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{SECT 0 {PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 256 "" 0 "" 
{TEXT -1 10 "EinfŸhrung" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 66 "Dynami
cs Routinen von Antonio Giraldo modifiziert von Udo Schmidt\n" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart;\nwith(plots):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
88 " f= family of functions depending of a parameter P,\n c= chosen va
lue of the parameter P," }}{PARA 0 "" 0 "" {TEXT -1 38 " p= point whos
e orbit we want compute," }}{PARA 0 "" 0 "" {TEXT -1 36 " n= number of
 elements of the orbit," }}{PARA 0 "" 0 "" {TEXT -1 40 " [d,e]= interv
al where f is restricted,\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
62 "orbn:=proc(f,c,p,n,d,e)\nlocal p0,g,cr,u,v,s,i,t,p1,l,a,b,k,p2;" }
}{PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "  p0:=plot([[p,0]], x=d..e,style=p
oint,symbol=BOX,scaling=constrained);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 17 "  g:=subs(P=c,f);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "  u:=eva
lf(d);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "  v:=evalf(e);" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 46 "    p1:=plot(\{g,x\}, x=u..v, y=u..v,colo
r=red):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "  a:=evalf(p);" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 13 "  l:=[[a,0]];" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 41 "  for k from 1 to n while u<a and a<v do;" }}{PARA 0 
"> " 0 "" {MPLTEXT 1 0 26 "    b:=evalf(subs(x=a,g));" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 27 "    l:=[op(l),[a,b],[b,b]];" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 9 "    a:=b;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "  od;
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "  p2:=plot(l,u..v,style=line):
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "  plots[display](\{p0,p1,p2\});
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 51 "Prozedur stellt schrittweise die Prozedur orbn dar." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "orbitera:=proc(f,c,p,n,d,e)
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "local l,i;" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 37 "  l:=[seq(orbn(f,c,p,i,d,e),i=0..n)]:" }}{PARA 0 ">
 " 0 "" {MPLTEXT 1 0 36 "  plots[display](l,insequence=true);" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 49 "Nichthyperb
olische Punkte und graphische Analyse\n" }}{PARA 5 "" 0 "" {TEXT -1 0 
"" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "orbitera(P+x-1/2*x^3,0,
0.8,22,-2,2);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 32 "orbitera(P+x+x^3,0,0.2,22,-2,2);" }}}{PARA 13 "" 1 "" {TEXT -1 
0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 12 "Cantor-Menge" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "restart:\nf:=(x)->4.5*x*(1-x);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "plot([f(x),1],x=-0.1..1.1,ti
tle=\" Erste Iteration\");" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
68 "plot([f(f(x)),1],x=-0.1..1.1,y=-0.7..1.5,title=\" Zweite Iteration
\");" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "plot([f(f(f(x))),1]
,x=-0.1..1.1,y=-0.7..2,numpoints=200,title=\" Dritte Iteration\");" }
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 
13 "" 1 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 
{PARA 4 "" 0 "" {TEXT 259 21 "Bifurkation Beispiele" }}{SECT 1 {PARA 
4 "" 0 "" {TEXT -1 65 "Dynamics Routinen von Antonio Giraldo modifizie
rt von Udo Schmidt" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restar
t;\nwith(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 88 " f= family of functions depending of a pa
rameter P,\n c= chosen value of the parameter P," }}{PARA 0 "" 0 "" 
{TEXT -1 38 " p= point whose orbit we want compute," }}{PARA 0 "" 0 "
" {TEXT -1 36 " n= number of elements of the orbit," }}{PARA 0 "" 0 "
" {TEXT -1 40 " [d,e]= interval where f is restricted,\n" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "orbn:=proc(f,c,p,n,d,e)\nlocal p0,g
