ENREDANDO 2026– UNIVERSIDADE DE VIÇOSA 2026, MG, BRAZIL

26 – 31 JULY 2026

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Dynamics of Systems of Swarmalators

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Hilda A. Cerdeira

Instituto de Física Teórica –IFT UNESP

and ICTP-SAIFR

São Paulo, BRAZIL

WHAT ARE SWARMALATORS?

Swarmalators represent a system of oscillators whose phase and spatial dynamics are coupled and they are used to describe the dynamics of living systems

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Xavi Bou

SWARMALATORS PROPERTIES

• Decentralized Control: No central authority governs agent behavior.
• Self-Organization: Agents autonomously organize and respond to environmental changes without external control.
• Stigmergy: Indirect communication where agents modify their environment without external control.
• Emergent Behavior: Complex global patterns and solutions arise from local interactions among agents.

• Sensitivity analyses play a crucial to understand the algorithm performance:
• Parameter Sensitivity.
• Initial Condition Sensitivity

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THE MODEL

EXEMPLO DE TEXTO DE RODAPÉ

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and

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a) Static Async; b) Active Phase Wave; c) Splintered Phase Wave; d) Static Phase Wave; e) Static Sync

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With the help of the Kuramoto Order Parameter and the Internal and

Spatial Phase correlation, we identified the transition to Synchronization

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We study the effect that the internal dynamics has on the spatial distribution of the swarmalators and vice versa. We find that the expected explosive synchronization, a characteristic of the transition to synchronization in many networks is not a necessary condition of these system. After studying these transitions we discovered several other phases which we will present in what follows.

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Near synchronization the distance between swarmalators is practically constant and the energy can be approximated by:

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Above synchronization, the energy can be approximated by:

New Phases before complete synchronization:

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Splintered Phase Wave to Static Wing Phase Wave

both of them rotating phases.

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Explosive or non Explosive Synchronization?

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1. Cuesta and Sanchez essentially lifted the ban imposed by the theorem of Van Hove.
2. Hong and Martens studied phase transitions in an XY model related to a variant of the Kuramoto model for coupled oscillators. They found that for the case without noise, the system shows features of a first-order phase transition at complete synchronization, while this transition is found to be continuous for the noisy case.
3. Gomez Gardeñes et al. have shown that there is an explosive transition in a scale free network of Kuramoto oscillators when the natural frequency and the degree of a node are equal.
4. Skardal et al. observed that adding infinitesimal disorder induces explosive transition to synchronization for the case discussed in (b).
5. Leyva et al extended results when natural frequencies cannot be equal between themselves or to the mean.

J.A. Cuesta and A. Sanchez, J. Stat. Phys. 115, 869 (2004)

H. Hong and E. A. Martens, Chaos 32, 063125 (2022).�J. Gómez-Gardeñes, S. Gómez, A. Arenas, and Y. Moreno, Phys. Rev. Lett. 106, 128701 (2011).�P. S. Skardal and A. Arenas, Phys. Ver. E 89, 062811 (2014).

I. Leyva, A. Navas, I. Sendinha Nadal, J. A. Almendral, J. M Buldú, M. Zanin, D. Papo, S. Boccaletti, Sci. Rep. 3, 1281 (2013).

Case of Two Layers:

Snapshots of all internal phases (Upper layer) and nodes positions (lower layer) showing the changes of cluster positions for diferente times when frequencies are distributed in two groups. (Rotating Splintered Phase Wave, spatial and phase rotations.)

Swarmalators with delayed internal phases

HERE =

Double Explosive Synchronization appears in delayed systems

EXEMPLO DE TEXTO DE RODAPÉ

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SA

SpPW

Boiling State

Boiling Chimera State

BS/ Phases distribution

BCS/ Phases distribution

A POSSIBLE SOLUTION TO THE PROBLEM:

CALCULATION OF THE TOPOLOGICAL CHARGE AS A FUNCTION OF THE COUPLING CONSTANT

Topological charge

Helicity

Global Helicity Variance

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Original case near the

transition to synchronization

Case of two layers

Two layers

Blum et al varying

delay time

DELAYED SYSTEM LAMBU ET AL.

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EXEMPLO DE TEXTO DE RODAPÉ

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Lambu et al., varying K

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Therefore we have presented some parameters which we have taken from the theory of turbulences, to complete the description

We hope thsi definitions will complemente the studies of complex networks.

Our publications on the subject

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Phase transitions on a multiplex of swarmalators

• S. J. Kongni, V. Nguefoue, T. Njougouo, P. Louodop, F. F. Ferreira, R. Tchitnga, and H. A. Cerdeira
• Phys. Rev. E 108, 034303 (2023). DOI: 10.1103/PhysRevE.108.034303

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Expected and unexpected routes to synchronization in a system of swarmalators

S. J. Kongni, T. Njougouo, P. Louodop, R. Tchitnga, F. F. Ferreira and H. A. Cerdeira

Phys. Rev. E 110, L062301, 2024 DOI: https://doi.org/10.1103/PhysRevE.110.L062301

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Attractive-repulsive challenge in swarmalators with time-dependent speed.

Steve J. Kongni, T. Njougouo, G. R. Simo, P. Louodop, R. Tchitnga, and H. A. Cerdeira

WIVACE 2024, Springer

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Delay induced double explosive transition in a swarmalator system.

Carmel T. Lambu. Romuald Mbonwouo, Delors A. Jiofack, Steve J. Kongni, G. R. Simo, P. Louodop, and H. A. Cerdeira, Phys.Rev. E 113, 024203 (2026).

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Topological transitions in swarmalators systems

Patrick Louodop, Michael N. Jipdi, Gael R. Simo, Steve J. Kongni, Carmel Lambu, Thierry Njougouo, Pablo D. Mininni, Kevin O’Keeffe, and Hilda A. Cerdeira, Phys. Rev. E Letter 114, L012201 (2026)

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Steve Kongni

Thierry Njougouo Gilles Koudafoke

Gael R. Simo

Carmel Lambu

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Michael Jipdi Kevin O’Keeffe Pablo Minnini Patrick Louodop Robert Tchitnga

COLLABORATORS

CONCLUSIONS

• We studied a system of swarmalators that presents different phases.
• Analyzing the energy we noticed that near the transition to complete synchronization, it can be represented by a modified XY model which at first sight presents a first order first transition.
• A closer analysis took us into un unexpected track which ended on finding diferente topological phases for the first time in these kind of systems. We found some new states, and found that as important the Kuramoto order parameter is, it is not enough for some systems.

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Thank you!

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EDITORS: PATRICK LOUODOP, ALBERT DIAZ GUILERA AND HILDA CERDEIRA

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