
Institute for Computational Neuroscience 
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Universal Scaling Behaviors in Symmetrically Coupled Systems [PPT] We generalize the critical scaling results of the period ntuplings, intermittency, and quasiperiodicity for the case of lowdimensional systems to higherdimensional 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 ntuplings (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 areapreserving 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 ntuplings
in 4dimensional volumepreserving maps," J. Korean Phys. Soc. 22, 406414
(1989). 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 perioddoubling
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 EigenvalueMatching RG Method,
and find a new kind of nonFeigenbaum
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 perioddoubling subsystems. For more details, see the following publications: ¡¡ Tricritical Behavior of Period Doublings in Unidirectionally Coupled Maps [PPT] We study the scaling behavior in two unidirectionally coupled onedimensional 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 perioddoubling cascades to chaos and multistability (associated with saddlenode bifurcations) occur in any neighborhood of the tricritical point. For this tricritical case, the response subsystem exhibits a type of nonFeigenbaum codimension2 scaling behavior, while the drive subsystem is in a periodic state. To analyze the tricritical behavior, we develop an eigenvaluematching renormalizationgroup (RG) method, and obtain the scaling factors. These RG results agree well with those of previous works. [1] W. Lim and S.Y. Kim, "EigenvalueMatching renormaligation analysis of tricritical behavior in unidirectionally coupled maps," J. Korean Phys. Soc. 48, s152 (2006). ¡¡ 
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