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m">1</A></TD>
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m">6</A></TD>
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m">7</A></TD>
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m">8</A></TD>
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m">9</A></TD>
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tm">10</A></TD>
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">=CE=B2=C9=F9</A></TD>
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href=3D"http://www.phil.pku.edu.cn/personal/huajie/fractalart/html/REF.ht=
m">=B2=CE=BF=BC=CE=C4=CF=D7</A></TD>
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href=3D"http://www.phil.pku.edu.cn/personal/huajie/fractalart/html/EE.htm=
">=BA=F3=BC=C7</A></TD>
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tm">=CB=F7=D2=FD</A></TD></TR></TBODY></TABLE></FONT>
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    <TD><A=20
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href=3D"http://www.phil.pku.edu.cn/personal/huajie/fractalart/html/801.ht=
m">8.1</A></TD>
    <TD><A=20
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href=3D"http://www.phil.pku.edu.cn/personal/huajie/fractalart/html/802.ht=
m">8.2</A></TD>
    <TD><A=20
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href=3D"http://www.phil.pku.edu.cn/personal/huajie/fractalart/html/803.ht=
m">8.3</A></TD>
    <TD><A=20
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href=3D"http://www.phil.pku.edu.cn/personal/huajie/fractalart/html/804.ht=
m">8.4</A></TD>
    <TD><A=20
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href=3D"http://www.phil.pku.edu.cn/personal/huajie/fractalart/html/805.ht=
m">8.5</A></TD>
    <TD><A=20
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href=3D"http://www.phil.pku.edu.cn/personal/huajie/fractalart/html/806.ht=
m">8.6</A></TD>
    <TD><A=20
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href=3D"http://www.phil.pku.edu.cn/personal/huajie/fractalart/html/807.ht=
m">8.7</A></TD></TR></TBODY></TABLE></FONT></CENTER>
<HR width=3D"100%">

