@article{Omel'chenko2022,
  author    = {Omel'chenko, Oleh},
  title     = {Mathematical framework for breathing chimera states},
  series = {Journal of nonlinear science},
  volume    = {32},
  journal   = {Journal of nonlinear science},
  number    = {2},
  publisher = {Springer},
  address   = {New York},
  issn      = {0938-8974},
  doi       = {10.1007/s00332-021-09779-1},
  pages     = {34},
  year      = {2022},
  abstract  = {About two decades ago it was discovered that systems of nonlocally coupled oscillators can exhibit unusual symmetry-breaking patterns composed of coherent and incoherent regions. Since then such patterns, called chimera states, have been the subject of intensive study but mostly in the stationary case when the coarse-grained system dynamics remains unchanged over time. Nonstationary coherence-incoherence patterns, in particular periodically breathing chimera states, were also reported, however not investigated systematically because of their complexity. In this paper we suggest a semi-analytic solution to the above problem providing a mathematical framework for the analysis of breathing chimera states in a ring of nonlocally coupled phase oscillators. Our approach relies on the consideration of an integro-differential equation describing the long-term coarse-grained dynamics of the oscillator system. For this equation we specify a class of solutions relevant to breathing chimera states. We derive a self-consistency equation for these solutions and carry out their stability analysis. We show that our approach correctly predicts macroscopic features of breathing chimera states. Moreover, we point out its potential application to other models which can be studied using the Ott-Antonsen reduction technique.},
  language  = {en}
}
@article{TurukinaPikovskij2011,
  author    = {Turukina, L. V. and Pikovskij, Arkadij},
  title     = {Hyperbolic chaos in a system of resonantly coupled weakly nonlinear oscillators},
  series = {Modern physics letters : A, Particles and fields, gravitation, cosmology, nuclear physics},
  volume    = {375},
  journal   = {Modern physics letters : A, Particles and fields, gravitation, cosmology, nuclear physics},
  number    = {11},
  publisher = {Elsevier},
  address   = {Amsterdam},
  issn      = {0375-9601},
  doi       = {10.1016/j.physleta.2011.02.017},
  pages     = {1407 -- 1411},
  year      = {2011},
  abstract  = {We show that a hyperbolic chaos can be observed in resonantly coupled oscillators near a Hopf bifurcation, described by normal-form-type equations for complex amplitudes. The simplest example consists of four oscillators, comprising two alternatively activated, due to an external periodic modulation, pairs. In terms of the stroboscopic Poincare map, the phase differences change according to an expanding Bernoulli map that depends on the coupling type. Several examples of hyperbolic chaos for different types of coupling are illustrated numerically.},
  language  = {en}
}
@article{PikovskijRosenblum2011,
  author    = {Pikovskij, Arkadij and Rosenblum, Michael},
  title     = {Dynamics of heterogeneous oscillator ensembles in terms of collective variables},
  series = {Physica :D, Nonlinear phenomena},
  volume    = {240},
  journal   = {Physica :D, Nonlinear phenomena},
  number    = {9-10},
  publisher = {Elsevier},
  address   = {Amsterdam},
  issn      = {0167-2789},
  doi       = {10.1016/j.physd.2011.01.002},
  pages     = {872 -- 881},
  year      = {2011},
  abstract  = {We consider general heterogeneous ensembles of phase oscillators, sine coupled to arbitrary external fields. Starting with the infinitely large ensembles, we extend the Watanabe-Strogatz theory, valid for identical oscillators, to cover the case of an arbitrary parameter distribution. The obtained equations yield the description of the ensemble dynamics in terms of collective variables and constants of motion. As a particular case of the general setup we consider hierarchically organized ensembles, consisting of a finite number of subpopulations, whereas the number of elements in a subpopulation can be both finite or infinite. Next, we link the Watanabe-Strogatz and Ott-Antonsen theories and demonstrate that the latter one corresponds to a particular choice of constants of motion. The approach is applied to the standard Kuramoto-Sakaguchi model, to its extension for the case of nonlinear coupling, and to the description of two interacting subpopulations, exhibiting a chimera state. With these examples we illustrate that, although the asymptotic dynamics can be found within the framework of the Ott-Antonsen theory, the transients depend on the constants of motion. The most dramatic effect is the dependence of the basins of attraction of different synchronous regimes on the initial configuration of phases.},
  language  = {en}
}
@article{BurylkoPikovskij2011,
  author    = {Burylko, Oleksandr and Pikovskij, Arkadij},
  title     = {Desynchronization transitions in nonlinearly coupled phase oscillators},
  series = {Physica :D, Nonlinear phenomena},
  volume    = {240},
  journal   = {Physica :D, Nonlinear phenomena},
  number    = {17},
  publisher = {Elsevier},
  address   = {Amsterdam},
  issn      = {0167-2789},
  doi       = {10.1016/j.physd.2011.05.016},
  pages     = {1352 -- 1361},
  year      = {2011},
  abstract  = {We consider the nonlinear extension of the Kuramoto model of globally coupled phase oscillators where the phase shift in the coupling function depends on the order parameter. A bifurcation analysis of the transition from fully synchronous state to partial synchrony is performed. We demonstrate that for small ensembles it is typically mediated by stable cluster states, that disappear with creation of heteroclinic cycles, while for a larger number of oscillators a direct transition from full synchrony to a periodic or a quasiperiodic regime occurs.},
  language  = {en}
}