Angular Frequency Synchronization and Localized Dynamics in Symmetrically Coupled Oscillators

Christian Bick (Exeter)

Frank Adams 1,

The emergence of collective behavior is a fascinating feature of interacting
oscillatory units in nature and technology. Apart from full synchronization,
solutions where a localized subset of oscillators are synchronized have at-
tracted an enormous amount of attention recently. Weak chimerasoriginally
de ned for networks of phase oscillators where the state of each oscillator is
given by a single phase-like variableprovide a rigorous notion to describe such
dynamics in terms of frequency synchronization along trajectories. We ex-
tend the de nition to more general classes of oscillators by relating frequency
synchronization to the symmetry properties of the system. Moreover, we dis-
cuss some persistence results for weak chimeras in coupled phase oscillators.
In particular, we explicitly give coupling functions which give rise to chaotic
weak chimeras for which the underlying dynamically invariant sets have trivial
or nontrivial symmetries.

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