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.
