Hysteretic transitions in the Kuramoto model with inertia

Abstract

We report finite size numerical investigations and mean field analysis of a Kuramoto model with inertia for fully coupled and diluted systems. In particular, we examine for a Gaussian distribution of the frequencies the transition from incoherence to coherence for increasingly large system size and inertia. For sufficiently large inertia the transition is hysteretic and within the hysteretic region clusters of locked oscillators of various sizes and different levels of synchronization coexist. A modification of the mean field theory developed by Tanaka, Lichtenberg, and Oishi [Physica D, 100 (1997) 279] allows to derive the synchronization profile associated to each of these clusters. We have also investigated numerically the limits of existence of the coherent and of the incoherent solutions. The minimal coupling required to observe the coherent state is largely independent of the system size and it saturates to a constant value already for moderately large inertia values. The incoherent state is observable up to a critical coupling whose value saturates for large inertia and for finite system sizes, while in the thermodinamic limit this critical value diverges proportionally to the mass. By increasing the inertia the transition becomes more complex, and the synchronization occurs via the emergence of clusters of whirling oscillators. The presence of these groups of coherently drifting oscillators induces oscillations in the order parameter. We have shown that the transition remains hysteretic even for randomly diluted networks up to a level of connectivity corresponding to few links per oscillator. Finally, an application to the Italian high-voltage power grid is reported, which reveals the emergence of quasi-periodic oscillations in the order parameter due to the simultaneous presence of many competing whirling clusters.

34 pages, 18 figures, submitted to Physical Review E

References in corpus (6)

• Synchronization in complex networks
• Low Dimensional Behavior of Large Systems of Globally Coupled Oscillators
• Analysis of a power grid using the Kuramoto-like model
• Partially integrable dynamics of hierarchical populations of coupled oscillators
• Entrainment transition in populations of random frequency oscillators
• Mean field theory of assortative networks of phase oscillators

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• Power-law distributions of dynamic cascade failures in power-grid models
• Stability of rotatory solitary states in Kuramoto networks with inertia
• Solitary states in the mean-field limit
• Stochastic Kuramoto oscillators with discrete phase states
• Nonlinear transient waves in coupled phase oscillators with inertia
• Continuous and discontinuous transitions to synchronization
• Modelling power grids as pseudo adaptive networks
• Dynamics of coupled oscillator systems in presence of a local potential
• Chimera states for directed networks
• Kinetic description of one-dimensional stochastic dynamics with small inertia
• Path-dependent Dynamics Induced by Rewiring Networks of Inertial Oscillators
• Synchronization of coupled second-order Kuramoto-Sakaguchi oscillators
• Oscillatory states, standing waves, and additional synchronized clusters in networks of second-order oscillators: uncovering the role of inertia