Jinyeong Park

Remarks on the complete synchronization of Kuramoto oscillators

We present an improved exponential frequency synchronization estimate for globally coupled Kuramoto oscillators. For a sufficiently large coupling, it is numerically observed that Kuramoto oscillators exhibit relaxation toward the phase-locked state, independent of the initial configuration. This phenomenon has never been confirmed analytically in full generality. To date, the analytical treatment of complete frequency synchronization is restricted to initial configurations that are geometrically confined to the half-unit circle. We extend this previous work (Choi et al 2012 Physica D 241 735–54 and Chopra and Spong 2009 IEEE Trans. Autom. Control 54 353–7) to a class of initial configurations lying on an arc of length greater than π by exploiting the dynamics of the Kuramoto order parameter in a finite-dimensional setting.

Interplay of inertia and heterogeneous dynamics in an ensemble of Kuramoto oscillators

We study the dynamic interplay between inertia and heterogeneous dynamics in an ensemble of Kuramoto oscillators. When external fields and internal forces are exerted on a system of Kuramoto oscillators, each oscillator has its own distinct dynamics, so that there is no notion of collective dynamics in the ensemble, and complete synchronization is not observed in such systems. In this paper, we study a relaxed version of synchronization, namely the “practical synchronization”, of Kuramoto oscillators, emerging from the dynamic interplay between inertia and heterogeneous decoupled dynamics. We will show that for some class of initial configurations and parameters, the fluctuation of phases and frequencies around the average values will be proportional to the inverse of the coupling strength. We provide several numerical examples, and compare these with our analytical results.

Synchronization of Kuramoto Oscillators with Adaptive Couplings

We study the synchronization of Kuramoto oscillators with adaptive coupling in interacting networks. Network dynamics preserves the sum of all incoming pairwise coupling strengths and is designed to adaptively interact with system dynamics. For adaptive couplings, we use two adaptive coupling laws for the pairwise coupling strength. Kuramoto oscillators are assumed to be on the nodes of the networks. We present frameworks that guarantee the emergence of synchronization for various coupling feedback laws. Our results generalize earlier work on the synchronization of Kuramoto oscillators in fixed and symmetric networks.

Emergent Dynamics of Kuramoto Oscillators with Adaptive Couplings: Conservation Law and Fast Learning

We study an emergent dynamics of the Kuramoto oscillators with adaptive couplings. In the Kuramoto model, pairwise coupling strengths are assumed to be constant and uniform over all interaction pai...