Exploring the phase diagrams of multidimensional Kuramoto models

Fariello, Ricardo; de Aguiar, Marcus A.M.

doi:10.1016/j.chaos.2023.114431

Cited by 5 publications

(1 citation statement)

References 38 publications

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“…Usage of the term 'coupling matrix' has been pervasive in previous literatures in network science [53][54][55], or even sometimes referring to the connectivity matrix in brain theory [56], but caution is due where it stands for an adjacency matrix as the cases above that just arranges the scalar-valued pairwise coupling strengths into a matricial form. An explicit matrix effect on the diffusive coupling between high-dimensional Kuramoto oscillators was rarely studied save for a few exceptions [26,57], not to say that in the representation of equation ( 2), to the best knowledge of the authors.…”

Section: Discussionmentioning

confidence: 99%

Binary system modes of matrix-coupled multidimensional oscillators

Wang,

Shao,

Tan

et al. 2024

New J. Phys.

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The standard Kuramoto model has been instrumental in explaining synchronization and desynchronization, two emergent phenomena often observed in biological, neuronal, and physical systems. While the Kuramoto model has turned out effective with one-dimensional oscillators, real-world systems often involve high-dimensional interacting units, such as biological swarms, necessitating a model of multidimensional oscillators. However, existing high-dimensional generalizations of the Kuramoto model commonly rely on a scalar-valued coupling strength, which limits their ability to capture the full complexity of high-dimensional interactions. This work introduces a matrix, A, to couple the interconnected components of the oscillators in a d-dimensional space, leading to a matrix-coupled multidimensional Kuramoto model that approximates a prototypical swarm dynamics by its first-order harmonics. Moreover, the matrix A introduces an inter-dimensional higher-order interaction that partly accounts for the emergence of 2^{d} system modes in a d-dimensional population, where each dimension can either be synchronized or desynchronized, represented by a set of almost binary order parameters. The binary system modes capture characteristic swarm behaviors such as fish milling or polarized schooling. Additionally, our findings provides a theoretical analogy to cerebral activity, where the resting state and the activated state coexist unihemispherically. It also suggests a new possibility for information storage in oscillatory neural networks.

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