<H2>8.2 =C0=EF=D1=C5=C6=D5=C5=B5=B7=F2=D6=B8=CA=FD</H2>
<P></P>
<P>=C8=B7=B6=A8=BB=EC=E3=E7=D4=CB=B6=AF=D2=BB=B0=E3=D3=D0=B7=D6=CA=FD=CE=AC=
=CA=FD=A1=A2=C0=EF=D1=C5=C6=D5=C5=B5=B7=F2(A.M.Liapnov=A3=AC1857-1918)=D6=
=B8=CA=FD=A1=A2=CD=D8=C6=CB=EC=D8=A1=A2=B9=A6=C2=CA =
=C6=D7=B5=C8=BC=B8=B8=F6=B6=A8=C1=BF=D6=B8=B1=EA=A3=AC=B1=BE=BD=DA=CC=B8=C0=
=EF=D1=C5=C6=D5=C5=B5=B7=F2=D6=B8=CA=FD=A1=A3</P>
<P>=D2=BB=B0=E3=CF=B5=CD=B3=B5=C4=C0=EF=D1=C5=C6=D5=C5=B5=B7=F2=D6=B8=CA=FD=
=B2=A2=B2=BB=BA=DC=C8=DD=D2=D7=C7=F3=B3=F6=A3=AC=B5=AB=B6=D4=D3=DA=D2=BB=CE=
=AC=C2=DF=BC=AD=CB=B9=B5=D9=D3=B3=C9=E4=A3=AC=BF=C9=D2=D4=B8=F8=B3=F6=BD=E2=
=CE=F6=CA=BD=A3=AC=20
=B1=E3=D3=DA=CA=FD=D6=B5=BC=C6=CB=E3=A1=A3=C9=E8<I>=A6=CB</I>=CA=C7=C0=EF=
=D1=C5=C6=D5=C5=B5=B7=F2=D6=B8=CA=FD=A3=AC=B6=D4=D3=DA=D3=B3=C9=E4<I>x</I=
>=A1=FA<I>ax</I>(1-<I>=20
x</I>)=A3=AC<BR><I>f</I>=A1=E4(<I>x</I>)=3Dd<I>f</I>/d<I>x</I>=3D<I>a</I>=
-2<I>ax</I>=A3=AC=D4=F2</P>
<UL><I>=A6=CB</I>=3Dlim_(<I>N</I>=A1=FA=A1=DE)=A3=FB1/<I>N</I>=A1=C6^(<I>=
N</I>-1)_(<I>i</I>=3D1)ln=A3=FCd<I>f=20
  =
</I>/d<I>x</I>=A3=FC=A3=FD<BR>=3Dlim_(<I>N</I>=A1=FA=A1=DE)=A3=FB1/<I>N</=
I>=A1=C6^(<I>N</I>-1)_(<I>i</I>=3D1)=20
  =
ln=A3=FC<I>a</I>-2<I>ax</I>=A3=FC.</UL>=D4=DA=CA=B5=BC=CA=BC=C6=CB=E3=D6=D0=
<I>N</I>=C8=A1=D7=E3=B9=BB=B4=F3=BC=B4=BF=C9=A3=AC=BF=C9=D2=D4=C8=A1500=BB=
=F2=D5=DF2000=A1=A3=BC=C6=CB=E3=C0=EF=D1=C5=C6=D5=C5=B5=B7=F2=D6=B8=CA=FD=
<I>=A6=CB=20
</I>=B5=C4=D4=B4=B3=CC=D0=F2=C8=E7=CF=C2=A3=BA
<P></P>
<P>
<UL><PRE>{Lyapunov Exponent for logistic Map, Huajie, Oct.12,1993}
uses graph,crt;
const tonum=3D500;
var
m,k,i,np,winwidth,Gd,Gm:integer;
p0,x0,y0,x1,y1,lins,pmin,pmax,deltap,a,sum,sumsh:real;
begin
Gd:=3DVGA; Gm:=3DVGAHI;Initgraph(Gd,Gm,=A1=AFd:\\pascal=A1=AF);
winwidth:=3D500;line(10,30,10,460); line(10,70,15,70);
OutTextXY(22,70,=A1=AF1.0=A1=AF);OutTextXY(22,330,=A1=AF-1.0=A1=AF);
line(10,330,15,330);line(10,200,winwidth+20,200);
pmin:=3D3.40; pmax:=3D3.99;lins:=3Dln(2)*tonum;
deltap:=3D(pmax-pmin)/(winwidth-1);
for np:=3D0 to winwidth-1 do
begin
a:=3Dpmin+np*deltap;
x0:=3D0.5;
for i:=3D1 to 20 do begin
x1:=3Da*x0*(1-x0);x0:=3Dx1;
end;
sum:=3D0;
for i:=3D1 to tonum do  begin
x1:=3Da*x0*(1-x0);
sum:=3Dsum+ln(abs(a-2*a*x1));
x0:=3Dx1;
end;
sum:=3Dsum/lins; sum:=3Dsum*140;
if np=3D0 then sumsh:=3Dsum;
line(np+11,200-round(sum),np+10,200-round(sumsh));
sumsh:=3Dsum;
end;
readln;sound(500);delay(200);nosound;closegraph;
end.</PRE></UL><PRE></PRE>
<P></P>
<P>
<CENTER><IMG=20
src=3D"http://www.phil.pku.edu.cn/personal/huajie/fractalart/html/p803.gi=
f">
<P></P>
<P>=CD=BC8.3 =
=D2=BB=CE=AC=C2=DF=BC=AD=CB=B9=B5=D9=D3=B3=C9=E4=C0=EF=D1=C5=C6=D5=C5=B5=B7=
=F2=D6=B8=CA=FD=C6=D7</CENTER></P>
<P>=B5=B1<I>=A6=CB</I>=B4=F3=D3=DA=C1=E3=CA=B1=A3=AC=D2=BB=B0=E3=C8=CF=CE=
=AA=CF=B5=CD=B3=B4=A6=D3=DA=BB=EC=E3=E7=D4=CB=B6=AF=A3=AC=B5=B1<I>=A6=CB<=
/I>=D0=A1=D3=DA=C1=E3=CA=B1=A3=AC=C8=CF=CE=AA=CF=B5=CD=B3=B4=A6=D3=DA=20
=D6=DC=C6=DA=D4=CB=B6=AF=BB=F2=D5=DF=D7=BC=D6=DC=C6=DA=D4=CB=B6=AF=A1=A3=CD=
=BC8.3=D6=D0=A3=AC=C9=CF=CD=BC=D5=B9=CA=BE=B5=C4=CA=C7=B2=CE=CA=FD<I>a</I=
>=A1=CA=A3=DB3.40,3.99=A3=DD=C7=F8=BC=E4=C9=CF=C0=EF=20
=D1=C5=C6=D5=C5=B5=B7=F2=D6=B8=CA=FD=B5=C4=C7=E9=BF=F6=A3=AC=D6=D0=CD=BC=BA=
=CD=CF=C2=CD=BC=CA=BE=D2=E2=C1=CB=B2=CE=CA=FD<I>a</I>=A1=CA=A3=DB3.60,3.6=
5=A3=DD=BA=CD<I>a</I>=A1=CA=20
=A3=DB3.62,3.64=A3=DD=C7=F8=BC=E4=B5=C4=C7=E9=BF=F6=A3=AC=BA=E1=D7=F8=B1=EA=
=B2=BB=B6=CF=B1=E4=B4=F3=A3=AC=D7=DD=D7=F8=B1=EA=CE=B4=B1=E4=A1=A3 </P>
<P>
<P>
<HR width=3D"100%">

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    <TD><A=20
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href=3D"http://www.phil.pku.edu.cn/personal/huajie/fractalart/html/pref1.=
htm">=D0=F2=D1=D4=D2=BB</A></TD>
    <TD><A=20
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href=3D"http://www.phil.pku.edu.cn/personal/huajie/fractalart/html/pref2.=
htm">=D0=F2=D1=D4=B6=FE</A></TD>
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t.htm">=C4=BF=C2=BC</A></TD>
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tm">10</A></TD>
    <TD><A=20
      =
12

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