Emergent Behaviors of Thermodynamic Kuramoto Ensemble on a Regular Ring Lattice
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Emergent Behaviors of Thermodynamic Kuramoto Ensemble on a Regular Ring Lattice Seung-Yeal Ha1,2 · Hansol Park3 · Tommaso Ruggeri4 · Woojoo Shim3 Received: 3 February 2020 / Accepted: 6 July 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract The temporal evolution of Kuramoto oscillators influenced by the temperature field often appears in biological oscillator ensembles. In this paper, we propose a generalized Kuramoto type lattice model on a regular ring lattice with the equal spacing assuming that each oscillator has an internal energy (temperature). Our lattice model is derived from the thermodynamical Cucker–Smale model for flocking on the 2D free space under the assumption that the ratio between velocity field and temperature field at each lattice point has a uniform magnitude over lattice points. The proposed model satisfies an entropy principle and exhibits emergent dynamics under some sufficient frameworks formulated in terms of initial data and system parameters. Moreover, the phase-field tends to the Kuramoto phase-field asymptotically. Keywords Emergence · Entropy principle · Kuramoto model · Thermodynamics Mathematics Subject Classification 82C10 · 82C22 · 35B37
Communicated by Michael Kiessling.
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Woojoo Shim [email protected] Seung-Yeal Ha [email protected] Hansol Park [email protected] Tommaso Ruggeri [email protected]
1
Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul 08826, Republic of Korea
2
Korea Institute for Advanced Study, Hoegiro 85, 02455 Seoul, Republic of Korea
3
Department of Mathematical Sciences, Seoul National University, Seoul 08826, Republic of Korea
4
Department of Mathematics and Alma Mater Research Center on Applied Mathematics AM2, University of Bologna, Bologna, Italy
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S-Y. Ha et al.
1 Introduction Synchronization of phase-coupled oscillators often appears in our biological and chemical complex oscillatory systems [1,4,10,30,31,44]. In spite of its ubiquity in our nature, a rigorous mathematical study based on mathematical models has been initiated by two pioneers Arthur Winfree [43] and Yoshiki Kuramoto [23] about a half-century ago. In this paper, we present a generalized Kuramoto lattice model on a ring lattice for the phase evolution of Kuramoto oscillators assuming that each oscillator has its own internal temperature, i.e., a coupled phase-temperature model on the ring lattice. For the mathematical modeling for such a situation, we consider a finite size ensemble of Kuramoto oscillators (or rotators) on the unit circle under the influence of a temperature field. Interactions between Kuramoto oscillators and temperature fields were already discussed in biology literature, e.g., [29,36,45] without any explicit mathematical models. We can reasonably argue that the synchronizability of Kuramoto oscillators will be inversely proportional to the size of temperature, i.e., as the temperature becomes higher, the synchronizability of oscillator ensemble will b
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