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Critical Scaling Behaviors of Period nTuplings in
AreaPreserving Maps
Using both a direct numerical method and a
renormalizationgroup method, we studied the critical scaling behaviors
associated with
Multifurcations through Period nTuplings
(n=3, 4, 5, ...) in areapreserving maps. At the accumulation point of each
period ntuplings, an infinitely nested islands of all classes exist and they
exhibit an asymptotically selfsimilar structure. Unlike the perioddoubling
(n=2) case, these islands play the role "trap," near which trajectories may have
longtime correlations. It is thus found that the pattern of islands repeat
itself asymptotically from one class k to the next class k+1 for even ntuplings
and to every other class k+2 for odd ntuplings. Furthermore, we also studied
the global scaling of the asymptotically selfsimilar pattern by obtaining the
power spectrum, $f(\alpha)$ spectrum, and the generalized dimensions.
For more details, see the following publications:
[1] K.C. Lee, S.Y.Kim, and D.I. Choi, "Universality of $k 3^n$ and $k 4^n$
bifurcations in areapreserving maps," Phys. Lett. A 103, 225228 (1984).
[2] K.C. Lee, S.Y. Kim, and D.I. Choi, "Scaling behavior of period ntupling
bifurcations with high n in areapreserving maps," J. Korean Phys. Soc. 18,
243247 (1985).
[3] S.Y. Kim, K.C. Lee, and D.I. Choi, "Renormalization analysis of
m/nbifurcations and invariant curves in areapreserving maps," J. Korean Phys.
Soc. 19, 249261 (1986).
[4] S.Y. Kim and B. Hu, "Singularity spectrum for period ntupling in
areapreserving maps," Phys. Rev. A 38, 15341537 (1988).
[5] B. Hu, J. Shi, and S.Y. Kim, "Power spectra of higher period multipling in
areapreserving maps," Phys. Lett. A 140, 158160 (1989).
[6] K.C. Lee, S.Y. Kim, and D.I. Choi, "Bifurcations in 2D areapreserving
maps," in the proceeding of the 6th Kyoto summer institute, edited by Y.
Kuramoto, pp. 170174 (SpringerVerlag, NewYork, 1984).
[7] K.C. Lee, S.Y. Kim, and D.I. Choi, "Scaling behaviors in the resonance
multifurcations in 2D areapreserving maps," in the proceeding of the 14th
ICGTMP, edited by Y. M. Cho, pp. 425427 (World Scientific, Singapore, 1986).
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