# Remarks on the complete synchronization for the Kuramoto model with frustrations

[Analysis and Applications, published in online]

This is a joint work with Seung-Yeal Ha and Hwa Kil Kim.

The synchronous dynamics of many limit-cycle oscillators can be described by phase models. The Kuramoto model serves as a prototype model for phase synchronization and has been extensively studied in the last 40 years. In this paper, we deal with the complete synchronization problem of the Kuramoto model with frustrations on a complete graph. We study the robustness of complete synchronization with respect to the network structure and the interaction frustrations, and provide sufficient frameworks leading to the complete synchronization, in which all frequency differences of oscillators tend to zero asymptotically. For a uniform frustration and unit capacity, we extend the applicable range of initial configurations for the complete synchronization to be distributed on larger arcs than a half circle by analyzing the detailed dynamics of the order parameters. This improves the earlier results [S.-Y. Ha, H. Kim and J. Park, Remarks on the complete frequency synchronization of Kuramoto oscillators, Nonlinearity, 28 (2015) 1441–1462; Z. Li and S.-Y. Ha, Uniqueness and well-ordering of emergent phase-locked states for the Kuramoto model with frustration and inertia, Math. Models Methods Appl. Sci. 26 (2016) 357–382.] which can be applicable only for initial configurations confined in a half circle.

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

[Analysis and Applications, 15 (2017), 837.]

This is a joint work with Seung-Yeal Ha and Se Eun Noh.
(To see the Kuramoto model)

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.

# On the global well-posedness of BV weak solutions to the Kuramoto-Sakaguchi equation

[Journal of Differential Equations, published in Journal of Differential Equations, 262 (2017), no. 2, 978 – 1022.]

This is a joint work with Debora Amadori and Seung-Yeal Ha.

The Kuramoto model is a prototype phase model describing the synchronous behavior of weakly coupled limit-cycle oscillators. When the number of oscillators is sufficiently large, the dynamics of Kuramoto ensemble can be effectively approximated by the corresponding mean-field equation, namely “the Kuramoto–Sakaguchi (KS) equation”. This KS equation is a kind of scalar conservation law with a nonlocal flux function due to the mean-field interactions among oscillators. In this paper, we provide a unique global solvability of bounded variation (BV) weak solutions to the kinetic KS equation for identical oscillators using the method of front-tracking in hyperbolic conservation laws. Moreover, we also show that our BV weak solutions satisfy local-in-time $L^1$-stability with respect to BV-initial data. For the ensemble of identical Kuramoto oscillators, we explicitly construct an exponentially growing BV weak solution generated from BV perturbation of incoherent state for any positive coupling strength. This implies the nonlinear instability of incoherent state in a positive coupling strength regime. We provide several numerical examples and compare them with our analytical results.

# Emergence of partial locking states from the ensemble of Winfree oscillators

[Published in Quarterly of Applied Mathematics, 75 (2017), 39 – 68.]

This is a joint work with Seung-Yeal Ha, Dongnam Ko, and Sang Woo Ryoo.

(To see the Winfree model)

We study the emergence of partial locking states for a subsystem whose dynamics is governed by the Winfree model. The Winfree model is the first mathematical model for synchronization. Thanks to the lack of conservation laws except for the number of oscillators, it exhibits diverse asymptotic nonlinear patterns such as partial and complete phase locking, partial and complete oscillator death, and incoherent states. In this paper, we present two sufficient frameworks for a majority sub-ensemble to evolve to the phase-locked state asymptotically. Our sufficient frameworks are characterized in terms of the mass ratio of the subsystem compared to the total system, ratio of the coupling strength to the natural frequencies, and the phase diameter of the subsystem. We also provide several numerical simulations and compare their results to the analytical results.

# Collective synchronization of classical and quantum oscillators

[Published in EMS Surveys in Mathematical Sciences, 3 (2016), no. 2, 209 – 267.]

This is a joint work with Seung-Yeal Ha, Dongnam Ko, and Xiongtao Zhang.

Synchronization of weakly coupled oscillators is ubiquitous in biological and chemical complex systems. Recently, research on collective dynamics of many-body systems has been received much attention due to their possible applications in engineering. In this survey paper, we mainly focus on the large-time dynamics of several synchronization models and review state-of-art results on the collective behaviors for synchronization models. Following a chronological order, we begin our discussion with two classical phase models (Winfree and Kuramoto models), and two quantum synchronization models (Lohe and Schrödinger–Lohe models). For these models, we present several sufficient conditions for the emergence of synchronization using mathematical tools from dynamical systems theory, kinetic theory and partial differential equations in a unified framework.