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 breathe moderately, no matter if the dynamics take place near the Poisson manifold or far away from it.

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 combinatorial algorithms. Moreover, we show that, with the same computational cost, our method efficiently solves the harder problem of improper coloring of weighed graphs.

Low-dimensional dynamics in non-Abelian Kuramoto model on the 3-sphere.

This paper deals with the low-dimensional dynamics in the general non-Abelian Kuramoto model of mutually interacting generalized oscillators on the 3-sphere. If all oscillators have identical intrinsic generalized frequencies and the coupling is global, the dynamics is fully determined by several global variables. We state that these generalized oscillators evolve by the action of the group GH of (quaternionic) Möbius transformations that preserve S3 . The global variables satisfy a certain system of quaternion-valued ordinary differential equations, that is an extension of the Watanabe-Strogatz system. If the initial distribution of oscillators is uniform on S3 , additional symmetries arise and the dynamics can be restricted further to invariant submanifolds of (real) dimension four.