EMS Surveys in Mathematical Sciences


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Volume 3, Issue 2, 2016, pp. 209–267
DOI: 10.4171/EMSS/17

Collective synchronization of classical and quantum oscillators

Seung-Yeal Ha[1], Dongnam Ko[2], Jinyeong Park[3] and Xiongtao Zhang[4]

(1) Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, GwanAkRo 1, Gwanak-Gu, 08826, SEOUL, SOUTH KOREA
(2) Department of Mathematical Sciences, Seoul National University, GwanAkRo 1, Gwanak-Gu, 08826, SEOUL, SOUTH KOREA
(3) Departamento de Matemática Aplicada, Universidad de Granada, Campus de Fuentenueva, 18071, GRANADA, SPAIN
(4) Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, GwanAkRo 1, Gwanak-Gu, 08826, SEOUL, SOUTH KOREA

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.

Keywords: Kuramoto oscillators, large-time dynamics, Lohe oscillators, synchronization and Winfree oscillators

Ha Seung-Yeal, Ko Dongnam, Park Jinyeong, Zhang Xiongtao: Collective synchronization of classical and quantum oscillators. EMS Surv. Math. Sci. 3 (2016), 209-267. doi: 10.4171/EMSS/17