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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 256 58 "MISE EN EVIDENCE DES PR
OPRIETES CARACTERISTIQUES DU CHAOS\n" }}}{EXCHG {PARA 257 "" 0 "" 
{TEXT 257 36 "D\351finition de l'it\351rateur du syst\350me" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 105 "with(plots):\nf:=x->4*x*(1-x): #parabole
 d'it\351ration\n#le param\350tre de l'it\351rateur quadratique est fi
x\351 \340 4" }}}{EXCHG {PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "
" 0 "" {TEXT 258 13 "La proc\351dure " }{TEXT 259 6 "orbite" }{TEXT 
260 21 " et quelques exemples" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 255 "orbite:=proc(n,x0) local k,x,s;\n#n est le nombre d'it\351ratio
ns \340 repr\351senter\n#x0 est la valeur initiale, comprise entre 0 e
t 1\nx[0]:=x0;s:=[0,x0];\nfor k from 1 to n\n    do x[k]:=f(x[k-1]); s
:=s,[k,x[k]]\n    od;\nplot([s],x=0..n,style=LINE,symbol=POINT)\nend:
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "orbite(200,0.502);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "orbite(15,(sin(Pi/7))^2);" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "orbite(80,evalf((sin(Pi/7)
)^2));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" 
{TEXT 261 13 "La proc\351dure " }{TEXT 262 11 "comparaison" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 375 "comparaison:=proc(n,x0,epsilon)\nl
ocal x,y,s,k;\n#n est le nombre d'it\351rations \340 effectuer\n#x0 es
t la valeur initiale, comprise entre 0 et 1\n#epsilon est l'erreur sur
 la valeur initiale\nx[0]:=x0; y[0]:=x0+epsilon; s:=[0,y[0]-x[0]];\nfo
r k from 1 to n\n    do x[k]:=f(x[k-1]); y[k]:=f(y[k-1]);             \+
  s:=s,[k,y[k]-x[k]]\n    od;\nplot([s],x=0..n,style=LINE,color=GREEN)
;\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "comparaison(200,
0.502,0.00001);" }}}{EXCHG {PARA 260 "" 0 "" {TEXT -1 0 "" }}{PARA 
260 "" 0 "" {TEXT 263 13 "La proc\351dure " }{TEXT 264 7 "m\351lange" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 121 "m\351lange:=proc(x0,gamm
a,delta)\nlocal x,n;\nx:=x0; n:=0;\nwhile (x<=gamma or x>=delta) do x:
=f(x); n:=n+1 od;\nprint(n,x)\nend:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 31 "m\351lange(0.342,0.57581,0.57582);" }}}{EXCHG {PARA 
261 "" 0 "" {TEXT -1 0 "" }}{PARA 261 "" 0 "" {TEXT 265 13 "La proc
\351dure " }{TEXT 266 11 "histogramme" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 464 "histogramme:=proc(x0,n,pr\351cision)\nlocal k,t,x,j,
i;\n#n est le nombre d'it\351rations \340 repr\351senter\n#x0 est la v
aleur initiale, comprise entre 0 et 1\n#pr\351cision est le nombre de \+
subdivisions de [0;1]\n#consid\351r\351es pour former le tableau des r
\351sultats\nfor k from 0 to pr\351cision do t[k]:=0 od;\nx:=x0;\nfor \+
j from 1 to n do x:=f(x); i:=floor(pr\351cision*x); t[i]:=t[i]+1 od;\n
plot([seq([k/pr\351cision,t[k]],k=0..pr\351cision-1)],style=point,symb
ol=POINT,thickness=0,axes=frame)\nend:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 28 "histogramme(0.51,50000,500);" }}}}{MARK "14 0 0" 264 
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