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Explosive entrainment in Arnold tongues

Jiangsheng Wang1,2,3, Changgui Gu1,*, Wei Zou4, Haiying Wang1, Huijie Yang1, Ludovico Minati5,6,7,†, and Stefano Boccaletti8,9,10

  • *Contact author: gu_changgui@163.com
  • †Contact author: lminati@uestc.edu.cn; lminati@ieee.org
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Phys. Rev. Research 7, 043156 – Published 12 November, 2025

DOI: https://doi.org/10.1103/9crr-yd6r

Abstract

Entrainment refers to the unidirectional synchronization of an oscillator system to external periodic forcing, with phase-locking regions forming Arnold tongues. In some specific models, the hysteresis phenomenon of Arnold tongue has been observed, but methods to actively induce it across models are still lacking. In this study, we show that adaptive responses to periodic forcing can induce explosive entrainment, a phenomenon of sudden locking to an external frequency accompanied by significant hysteresis. In contrast to explosive synchronization, where synchronization emerges suddenly due to internal coupling, explosive entrainment describes the sudden locking to a given frequency caused by external forcing. Therefore, explosive entrainment is more likely to occur in systems where internal couplings are weak or difficult to estimate, but external forcing can be applied. Furthermore, we demonstrate the universal emergence of explosive entrainment in the Kuramoto model, Poincaré model, chaotic Rössler model, and complex Ginzburg-Landau equation. Theoretical hysteresis bounds in the Kuramoto model are determined based on analytical results of the Ott-Antonsen reduction method. This study provides ideas for actively inducing hysteresis through external forcing.

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Phys. Rev. Research 7, 043156– Published 12 November, 2025
  • Received 27 April 2025
  • Accepted 7 October 2025
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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