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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 38 "Lyapunov Exponents of Qu
adratic Family" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 123 "Calculate the Lyapunov exponen
t for  f(x) = r*x*(1-x)\n\nUse  r = 3.80, 3.81, 3.82, 3.83, 3.84, 3.85
, 3.86, 3.87, 3.88, 3.89." }}{PARA 0 "" 0 "" {TEXT -1 54 "Use the init
ial condition  x0 := 0.5.  \nThe variable  " }}{PARA 0 "" 0 "" {TEXT 
-1 516 "        s = s(x0,i) :=  ln(|df(xi)|) + s(i-1)  as the sum of t
he logarithms of the first  i   iterates.\nWe take  s(x0,0) = 0, so wh
en we take x0 = 0.5 we do not calculate the derivative at 0.5.\nThe va
riable  \n       h = h(x0,i) := s(x0,i)/(i+1)\n  is the estimate of th
e Lyapunov exponent starting with the initial condition  x0 after  i  \+
iterates.  \n(In the text, we write the dependence of s  and  h   on t
he initial condition  x0 and the iterate  i, \nbut in the calculation \+
we only use  the variables  s  and   h.)" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 84 "For the first  n1=20000  iterates \+
the calculation for  h(x0,i)   is not printed out." }}{PARA 0 "" 0 "" 
{TEXT -1 129 "For the next  n2 = 20  iterates the calculation for h(x0
,i)  is printed out, which is the estimate for the\nLyapunov  exponent
.  \n" }}{PARA 0 "" 0 "" {TEXT -1 39 "f        The function  f(x) = r*
x*(1-x)" }}{PARA 0 "" 0 "" {TEXT -1 35 "df     The derivative df(x) = \+
f'(x)" }}{PARA 0 "" 0 "" {TEXT -1 104 "r:      Parameter value.  \nn1 \+
   The number of iterates for which the estimate of h(x0) is not displ
ayed" }}{PARA 0 "" 0 "" {TEXT -1 105 "n2    The number of iterates for
 which the estimate of h(x0) is displayed\nx0:   The  initial conditio
n  \n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "r := 3.87;\nf:= x \+
-> r*x*(1-x);\ndf := x -> r - 2*r*x;\nn1  := 20000;\nn2 := 100;" }
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 323 "x0 := 0.5:
\nx := x0:\ns := 0:\nfor i from 1 to n1 do\n   x := f(x);\n   s := s +
 ln(abs(df(x))):\nod:\nprintf(`\\n\\t r = %1.4f \\n\\n`, r);\nprintf(`
\\t x0 = %1.4f \\n\\n`, x0);\nprintf(`\\t i \\t\\t h(x0,i) \\n`);\nfor
 j from 1 to n2 do\n   x := f(x);\n   s := s + ln(abs(df(x))):\n   h :
= s/(j+n1):\n   printf(`\\t %d \\t %1.4f \\n`, j+n1, h );\nod:" }
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "r := 3.81;
\nf:= x -> r*x*(1-x);\ndf := x -> r - 2*r*x;\nn1  := 20000;\nn2 := 100
;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 291 "x0 := 0.2:\nx := x0:
\ns := 0:\nfor i from 1 to n1 do\n   x := f(x);\n   s := s + ln(abs(df
(x))):\nod:\nprintf(`\\n\\t x0 = %1.4f \\n\\n`, x0);\nprintf(`\\t i \\
t\\t h(x0,i) \\n`);\nfor j from 1 to n2 do\n   x := f(x);\n   s := s +
 ln(abs(df(x))):\n   h := s/(j+n1):\n   printf(`\\t %d \\t %1.4f \\n`,
 j+n1, h );\nod:" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 77 "r := 3.81;\nf:= x -> r*x*(1-x);\ndf := x -> r - 2*r*x;\nn1  :=
 20000;\nn2 := 100;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 291 "x0 \+
:= 0.3:\nx := x0:\ns := 0:\nfor i from 1 to n1 do\n   x := f(x);\n   s
 := s + ln(abs(df(x))):\nod:\nprintf(`\\n\\t x0 = %1.4f \\n\\n`, x0);
\nprintf(`\\t i \\t\\t h(x0,i) \\n`);\nfor j from 1 to n2 do\n   x := \+
f(x);\n   s := s + ln(abs(df(x))):\n   h := s/(j+n1):\n   printf(`\\t \+
%d \\t %1.4f \\n`, j+n1, h );\nod:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 77 "r := 3.81;\nf:= x -> r*x*(1-x);\ndf := x -> r - 2*r*x
;\nn1  := 20000;\nn2 := 100;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 292 "x0 := 0.49:\nx := x0:\ns := 0:\nfor i from 1 to n1 do\n   x := \+
f(x);\n   s := s + ln(abs(df(x))):\nod:\nprintf(`\\n\\t x0 = %1.4f \\n
\\n`, x0);\nprintf(`\\t i \\t\\t h(x0,i) \\n`);\nfor j from 1 to n2 do
\n   x := f(x);\n   s := s + ln(abs(df(x))):\n   h := s/(j+n1):\n   pr
intf(`\\t %d \\t %1.4f \\n`, j+n1, h );\nod:" }}}{EXCHG {PARA 0 "> " 
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