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Interplay of inertia and heterogeneous dynamics in an ensemble of Kuramoto oscillators
Ha, Seung-Yeal,Noh, Se Eun,Park, Jinyeong WORLD SCIENTIFIC 2017 ANALYSIS AND APPLICATIONS Vol.15 No.6
<P>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 “<I>practical synchronization</I>”, 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.</P>
On the basin attractions to the Kuramoto system with unidirectionally coupled ring topology
Seung-Yeal Ha,Moon-Jin Kang 한국산업응용수학회 2010 한국산업응용수학회 학술대회 논문집 Vol.5 No.1
In this short note, we discuss the basin of attractions for many Kuramoto oscillators with unidirectional couplings. For attractive and repulsive couplings, we present proper subsets for the basin of attractions, which are independent of the number of oscillators.
Euler-Poisson system and its application to plasma physics
Seung-Yeal Ha 한국산업응용수학회 2006 한국산업응용수학회 학술대회 논문집 Vol.1 No.1
In this talk, I will present a mathematical modelling of the plasma sheath using the Euler-Poisson system arising from plasma physics such as semiconductor industry. I will derive a hyperbolic type interface system using axiomatic approach and show the local existence of approximate interface system. This is ajoint work with Mikhail Feldman and Marshall Slemrod [5].
The Boltzmann equation near a local Maxwellian
Seung-Yeal Ha,Seok-Bae Yun 한국산업응용수학회 2005 한국산업응용수학회 학술대회 논문집 Vol.- No.-
We present two robust nonlinear functionals measuring future possible collisions and L¹-distance between two mild solutions to the Boltzmann equation near a close-to-a local Maxwellian regime. Through the explicit local cancellation of the collision mechanism due to the special structure of a local Maxwellian, we show that the nonlinear functionals in [4] satisfy stability estimates. These functionals can be employed to the study of L¹-scattering and uniform L¹-stability estimate of mild solutions for the moderately soft interaction potential.
Emergence of partial locking states from the ensemble of Winfree oscillators
Ha, Seung-Yeal,Ko, Dongnam,Park, Jinyeong,Ryoo, Sang Woo Brown University, Division of Applied Mathematics 2017 Quarterly of applied mathematics Vol.75 No.1
<P>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.</P>