@article{TyulkinaGoldobinKlimenkoetal.2018,
  author    = {Tyulkina, Irina and Goldobin, Denis S. and Klimenko, Lyudmila S. and Pikovskij, Arkadij},
  title     = {Dynamics of noisy oscillator populations beyond the Ott-Antonsen Ansatz},
  series = {Physical review letters},
  volume    = {120},
  journal   = {Physical review letters},
  number    = {26},
  publisher = {American Physical Society},
  address   = {College Park},
  issn      = {0031-9007},
  doi       = {10.1103/PhysRevLett.120.264101},
  pages     = {6},
  year      = {2018},
  abstract  = {We develop an approach for the description of the dynamics of large populations of phase oscillators based on "circular cumulants" instead of the Kuramoto-Daido order parameters. In the thermodynamic limit, these variables yield a simple representation of the Ott-Antonsen invariant solution [E. Ott and T. M. Antonsen, Chaos 18, 037113 (2008)] and appear appropriate for constructing perturbation theory on top of the Ott-Antonsen ansatz. We employ this approach to study the impact of small intrinsic noise on the dynamics. As a result, a closed system of equations for the two leading cumulants, describing the dynamics of noisy ensembles, is derived. We exemplify the general theory by presenting the effect of noise on the Kuramoto system and on a chimera state in two symmetrically coupled populations.},
  language  = {en}
}
@article{GoldobinTyulkinaKlimenkoetal.2018,
  author    = {Goldobin, Denis S. and Tyulkina, Irina V. and Klimenko, Lyudmila S. and Pikovskij, Arkadij},
  title     = {Collective mode reductions for populations of coupled noisy oscillators},
  series = {Chaos : an interdisciplinary journal of nonlinear science},
  volume    = {28},
  journal   = {Chaos : an interdisciplinary journal of nonlinear science},
  number    = {10},
  publisher = {American Institute of Physics},
  address   = {Melville},
  issn      = {1054-1500},
  doi       = {10.1063/1.5053576},
  pages     = {6},
  year      = {2018},
  abstract  = {We analyze the accuracy of different low-dimensional reductions of the collective dynamics in large populations of coupled phase oscillators with intrinsic noise. Three approximations are considered: (i) the Ott-Antonsen ansatz, (ii) the Gaussian ansatz, and (iii) a two-cumulant truncation of the circular cumulant representation of the original system's dynamics. For the latter, we suggest a closure, which makes the truncation, for small noise, a rigorous first-order correction to the Ott-Antonsen ansatz, and simultaneously is a generalization of the Gaussian ansatz. The Kuramoto model with intrinsic noise and the population of identical noisy active rotators in excitable states with the Kuramoto-type coupling are considered as examples to test the validity of these approximations. For all considered cases, the Gaussian ansatz is found to be more accurate than the Ott-Antonsen one for high-synchrony states only. The two-cumulant approximation is always superior to both other approximations. Synchrony of large ensembles of coupled elements can be characterised by the order parameters—the mean fields. Quite often, the evolution of these collective variables is surprisingly simple, which makes a description with only a few order parameters feasible. Thus, one tries to construct accurate closed low-dimensional mathematical models for the dynamics of the first few order parameters. These models represent useful tools for gaining insight into the underlaying mechanisms of some more sophisticated collective phenomena: for example, one describes coupled populations by virtue of coupled equations for the relevant order parameters. A regular approach to the construction of closed low-dimensional systems is also beneficial for dealing with phenomena, which are beyond the applicability scope of these models; for instance, with such an approach, one can determine constraints on clustering in populations. There are two prominent types of situations, where the low-dimensional models can be constructed: (i) for a certain class of ideal paradigmatic systems of coupled phase oscillators, the Ott-Antonsen ansatz yields an exact equation for the main order parameter and (ii) the Gaussian approximation for the probability density of the phases, also yielding a low-dimensional closure, is frequently quite accurate. In this paper, we compare applications of these two model reductions for situations, where neither of them is perfectly accurate. Furthermore, we construct a new reduction approach which practically works as a first-order correction to the best of the two basic approximations.},
  language  = {en}
}