,cr,u,v,s,i,t,p1,l,a,b,k,p2;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "  p
0:=plot([[p,0]], x=d..e,style=point,symbol=BOX,scaling=constrained);" 
}}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "  g:=subs(P=c,f);" }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 14 "  u:=evalf(d);" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 14 "  v:=evalf(e);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "    p1:=p
lot(\{g,x\}, x=u..v, y=u..v,color=red):" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 14 "  a:=evalf(p);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "  l:=[[a,
0]];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "  for k from 1 to n while u
<a and a<v do;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "    b:=evalf(subs
(x=a,g));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "    l:=[op(l),[a,b],[b
,b]];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "    a:=b;" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 5 "  od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "  p2:
=plot(l,u..v,style=line):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "  plot
s[display](\{p0,p1,p2\});" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "Prozedur stellt schrittweise die
 Prozedur orbn dar." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "orbi
tera:=proc(f,c,p,n,d,e)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "local l,
i;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "  l:=[seq(orbn(f,c,p,i,d,e),i
=0..n)]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "  plots[display](l,inse
quence=true);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 257 "" 0 "" {TEXT -1 0 "
" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 21 "Tangentenbifurkation\n" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
153 "f:=P*exp(x);\nw:=x;\nK:=plot(w,x=-0.5..3,color=black):\nwith(plot
s):\nA:=animate(f,x=-0.5..3,P=1/(2*exp(1))..1.2/exp(1),frames=15,color
=red):\ndisplay(\{K,A\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
34 "orbitera(f,1.2/exp(1),0.1,22,0,3);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 38 "orbitera(f,1.7/(2*exp(1)),0.1,22,0,2);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "orbitera(f,1.7/(2*exp(1)),1.6,22,0,
2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "orbitera(f,1.7/(2*ex
p(1)),1.7,22,0,2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "impli
citplot(P*exp(x) = x,P=1/(2*exp(1))..1.2/exp(1),x=0..3,title=Tangenten
bifurkation);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}
{SECT 1 {PARA 4 "" 0 "" {TEXT -1 32 "Periodenverdopplungs Bifurkation
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "f:=x->P*exp(x);\nw:=x;\n
K:=plot(w,x=-7..0,color=black):\nwith(plots):" }{TEXT -1 0 "" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 91 "A:=animate(f(f(x)),x=-7..0,P=-2.5*exp(1).
.-1/2*exp(1),frames=15,color=red):\ndisplay(\{K,A\});" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "orbitera(f(x),-0.5*exp(1),-0.5,22,-
2,0);\norbitera(f(f(x)),-0.5*exp(1),-0.5,22,-2,0);\n" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 78 "orbitera(f(x),-exp(1),-0.8,22,-2,0);\norb
itera(f(f(x)),-exp(1),-0.8,22,-2,0);\n\n" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 176 "orbitera(f(x),-1.3*exp(1),-0.8,22,-2,0);\norbitera(f
(f(x)),-1.3*exp(1),-0.8,22,-2,0);\norbitera(f(f(x)),-1.3*exp(1),-1.2,2
2,-3,0);\norbitera(f(f(x)),-1.3*exp(1),-0.12,22,-0.4,0);\n" }}{PARA 
13 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "
f:=(x,P)->(-4+abs(P))*exp(x);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 125 "implicitplot(f(f(x,P),P) = x,P=-1.5..1.5,x=-7..0,numpoints=80
00,title=\"Pathologische Periodenverdopplungs Bifurkation n=2 \");" }}
}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT 257 
29 "Chaos und Periodenverdopplung" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 
4 "Def:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "restart;\nf:=(x,P
