Hysteretic transitions and chaotic chimera states in networks of Kuramoto oscillators with inertia

We report finite size numerical investigations and mean field analysisof a Kuramoto model with inertia for fully coupled and diluted sys-tems. In particular, we examine for a Gaussian distribution of the fre-quencies the transition from incoherence to coherence for increasinglylarge system size and inertia. For sufficiently large inertia the tran-sition is hysteretic and within the hysteretic region clusters of lockedoscillators of various sizes and different levels of synchronization co-exist. A modification of the mean field theory developed by Tanaka,Lichtenberg, and Oishi [Physica D, 100 (1997) 279] allows to derivethe synchronization profile associated to each of these clusters. Byincreasing the inertia the transition becomes more complex, and thesynchronization occurs via the emergence of clusters of whirling oscil-lators. The presence of these groups of coherently drifting oscillatorsinduces oscillations in the order parameter. We have shown that thetransition remains hysteretic even for randomly diluted networks upto a level of connectivity corresponding to few links per oscillator. Fi-nally an extension to a system of two symmetrically coupled networksof Kuramoto oscillators with inertia is reported. In this system theexistence and the dynamical properties of novel chaotic chimera statesare investigated, concentrating both on the microscopic dynamics andthe macroscopic behavior.