The General Non-Abelian Kuramoto Model on the 3-sphere

Vladimir Jaćimović; Aladin Crnkić; Vladimir Jaćimović; Aladin Crnkić

doi:10.3934/nhm.2020005

Networks and Heterogeneous Media

2020, Volume 15, Issue 1: 111-124. doi: 10.3934/nhm.2020005

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The General Non-Abelian Kuramoto Model on the 3-sphere

Vladimir Jaćimović ^{1
,},
Aladin Crnkić ^{2
,
,}

1.
Faculty of Natural Sciences and Mathematics, University of Montenegro, 81000 Podgorica, Montenegro
2.
Faculty of Technical Engineering, University of Bihać, 77000 Bihać, Bosnia and Herzegovina

Received: 01 March 2019 Revised: 01 July 2019
Primary: 34M45, 34C15, 92B25

We introduce non-Abelian Kuramoto model on $ S^3 $ in the most general form. Following an analogy with the classical Kuramoto model (on the circle $ S^1 $), we study some interesting variations of the model on $ S^3 $ that are obtained for particular coupling functions. As a partial case, by choosing "standard" coupling function one obtains a previously known model, that is referred to as Kuramoto-Lohe model on $ S^3 $.
We briefly address two particular models: Kuramoto models on $ S^3 $ with frustration and with external forcing. These models on higher dimensional manifolds have not been studied so far. By choosing suitable values of parameters we observe different nontrivial dynamical regimes even in the simplest setup of globally coupled homogeneous population.
Although non-Abelian Kuramoto models can be introduced on various symmetric spaces, we restrict our analysis to the case when underlying manifold is the 3-sphere. Due to geometric and algebraic properties of this specific manifold, variations of this model are meaningful and geometrically well justified.
- Non-Abelian Kuramoto model,
- 3-sphere,
- Kuramoto-Lohe oscillators,
- Kuramoto-Sakaguchi model,
- Quaternion-valued Riccati equation
Citation: Vladimir Jaćimović, Aladin Crnkić. The General Non-Abelian Kuramoto Model on the 3-sphere[J]. Networks and Heterogeneous Media, 2020, 15(1): 111-124. doi: 10.3934/nhm.2020005

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Abstract

We introduce non-Abelian Kuramoto model on $ S^3 $ in the most general form. Following an analogy with the classical Kuramoto model (on the circle $ S^1 $), we study some interesting variations of the model on $ S^3 $ that are obtained for particular coupling functions. As a partial case, by choosing "standard" coupling function one obtains a previously known model, that is referred to as Kuramoto-Lohe model on $ S^3 $.

We briefly address two particular models: Kuramoto models on $ S^3 $ with frustration and with external forcing. These models on higher dimensional manifolds have not been studied so far. By choosing suitable values of parameters we observe different nontrivial dynamical regimes even in the simplest setup of globally coupled homogeneous population.

Although non-Abelian Kuramoto models can be introduced on various symmetric spaces, we restrict our analysis to the case when underlying manifold is the 3-sphere. Due to geometric and algebraic properties of this specific manifold, variations of this model are meaningful and geometrically well justified.

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