Möbius group actions in the solvable chimera model

World Scientific Pub Co Pte Ltd

2024

We study actions of Möbius group on two sub-populations in the solvable chimera model proposed by Abrams et al. Dynamics of global variables are given by two coupled Watanabe–Strogatz systems, one for each sub-population. At the first glance, asymptotic dynamics in the model seem to be very simple. For instance, in the stable chimera state distributions of oscillators perform a simple rotation after a certain (sufficiently large) moment. However, a closer look unveils that dynamics are subtler that what can be observed from evolution of densities of oscillators’ phases. In order to gain the full picture, one needs to investigate dynamics on the transformation group that acts on these densities. Such an approach emphasizes impact of the “hidden” variable that is not visible on macroscopic level. More precisely, we demonstrate that the chimera model is an intriguing example of the classical system that exhibits the holonomy in fiber bundles of the group of Möbius transformations.

AIP Publishing

Phase holonomy underlies puzzling temporal patterns in Kuramoto models with two sub-populations

We present a geometric investigation of curious dynamical behaviors previously reported in Kuramoto models with two sub-populations. Our study demonstrates that chimeras and traveling waves in such models are associated with the birth of geometric phase. Although manifestations of geometric phase are frequent in various fields of physics, this is the first time (to our best knowledge) that such a phenomenon is exposed in ensembles of Kuramoto oscillators or, more broadly, in complex systems.

Download 2024 Aladin Crnkić , Vladimir Jaćimović

World Scientific Pub Co Pte Ltd

Möbius group actions in the solvable chimera model

Download 2024 Aladin Crnkić , Vladimir Jaćimović

AIP Publishing

Chimeras and traveling waves in ensembles of Kuramoto oscillators off the Poisson manifold

We examine how perturbations off the Poisson manifold affect chimeras and traveling waves (TWs) in Kuramoto models with two sub-populations. Our numerical study is based on simulations on invariant manifolds, which contain von Mises probability distributions. Our study demonstrates that chimeras and TWs off the Poisson manifold always “breathe”, and the effect of breathing is more pronounced further from the Poisson manifold. On the other side, TWs arising in similar models on the sphere always...

Download 2024 Aladin Crnkić , Vladimir Jaćimović

AIP Publishing

Collective dynamics of phase-repulsive oscillators solves graph coloring problem

We show how to couple phase-oscillators on a graph so that collective dynamics “searches” for the coloring of that graph as it relaxes toward the dynamical equilibrium. This translates a combinatorial optimization problem (graph coloring) into a functional optimization problem (finding and evaluating the global minimum of dynamical non-equilibrium potential, done by the natural system’s evolution). Using a sample of graphs, we show that our method can serve as a viable alternative to the traditional...

Download 2020 Janez Povh , Aladin Crnkić , Zoran Levnajić , Vladimir Jaćimović

Informa UK Limited

Consensus and coordination on groups SO(3) and S3 over constant and state-dependent communication graphs

Download 2020 Aladin Crnkić , Vladimir Jaćimović , Milojica Jaćimović , Nevena Mijajlović