Institute for Computational Neuroscience

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Critical Scaling Behaviors of Period n-Tuplings in Area-Preserving Maps

Using both a direct numerical method and a renormalization-group method, we studied the critical scaling behaviors associated with Multifurcations  through Period n-Tuplings (n=3, 4, 5, ...) in area-preserving maps. At the accumulation point of each period n-tuplings, an infinitely nested islands of all classes exist and they exhibit an asymptotically self-similar structure. Unlike the period-doubling (n=2) case, these islands play the role "trap," near which trajectories may have long-time 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 n-tuplings and to every other class k+2 for odd n-tuplings. Furthermore, we also studied the global scaling of the  asymptotically self-similar 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 area-preserving maps," Phys. Lett. A 103, 225-228 (1984).
[2] K.-C. Lee, S.-Y. Kim, and D.-I. Choi, "Scaling behavior of period n-tupling bifurcations with high n in area-preserving maps," J. Korean Phys. Soc. 18, 243-247 (1985).
[3] S.-Y. Kim, K.-C. Lee, and D.-I. Choi, "Renormalization analysis of m/n-bifurcations and invariant curves in area-preserving maps," J. Korean Phys. Soc. 19, 249-261 (1986).
[4] S.-Y. Kim and B. Hu, "Singularity spectrum for period n-tupling in area-preserving maps," Phys. Rev. A 38, 1534-1537 (1988).
[5] B. Hu, J. Shi, and S.-Y. Kim, "Power spectra of higher period multipling in area-preserving maps," Phys. Lett. A 140, 158-160 (1989).
[6] K.-C. Lee, S.-Y. Kim, and D.-I. Choi, "Bifurcations in 2D area-preserving maps," in the proceeding of the 6th Kyoto summer institute, edited by Y. Kuramoto, pp. 170-174 (Springer-Verlag, New-York, 1984).
[7] K.-C. Lee, S.-Y. Kim, and D.-I. Choi, "Scaling behaviors in the resonance multifurcations in 2D area-preserving maps," in the proceeding of the 14th ICGTMP, edited by Y. M. Cho, pp. 425-427 (World Scientific, Singapore, 1986).

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