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Synchronism in Large Networks of Oscillators
Large networks of
many coupled oscillators have been used as models for phenomena in
diverse systems. Examples include flashing fire flies, control of
animals' circadian rhythm by the suprachiasmatic nucleus, neuronal
dynamics, synchronization of cardiac pace-maker cells, animals that
rest with half their brain in the sleep state, chemical oscillations,
oscillations in bubbly fluids, synchronization of pedestrians walking
on foot bridges, clapping audiences, Josephson junction circuits,
arrays of coupled lasers, and many others. This talk will demonstrate
and illustrate a recently developed, very general technique  that
reduces the study of the long time temporal behavior of these models
to that of low dimensional dynamical systems (e.g., to a small number
of ODEs). In particular, it is shown  that all the possible
long-time behaviors and their bifurcations are captured by this
technique. Discussion of the basic mechanism responsible for this
reduction and of extensions (e.g., spatio-temporal patterning models)
will also be presented.
 E.Ott and T.M.Antonsen, Chaos (2008).
 E.Ott and T.M.Antonsen, Chaos (2009).