<p> Kuramoto oscillators coupled through a graph provide one of the most influential models for the study of collective synchronizations. We propose the first sufficient synchronization conditions for lattice models. This sufficient condition is optimal as an <i>L</i><sup>∞</sup> synchronization condition. We also propose a novel continuum limit of the Kuramoto oscillators on lattices, by viewing the lattice as a discretization of the space. In the continuous model we have an analogous synchronization condition. To prove the continuous (discrete) synchronization conditions, we show the existence of solutions to some (discrete) elliptic PDE of divergence form. The two main ingredients in the proof are variational methods and gradient estimates for (discrete) elliptic PDE's.</p><p>
| Identifer | oai:union.ndltd.org:PROQUEST/oai:pqdtoai.proquest.com:13421111 |
| Date | 02 April 2019 |
| Creators | Wu, Tianqi |
| Publisher | New York University |
| Source Sets | ProQuest.com |
| Language | English |
| Detected Language | English |
| Type | thesis |
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