Conformists and contrarians on spheres. (English) Zbl 1539.37028
This paper investigates models of identical Kuramoto coupled oscillators on spheres. In the seminal paper [Physica D 74, 197–253 (1994; Zbl 0812.34043)] S. Watanabe and S. S. Strogatz used symmetries to reduce the dimensions of large ensembles of coupled oscillators. A model of this kind, described in [H. Hong and S. Strogatz, Phys. Rev. Lett. 106, No. 5, Article ID 054102, 4 p. (2011; doi:10.1103/PhysRevLett.106.054102)], examines the interactions between two sub-populations. The first sub-population – the conformists – consists of oscillators that are attracted to the mean field, and the second sub-population – the contrarians – are repulsed from the others. A distinctive feature is that pairwise interactions between oscillators from different sub-populations are not symmetric. In this paper, the authors reduce the conformists-contrarians model to a three-dimensional system of ordinary differential equations and find nontrivial equilibrium states and bifurcations.
The paper examines an extension of the conformists-contrarians model with identical oscillators to spheres. They take the dynamics of a large ensemble of generalized Kuramoto oscillators to a three-dimensional system. A bifurcation analysis shows that all phenomena from the classical model also appear in higher dimensions with different parameter values. In particular, they note that models on spheres show traveling waves consisting of contrarians.
The paper examines an extension of the conformists-contrarians model with identical oscillators to spheres. They take the dynamics of a large ensemble of generalized Kuramoto oscillators to a three-dimensional system. A bifurcation analysis shows that all phenomena from the classical model also appear in higher dimensions with different parameter values. In particular, they note that models on spheres show traveling waves consisting of contrarians.
Reviewer: William J. Satzer Jr. (St. Paul)
MSC:
| 37C10 | Dynamics induced by flows and semiflows |
| 37C75 | Stability theory for smooth dynamical systems |
| 37G10 | Bifurcations of singular points in dynamical systems |
| 37G15 | Bifurcations of limit cycles and periodic orbits in dynamical systems |
| 34C15 | Nonlinear oscillations and coupled oscillators for ordinary differential equations |
| 34C40 | Ordinary differential equations and systems on manifolds |
| 92D25 | Population dynamics (general) |
Keywords:
Kuramoto models on spheres; traveling wave on sphere; opinion dynamics; isometries of the ballCitations:
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