Emergent behaviors of Kuramoto model with frustration under switching topology

In this paper, we study the emergent behavior of Kuramoto model with switching topology under the effect of uniform frustration. In our frameworks, the switching interaction topology contains a spanning tree in any switching mode. For the initial configuration distributed in an open half circle, we first exploit a similar procedure in [T. Zhu, Netw. Heterog. Media, 17(2):255–291, 2022] to conclude that the Kuramoto oscillators will be pushed into a small region at some instant before the first network switching. Then in a large coupling and small frustration regime, we lift the Kuramoto model to the second-order formulation and apply the matrix theory-based approach in [J.-G. Dong, S.-Y. Ha and D. Kim, Anal. Appl., 19(2):305–342, 2021] to derive the exponential fast frequency synchronization.

Keywords

synchronization, Kuramoto model, frustration, switching topology, spanning tree

2010 Mathematics Subject Classification

34C15, 34D06, 34K33, 93C15

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The work of T. Zhu is supported by the Talent Fund of Hefei University, China (Grant no. 21-22RC23), by the National Natural Science Foundation of China (Grant no. 12201172), and by the Natural Science Foundation for Colleges and Universities in Anhui Province, China (Grant nos. 2022AH051790 and KJ2021A0996).

Received 21 August 2021

Received revised 14 May 2022

Accepted 6 June 2022

Published 1 February 2023