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The classical Kuramoto model is studied in the setting of an infinite horizon mean field game. The system is shown to exhibit both syn- chronization and phase transition. Incoherence below a critical value of the interaction parameter is demonstrated by the stability of the uni- form distribution. Above this value, the game bifurcates and develops self-organizing time homogeneous Nash equilibria. As interactions get stronger, these stationary solutions become fully synchronized. Results are proved by an amalgam of techniques from nonlinear partial dif- ferential equations, viscosity solutions, stochastic optimal control and stochastic processes. ARTICLE HISTORY Received 23 October 2022 Accepted 17 June 2023 KEYWORDS Mean field games; Kuramoto model; synchronization; viscosity solutions 2020 MATHEMATICS SUBJECT CLASSIFICATION 35Q89; 35D40; 39N80; 91A16; 92B25 1. Introduction Originally motivated by systems of chemical and biological oscillators, the classical Kuramoto model [1] has found an amazing range of applications from neuroscience to Josephson junctions in superconductors, and has become a key mathematical model to describe self organization in complex systems. These autonomous oscillators are coupled through a nonlinear interaction term which plays a central role in the long time behavior of the system. While the system is unsynchronized when this term is not sufficiently strong, fascinatingly they exhibit an abrupt transition to self organization above a critical value of the interaction parameter. Synchronization is an emergent property that occurs in a broad range of complex systems such as neural signals, heart beats, fire-fly lights and circadian rhythms, and the Kuramoto dynamical system is widely used as the main phenomenological model. Expository papers [2, 3] and the references therein provide an excellent introduction to the model and its applications. The analysis of the coupled Kuramoto oscillators through a mean field game formalism is first explored by [4, 5] proving bifurcation from incoherence to coordination by a formal linearization and a spectral argument. [6] further develops this analysis in their application to a jet-lag recovery model. We follow these pioneering studies and analyze the Kuramoto model as a discounted infinite horizon stochastic game in the limit when the number of oscillators goes to infinity. We treat the system of oscillators as an infinite particle system, but instead of positing the dynamics of the particles, we let the individual particles endogenously determine their behaviors by minimizing a cost functional and hopefully, settling in a Nash equilibrium. Once the search for equilibrium is recast in this way, equilibria are given by solutions of nonlinear systems. Analytically, they are characterized by a backward dynamic CONTACT H. Mete Soner soner@princeton.edu Department of Operations Research and Financial Engineering, Prince- ton University, Princeton, NJ, 08540, USA. © 2023 Taylor & Francis Group, LLC
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Geometry unites synchrony, chimeras, and waves in nonlinear oscillator networks
One of the simplest mathematical models in the study of nonlinear systems is the Kuramoto model, which describes synchronization in systems from swarms of insects to superconductors. We have recently found a connection between the original, real-valued nonlinear Kuramoto model and a corresponding complex-valued system that permits describing the system in terms of a linear operator and iterative update rule. We now use this description to investigate three major synchronization phenomena in Kuramoto networks (phase synchronization, chimera states, and traveling waves), not only in terms of steady state solutions but also in terms of transient dynamics and individual simulations. These results provide new mathematical insight into how sophisticated behaviors arise from connection patterns in nonlinear networked systems.
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- Award ID(s):
- 2015276
- PAR ID:
- 10588624
- Publisher / Repository:
- American Institute of Physics
- Date Published:
- Journal Name:
- Chaos: An Interdisciplinary Journal of Nonlinear Science
- Volume:
- 32
- Issue:
- 3
- ISSN:
- 1054-1500
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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