)+z(t)),D(y)(t)=x(t)+0.2*y(t),D(z)(t)
=0.2+x(t)*z(t)-5.7*z(t),y(0)=0,x(0)=-1,z(0)=0\},\{x(t),y(t),z(t)\},typ
e=numeric,method=classical):\nl:=[seq(subs(rossler(20+0.05*k),z(t)),k=
0..2000)]:\ns:=NULL:\nfor n to nops(l)-20 do s:=s,[l[n],l[n+10],l[n+20
]] od:\npointplot3d([s],style=LINE,color=BLACK,orientation=[50,60]);" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 424 "# une reconstruction de \+
l'attracteur de Lorenz par un la m\351thode des phases\nlorenz:=dsolve
(\{D(x)(t)=10*(-x(t)+y(t)),D(y)(t)=28*x(t)-y(t)-x(t)*z(t),D(z)(t)=-8/3
*z(t)+x(t)*y(t),x(0)=8.4,y(0)=8.4,z(0)=27\},\{x(t),y(t),z(t)\},type=nu
meric,method=classical):\nl:=[seq(subs(lorenz(20+0.01*k),x(t)),k=0..30
00)]:\ns:=NULL:\nfor n to nops(l)-10 do s:=s,[l[n],l[n+5],l[n+10]] od:
\npointplot3d([s],style=LINE,color=BLACK,orientation=[70,70]);" }}}}
{MARK "58 0 0" 14 }{VIEWOPTS 1 1 0 1 1 1803 }