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Universal Scaling Behaviors in Symmetrically Coupled Systems [PPT]

  We generalize the critical scaling results of the period n-tuplings, intermittency, and quasiperiodicity for the case of low-dimensional systems to higher-dimensional symmetrically coupled systems. In particular, we develop a Reduced RG Method to make an analysis of the critical scaling behaviors of period doublings for the synchronous orbits in the coupled 1D maps, and found three kinds of critical scaling  behaviors. These critical scaling behaviors of period doublings in the  abstract system of the coupled maps are also examined in the real systems of the coupled oscillators. We thus suppose that the critical scaling  behaviors of period doublings in the symmetrically coupled systems are  universal ones. The reduced RG methods are also applied successfully to  the cases of the higher period n-tuplings (n=3,4,5,...) and intermittency. In future, employing the reduced RG method, we will investigate the critical scaling behaviors associated with the quasiperiodic transition to chaos in  the coupled circle maps. Furthermore, we also investigate critical behaviors of period doublings in coupled area-preserving maps, and find very rich scaling behaviors that are different from those in the coupled 1D maps. So far, only the case of synchronous orbits is considered. Hence, investigation of the scaling behaviors of period doublings for the asynchronous orbits will be also interesting. For more details, see the following publications:

[1] S.-Y. Kim, "Critical phenomena for period n-tuplings in 4-dimensional volume-preserving maps," J. Korean Phys. Soc. 22, 406-414 (1989).
[2] S.-Y. Kim and B. Hu, "Scaling pattern of period doubling in four dimensions," Phys. Rev. A 41, 5431-5440 (1990).
[3] S.-Y. Kim and H. Kook, "Critical behavior in coupled nonlinear system," Phys. Rev. A 46, R4467-R4470 (1992).
[4] S.-Y. Kim and H. Kook, "Renormalization analysis of two coupled maps," Phys. Lett. A 178, 258-264 (1993).
[5] S.-Y. Kim and H. Kook, "Period doubling in coupled maps," Phys. Rev. E 48, 785-799 (1993).
[6] S.-Y. Kim, "Universality of period doubling in coupled maps," Phys. Rev. E. 49, 1745-1748 (1994).
[7] S.-Y. Kim, "Critical behavior of period doubling in coupled area-preserving maps," Phys. Rev. E 50, 1922-1929 (1994).
[8] S.-Y. Kim, "Extension of Renormalization of period doubling in symmetric four dimensional volume-preserving maps," Phys. Rev. E 50, 4237-4240 (1994).
[9] S.-Y. Kim and K. Lee, "Period doublings in coupled parametrically forced damped pendulums," Phys. Rev. E 54, 1237-1252 (1996).
[10] S.-Y. Kim, "Period p-tuplings in coupled maps," Phys. Rev. E 54, 3393-3418 (1996).
[11] S.-Y. Kim, S.-H. Shin, J. Yi, and C. Jang, "Bifurcations in a parametrically forced magnetic pendulum," Phys. Rev. E 56, 6613-6619 (1997).
[12] S.-Y. Kim, "Symmetry and dynamics of a magnetic oscillator," J. Korean Phys. Soc. 32, 735-738 (1998).
[13] S.-Y. Kim and B. Hu, "Bifurcations and transitions to chaos in an inverted pendulum," Phys. Rev. E 58, 3028-3035 (1998).
[14] S.-Y. Kim and B. Hu, "Critical behavior of period doublings in coupled inverted pendulums," Phys. Rev. E 58, 7231-7241 (1998).
[15] S.-Y. Kim, "Intermittent transition to chaos in coupled maps," J. Korean Phys. Soc. 34, 75-78 (1999).
[16] S.-Y. Kim, "Intermittency in coupled maps," Phys. Rev. E 59, 2887-2901 (1999).
[17] S.-Y. Kim, "Renormalization analysis of intermittency in two coupled maps," Int. J. Mod. Phys. B 13, 283-292 (1999).
[18] S.-Y. Kim, "Critical scaling behavior in coupled magnetic oscillators," Int. J. Mod. Phys. B 13, 2405-2429 (1999).
[19] S.-Y. Kim and H. Kook, "Universality in coupled maps," in the proceeding of "The First International Workshop on Nonlinear Dynamics and Chaos," pp.49-90 (Pohang Institute of Science and Technology, Pohang, 1993).
[20] S.-Y. Kim, "Nonlinear dynamics in a parametrically forced damped pendulum," Sae Mulli (New Physics) 38, S33-S37 (1998) (in Korean).
[21] S.-Y. Kim, "Transition to chaos in coupled dynamical systems," Sae Mulli (New Physics) 38, S38-S43 (1998) (in Korean).

Bicritical Scaling Behaviors in Unidirectionally Coupled Systems [PPT]

  We investigate universal scaling behaviors of period doublings in  two unidirectionally coupled 1D maps near a bicritical point where two critical lines of period-doubling transitions to chaos in both subsystems meet. When crossing the bicritical point, corresponding to a border of  chaos in both subsystems, a hyperchaotic attractor with two positive Lyapunov  exponents appear. To analyze this bicritical scaling behavior, we develop an Eigenvalue-Matching RG Method, and find a new kind of non-Feigenbaum  scaling behaviors in the second response subsystem. These bicritical scaling  behaviors are also examined in unidirectionally coupled oscillators. We thus suppose that the bicriticality may occur generically in coupled systems consisting of period-doubling subsystems. For more details, see the following publications:

[1] S.-Y. Kim, "Bicritical behavior of period doublings in unidirectionally-coupled maps," Phys. Rev. E 59, 6585-6592 (1999).
[2] S.-Y. Kim and W. Lim, "Bicritical scaling behavior in unidirectionally coupled oscillators," Phys. Rev. E 63, 036223 (2001).
[3] S.-Y. Kim, W. Lim, and Y. Kim, "Universal bicritical behavior of period doublings in unidirectionally coupled oscillators," Prog. Theor. Phys. 106, 17-37 (2001).
[4] W. Lim and S.-Y. Kim, "Universal scaling behavior in unidirectionally coupled systems," Sae Mulli (New Physics) 39, 357-360 (1999) (in Korean).
[5] W. Lim and S.-Y. Kim, "Bicritical behavior in unidirectionally coupled systems," in the AIP proceeding of "Stochastic Dynamics and Pattern Formation in Biological and Complex Systems," pp. 317-323 (AIP, 2000).

Tricritical Behavior of Period Doublings in Unidirectionally Coupled Maps [PPT]

  We study the scaling behavior in two unidirectionally coupled one-dimensional maps near tricritical points which lie at ends of Feigenbaum critical lines and near edges of the complicated parts of the boundary of chaos. Note that both period-doubling cascades to chaos and multistability (associated with saddle-node bifurcations) occur in any neighborhood of the tricritical point. For this tricritical case, the response subsystem exhibits a type of non-Feigenbaum codimension-2 scaling behavior, while the drive subsystem is in a periodic state. To analyze the tricritical behavior, we develop an eigenvalue-matching renormalization-group (RG) method, and obtain the scaling factors. These RG results agree well with those of previous works.

[1] W. Lim and S.-Y. Kim, "Eigenvalue-Matching renormaligation analysis of tricritical behavior in unidirectionally coupled maps," J. Korean Phys. Soc. 48, s152 (2006).

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