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Delay induced double explosive transition in a swarmalator system

Carmel T. Lambu1,2,*, Romuald T. Mbonwouo1,2, Gael R. Simo2,3, Delors A. Jiofack2,4, Steve J. Kongni1,2,†, Patrick Louodop1,2,5,6, and Hilda A. Cerdeira7

  • 1Research Unit Condensed Matter, Electronics and Signal Processing, University of Dschang, P.O. Box 67, Dschang, Cameroon
  • 2MoCLiS Research Group, Dschang, Cameroon
  • 3Laboratory of Electrotechnics, Automatics and Energy, Department of Maintenance, Higher Technical Teachers, Training College (HTTC) of Ebolowa, University of Ebolowa, Cameroon
  • 4Department of Physics - FFCLRP, University of São Paulo (USP), 14040-901 Ribeirão Preto - SP, Brazil
  • 5ICTP South American Institute for Fundamental Research, São Paulo State University (UNESP), Instituto de Física Teórica, Rua Dr. Bento Teobaldo Ferraz 271, Bloco II, Barra Funda, 01140-070 São Paulo, Brazil
  • 6Potsdam Institute for Climate Impact Research (PIK), Member of the Leibniz Association, P.O. Box 60 12 03, D-14412 Potsdam, Germany
  • 7São Paulo State University (UNESP), Instituto de Física Teórica, Rua Dr. Bento Teobaldo Ferraz 271, Bloco II, Barra Funda, 01140-070 São Paulo, Brazil and Epistemic, Gomez & Gomez Ltda. ME, Rua Paulo Franco 520, Vila Leopoldina, 05305-031 São Paulo, Brazil

  • *Contact author: lambucarmel@gmail.com
  • †Contact author: kongnisteve@yahoo.fr
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Phys. Rev. E 113, 024203 – Published 4 February, 2026

DOI: https://doi.org/10.1103/hhzm-zfxt

Abstract

Synchronization and aggregation are important phenomena in the world, and many systems exhibit them. The swarmalator system is a good example, arousing interest from many scientists recently. Despite its rising interest, many do not consider delay of any form. Knowing that delay makes the system more realistic, surely containing rich dynamics, we use a delay term in the internal phase dynamics. Spanning through the various parameters and domains of this model, with the order parameter in its local and complex form, shows very interesting dynamics like the boiling chimera state in various domains and the ring static sync, which are primer discoveries in the field of delayed swarmalators. Also, the existence of phase delay helps us highlight a double explosive transition—first from synchronous to asynchronous and, secondly, from asynchronous to synchronous states, which have been used to understand some brain diseases. In this study, we lay down an additional base to the understanding of delay where environmental factors account for it, and so, we pave the way for more studies considering the reality of the environment on mathematical models.

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Phys. Rev. E 113, 024203– Published 4 February, 2026
  • Received 26 May 2025
  • Accepted 19 December 2025
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