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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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