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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.
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(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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