)->x*P*(1-x);\nwith(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 22 "Bifurkationsdiagramme\n
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "implicitplot(f(x,P) = x
,P=0..3,x=0..1,numpoints=8000,title=\"Periodenverdopplungs Bifurkation
 n=1 \");\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "implicitplo
t(f(f(x,P),P) = x,P=0..4,x=0..1,numpoints=20000,title=\"Periodenverdop
plungs Bifurkation n=2 \");\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 118 "implicitplot(f(f(f(f(x,P),P),P),P) = x,P=0..3.8,x=0..1,numpoint
s=20000,title=\"Periodenverdopplungs Bifurkation n=4 \");" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 140 "implicitplot(f(f(f(f(f(f(f(f(x,P),
P),P),P),P),P),P),P) = x,P=0..3.65,x=0..1,numpoints=100000,title=\"Per
iodenverdopplungs Bifurkation n=8 \");" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 39 "a=1/2;\nevalf(solve(1/2*b*(1-1/2)=1,b));" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "evalf(sqrt(5));" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 111 "implicitplot(f(f(f(x,P),P),P) = x,P=0..4,x
=0..1,numpoints=20000,title=\"Periodenverdopplungs Bifurkation n=2 \")
;" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 24 "Renormalisationsoperator" 
}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 35 " Quadratische Familie 2te Itera
tion" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "restart;\nf:=(x,P)->
x*P*(1-x);\nwith(plots):\nF:=(x,P)->f(f(x,P),P):" }{TEXT -1 0 "" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "one_poly := [[(-1+P)/P,(-1+P)/P],[1
/P,(-1+P)/P],[1/P,1/P] ,[(-1+P)/P,1/P]]:" }{TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 120 "P:=3.0:\n\nK:=polygonplot(one_poly
):\nL:=plot(x,x=0..1,color=green):\nM:=plot(F(x,P),x=0..1):\n\ndisplay
(\{K,L,M\},title=\"P=3\");" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
124 "P:=3.5:\n\nK:=polygonplot(one_poly):\nL:=plot(x,x=0..1,color=gree
n):\nM:=plot(F(x,P),x=0..1):\n\ndisplay(\{K,L,M\},title=\"P=3.5\");\n
\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "P:=3.8:\n\nK:=polygo
nplot(one_poly):\nL:=plot(x,x=0..1,color=green):\nM:=plot(F(x,P),x=0..
1):\n\ndisplay(\{K,L,M\},title=\"P=3.8\");\n" }}}}{SECT 1 {PARA 5 "" 
0 "" {TEXT -1 24 "Renormalisationsoperator" }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 65 "restart:\nf:=(x,P)->x*P*(1-x):\nwith(plots):\nF:=(x
,P)->f(f(x,P),P):" }{TEXT -1 1 "\n" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 78 "L:=(x,P)->1/(1/P-(-1+P)/P)*(x-(-1+P)/P):\nIL:=(x,P)->
(1/P-(-1+P)/P)*x+(-1+P)/P:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
29 "RF:=(x,P)->L(F(IL(x,P),P),P):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 59 "plot([RF(x,3),x],x=0..1,title=\"Renormalisierungsoper
ator\");" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "plot([RF(x,3.5)
,x],x=0..1,title=\"Renormalisierungsoperator\");" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 68 "plot([RF(RF(x,3.5),3.5),x],x=0..1,title=\"Zwe
ite Iteration von RF \");" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}}{PARA 4 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 
5 "" 0 "" {TEXT 256 9 "Periode 3" }{TEXT 258 1 "\n" }{TEXT -1 1 "\n" }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "restart;\nf:=(x,P)->P*x*(1-
x);\nwith(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "impli
citplot(f(f(f(x,P),P),P) = x,P=0..5,x=0..1,numpoints=8000,title=\"Peri
odenverdopplungs Bifurkation n=3 \");" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 115 "implicitplot(f(f(f(f(x,P),P),P),P) = x,P=0..5,x=0..1
,numpoints=8000,title=\"Periodenverdopplungs Bifurkation n=4 \");" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "f:=(x,P)->x*P*(1-x);\nwith(p
lots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "K:=animate(f(f(f(
x,P),P),P),x=0..1,P=1..3.9,frames=15,numpoints=500,color=red):" }
{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "L:=plot(x,x=0..1,co
lor=black,title=\"Dritte Iteration von f\"):\ndisplay(\{K,L\});" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}}}{MARK "5" 0 }{VIEWOPTS 1 1 0 1 1 1803 }