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Critical Scaling Behavior near The Accumulation Point
Eui-Sun Lee
Department of Physics
Kangwon National University
The transition to chaos through an infinite sequence of period-double bifurcation occurs
at the accumulation point A∞ .
Bifurcation diagram
1
Parameter Scaling Factor
Follow the period–doubling bifurcation point An of the
period – 2n orbit (n=0,1,2,3…) with the stability
multiplier λn= -1.
An converges to its accumulation point A∞
geometrically with ratio 1/ δ ,
An A ~ n .
Define the parameter scaling factor of the level n,
An 1 An
n
An An 1
n 1 / n
and n Converges to as n → ∞ .
An 1 An
lim = 4.6692… .
n An An 1
2
Orbit Scaling Factor
Find a period–2n orbit point x n with the maximum
distance at A n .
x n converges to its origin point x = 0
geometrically with ratio 1/ α ,
xn ~ n .
Define the orbital scaling factor of the level n,
xn 1 xn
n
xn xn 1
d n / d n 1
and n Converges to as n → ∞ .
xn 1 xn
lim = - 2.5029… .
n x x
n n 1
3
Parameter and Orbital Scaling Factor
Parameter Scaling Factor Orbital Scaling Factor
n An δn n xn α n
0 0.7500 000 000 … 0 0.666 666 666 …
1 1.250 000 000 … 4.233 738 275 … 1 -0.165 685 424 … -3.669 849 486 …
2 1.368 098 939 … 4.551 506 947 … 2 0.061 122 811 … -2.664 010 443 …
3 1.394 046 156 … 4.645 807 517 … 3 -0.024 015 081 … -2.535 664 422 …
4 1.399 631 238 … 4.663 938 173 … 4 0.009 561 086 … -2.509 770 155 …
5 1.400 828 742 .. 4.668 103 913 … 5 -0.003 817 098 … -2.504 378 918 …
6 1.401 085 271 … 4.668 962 792 … 6 0.001 524 818 … -2.503 221 760 …
7 1.401 140 214 … 4.669 150 919 … 7 -0.000 609 197 … -2.502 975 166 …
8 0.000 243 394 … -2.502 922 272 …
8 1.401 151 982 … 4.669 190 690 …
9 -0.000 097 244 … -2.502 910 961 …
9 1.401 154 502 … 4.669 199 277 …
10 0.000 038 852 … -2.502 908 483 …
10 1.401 155 041 … 4.669 201 155 …
11 -0.000 015 522 … -2.502 907 107 …
11 1.401 155 157 … 4.669 200 799 …
12 1.401 155 182 … 4.669 211 235 … 12 0.000 006 201 … -2.502 907 114 …
13 1.401 155 187 … 13 -0.000 002 477 …
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