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Emergent Dynamics of a Generalized Lohe Model on Some Class of Lie Groups
Ha, Seung-Yeal,Ko, Dongnam,Ryoo, Sang Woo Plenum Press 2017 Journal of statistical physics Vol.168 No.1
<P>We introduce a Lohe group which is a new class of matrix Lie groups and present a continuous dynamical system for the synchronization of group elements in a Lohe group. The Lohe group includes classical Lie groups such as the orthogonal, unitary, and symplectic groups, and since Lohe groups need not be compact, global existence of ODEs may fail. The proposed dynamical system generalizes the Lohe model (Lohe in J Phys A 43:465301, 2010; Lohe in J Phys A 42:395101-395126, 2009) itself a nonabelian generalization of the Kuramoto model, and alongside we also generalize the analytical framework (Ha and Ryoo in J Stat Phys 163:411-439, 2016) of emergent and unique phase-locked states. For the construction of the phase-locked states, we introduce Lyapunov functions measuring the ensemble diameter and the dissimilarity between two Lohe flows, and derive Gronwall-type differential inequalities for them. The global existence of solutions then become a consequence of the boundedness of these Lyapunov functions. Our sufficient framework for the emergent dynamics is formulated in terms of coupling strength and initial states, and it leads to the global existence of solutions and the formation and uniqueness of a phase-locked asymptotic state. As a concrete example, we demonstrate how our theory can show emergent phenomenon on the Heisenberg group, where all initial configurations tend to a unique phase-locked state exponentially fast.</P>
On the Emergence and Orbital Stability of Phase-Locked States for the Lohe Model
Ha, Seung-Yeal,Ryoo, Sang Woo Springer-Verlag 2016 Journal of statistical physics Vol.163 No.2
<P>We study the emergence and orbital stability of phase-locked states of the Lohe model, which was proposed as a non-abelian generalization of the Kuramoto phase model for synchronization. Lohe introduced a first-order system of matrix-valued ordinary differential equations for quantum synchronization and numerically observed the asymptotic formation and orbital stability of phase-locked states of the Lohe model. In this paper, we provide an analytical framework to confirm Lohe's observations of emergent phase-locked states. This extends earlier special results on lower dimensions to any finite dimension. For the construction and orbital stability of phase-locked states, we introduce Lyapunov functions to measure the ensemble diameter and dissimilarity between two Lohe flows, and using the time-evolution estimates of these Lyapunov functions, we present an admissible set of initial states, and show that an admissible initial state leads to a unique phase-locked asymptotic state.</P>
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].
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>