On synchronization in Kuramoto models on spheres

Crnkić, Aladin; Jaćimović, Vladimir; Markovic, Marijan

doi:10.1007/s13324-021-00567-4

Cited by 11 publications

(9 citation statements)

References 20 publications

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“…Parallel to the studies mentioned above, the collective dynamics of ensembles of so-called generalized Kuramoto oscillators have attracted considerable attention [21][22][23][24][25][26][27][28][29][30]. In particular, Lohe detailed the dynamics of generalized Kuramoto models on a compact classical Lie group and provided insights into their notable characteristics [31,32].…”

Section: Introductionmentioning

confidence: 99%

Volcano transition in a system of generalized Kuramoto oscillators with random frustrated interactions

Lee,

Jeong,

Son

et al. 2024

J. Phys. A: Math. Theor.

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In a system of heterogeneous (Abelian) Kuramoto oscillators with random or `frustrated' interactions, transitions from states of incoherence to partial synchronization were observed. These so-called volcano transitions are characterized by a change in the shape of a local field distribution and were discussed in connection with an oscillator glass. In this paper, we consider a different class of oscillators, namely a system of (non-Abelian) SU(2)-Lohe oscillators that can also be defined on the 3-sphere, i.e., an oscillator is generalized to be defined as a unit vector in 4D Euclidean space. We demonstrate that such higher-dimensional Kuramoto models with reciprocal and nonreciprocal random interactions represented by a low-rank matrix exhibit a volcano transition as well. We determine the critical coupling strength at which a volcano-like transition occurs, employing an Ott-Antonsen ansatz. Numerical simulations provide additional validations of our analytical findings and reveal the differences in observable collective dynamics prior to and following the transition. Furthermore, we show that a system of unit 3-vector oscillators on the 2-sphere does not possess a volcano transition.

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Section: Introductionmentioning

confidence: 99%

Volcano transition in a system of generalized Kuramoto oscillators with random frustrated interactions

Lee,

Jeong,

Son

et al. 2024

J. Phys. A: Math. Theor.

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“…In order to derive equations for real order parameters, we need to relate vectors a (l) and b (l) . This is substantiated in the following Proposition 3.1 ( [17,18]). Suppose that initial distribution of oscillators from both subpopulations is uniform on the sphere S d−1 .…”

Section: Dynamics On Extended Oa Submanifoldsmentioning

confidence: 59%

“…This concludes reduction of the model (4) to the three-dimensional dynamical system ( 14), ( 15) and (18). Notice that real order parameters r A and r B do not enter this system, but are expressed from s A and s B through relations (10).…”

Section: Reduction To the Three-dimensional Systemmentioning

confidence: 75%

“…these two types of models generate non-equivalent dynamics. In the present paper we investigate conformists and contrarians for real Kuramoto models, as in this case dynamics of order parameters depend on the dimension, see [18]. Densities of the form (2) appear in real Kuramoto models, while in complex Kuramoto models the extended OA submanifold consists of different densities, see [18].…”

Section: Modelmentioning

confidence: 99%

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Conformists and contrarians on spheres

Crnkić,

Jaćimović,

Niu

2024

J. Phys. A: Math. Theor.

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We investigate the conformists-contrarians model of identical Kuramoto oscillators evolving on a sphere.Using group-theoretic and geometric approach, we reduce the model to the dynamical system on extended Ott-Antonsen manifold. Further reduction yields the system of three scalar ODE's for global variables. This three-dimensional dynamical system is studied analytically in order to investigate an interplay between conformists and contrarians on spheres. Our study demonstrates that conformists-contrarians models on spheres display the same types of equilibria and dynamical phenomena in all dimensions. However, critical combination of parameters, for which particular equilibrium states arise, does depend on the dimension.In particular, models on spheres exhibit traveling waves consisting of contrarians. We derive an exact formula for the relation between parameter values for which such waves arise in different dimensions.Finally, we take a closer look at trajectories of traveling waves on spheres, demonstrating subtleties of this dynamical phenomenon.

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“…Similar extensions of the Kuramoto model to higher dimensions were also considered in refs. [37][38][39].…”

Section: Introductionmentioning

confidence: 99%