Institute for Computational Neuroscience

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Collective Dynamics in an Ensemble of Globally Coupled Chaotic Systems

(1) Dynamical Route to Clustering and Scaling in an Ensemble of Globally Coupled Chaotic Systems [PPT]
  As a representative model for period doublings, we consider the logistic map, and study dynamical routes to clustering (or partial synchronization) in an ensemble of globally coupled logistic maps. By varying the nonlinearity parameter $a$ and the coupling parameter $c$, scaling for the route to periodic two-cluster states, associated with the period-doubling cascade of the logistic map, is particularly investigated in a reduced 2D map governing the two-cluster dynamics. For the case of symmetric distribution of elements between the two clusters, a dynamical route to two-cluster states, related to appearance of asynchronous anti-phase and conjugate-phase orbits, is found through a bifurcation analysis. Based on the renormalization results on the scaling, it is shown that similar cluster states of higher orders appear successively, as the set of parameters $(a,c)$ approaches the zero-coupling critical point $(a_\infty,0)$ $(a_\infty$: accumulation point of the period-doubling cascade of the logistic map). The effect of asymmetric distribution of elements on such dynamical routes to clusters are also discussed. The role of conjugate-phase orbits is thus found to become dominant as the distribution becomes more asymmetric. These results are also confirmed in an ensemble of globally coupled pendula and Rossler oscillators.

(2) Onset of Coherence and Scaling in an Ensemble of Globally Coupled Chaotic Systems [PPT]
 As a representative model for period-doubling systems, we consider the logistic map and study the onset of coherence in an ensemble of globally coupled noisy logistic maps. As the coupling strength passes a threshold value, a transition from an incoherent state with a zero mean field to a coherent state with a macroscopic mean field occurs. Scaling, associated with the onset of coherence, is particularly investigated by varying the nonlinearity parameter $a$, the coupling parameter $\varepsilon$, and the noise intensity $\sigma$. Based on the renormalization results on the scaling, it is shown that similar dynamical transitions of higher orders occur successively as the set of parameters $(a, \varepsilon, \sigma$) approaches the zero-coupling critical point $(a_\infty, 0, 0)$, where $a_\infty$ is the accumulation point of the period-doubling cascade of the logistic map. In addition to the noisy case, we also study scaling for the transition to coherence in the case of the parameter spread (i.e., distributed parameters). These results are confirmed in an ensemble of globally coupled noisy pendula
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