@article{BelykhOsipovKucklaenderetal.2005,
  author    = {Belykh, Vladimir N. and Osipov, Grigory V. and Kuckl{\"a}nder, Nina and Blasius, Bernd and Kurths, J{\"u}rgen},
  title     = {Automatic control of phase synchronization in coupled complex oscillators},
  year      = {2005},
  abstract  = {We present an automatic control method for phase locking of regular and chaotic non-identical oscillations, when all subsystems interact via feedback. This method is based on the well known principle of feedback control which takes place in nature and is successfully used in engineering. In contrast to unidirectional and bidirectional coupling, the approach presented here supposes the existence of a special controller, which allows to change the parameters of the controlled systems. First we discuss general principles of automatic phase synchronization (PS) for arbitrary coupled systems with a controller whose input is given by a special quadratic form of coordinates of the individual systems and its output is a result of the application of a linear differential operator. We demonstrate the effectiveness of our approach for controlled PS on several examples: (i) two coupled regular oscillators, (ii) coupled regular and chaotic oscillators, (iii) two coupled chaotic R"ossler oscillators, (iv) two coupled foodweb models, (v) coupled chaotic R"ossler and Lorenz oscillators, (vi) ensembles of locally coupled regular oscillators, (vii) ensembles of locally coupled chaotic oscillators, and (viii) ensembles of globally coupled chaotic oscillators.},
  language  = {en}
}
@article{BoccalettiKurthsOsipov2002,
  author    = {Boccaletti, Stefano and Kurths, J{\"u}rgen and Osipov, Grigory V.},
  title     = {The synchronization of chaotic systems},
  year      = {2002},
  language  = {en}
}
@article{BolotovBolotovSmirnovetal.2019,
  author    = {Bolotov, Dmitry and Bolotov, Maxim I. and Smirnov, Lev A. and Osipov, Grigory V. and Pikovskij, Arkadij},
  title     = {Twisted States in a System of Nonlinearly Coupled Phase Oscillators},
  series = {Regular and chaotic dynamics : international scientific journal},
  volume    = {24},
  journal   = {Regular and chaotic dynamics : international scientific journal},
  number    = {6},
  publisher = {Pleiades publishing inc},
  address   = {Moscow},
  issn      = {1560-3547},
  doi       = {10.1134/S1560354719060091},
  pages     = {717 -- 724},
  year      = {2019},
  abstract  = {We study the dynamics of the ring of identical phase oscillators with nonlinear nonlocal coupling. Using the Ott - Antonsen approach, the problem is formulated as a system of partial derivative equations for the local complex order parameter. In this framework, we investigate the existence and stability of twisted states. Both fully coherent and partially coherent stable twisted states were found (the latter ones for the first time for identical oscillators). We show that twisted states can be stable starting from a certain critical value of the medium length, or on a length segment. The analytical results are confirmed with direct numerical simulations in finite ensembles.},
  language  = {en}
}
@article{BolotovBolotovSmirnovetal.2022,
  author    = {Bolotov, Dmitry and Bolotov, Maxim I. and Smirnov, Lev A. and Osipov, Grigory V. and Pikovsky, Arkady},
  title     = {Synchronization regimes in an ensemble of phase oscillators coupled through a diffusion field},
  series = {Radiophysics and quantum electronics},
  volume    = {64},
  journal   = {Radiophysics and quantum electronics},
  number    = {10},
  publisher = {Springer},
  address   = {New York},
  issn      = {0033-8443},
  doi       = {10.1007/s11141-022-10173-4},
  pages     = {709 -- 725},
  year      = {2022},
  abstract  = {We consider an ensemble of identical phase oscillators coupled through a common diffusion field. Using the Ott-Antonsen reduction, we develop dynamical equations for the complex local order parameter and the mean field. The regions of the existence and stability are determined for the totally synchronous, partially synchronous, and asynchronous spatially homogeneous states. A procedure of searching for inhomogeneous states as periodic trajectories of an auxiliary system of the ordinary differential equations is demonstrated. A scenario of emergence of chimera structures from homogeneous synchronous solutions is described.},
  language  = {en}
}
@article{BolotovOsipovPikovskij2016,
  author    = {Bolotov, M. I. and Osipov, Grigory V. and Pikovskij, Arkadij},
  title     = {Marginal chimera state at cross-frequency locking of pulse-coupled neural networks},
  series = {Physical review : E, Statistical, nonlinear and soft matter physics},
  volume    = {93},
  journal   = {Physical review : E, Statistical, nonlinear and soft matter physics},
  publisher = {American Physical Society},
  address   = {College Park},
  issn      = {2470-0045},
  doi       = {10.1103/PhysRevE.93.032202},
  pages     = {6},
  year      = {2016},
  abstract  = {We consider two coupled populations of leaky integrate-and-fire neurons. Depending on the coupling strength, mean fields generated by these populations can have incommensurate frequencies or become frequency locked. In the observed 2:1 locking state of the mean fields, individual neurons in one population are asynchronous with the mean fields, while in another population they have the same frequency as the mean field. These synchronous neurons form a chimera state, where part of them build a fully synchronized cluster, while other remain scattered. We explain this chimera as a marginal one, caused by a self-organized neutral dynamics of the effective circle map.},
  language  = {en}
}
@article{BolotovSmirnovOsipovetal.2018,
  author    = {Bolotov, Maxim I. and Smirnov, Lev A. and Osipov, Grigory V. and Pikovskij, Arkadij},
  title     = {Simple and complex chimera states in a nonlinearly coupled oscillatory medium},
  series = {Chaos : an interdisciplinary journal of nonlinear science},
  volume    = {28},
  journal   = {Chaos : an interdisciplinary journal of nonlinear science},
  number    = {4},
  publisher = {American Institute of Physics},
  address   = {Melville},
  issn      = {1054-1500},
  doi       = {10.1063/1.5011678},
  pages     = {9},
  year      = {2018},
  abstract  = {We consider chimera states in a one-dimensional medium of nonlinear nonlocally coupled phase oscillators. In terms of a local coarse-grained complex order parameter, the problem of finding stationary rotating nonhomogeneous solutions reduces to a third-order ordinary differential equation. This allows finding chimera-type and other inhomogeneous states as periodic orbits of this equation. Stability calculations reveal that only some of these states are stable. We demonstrate that an oscillatory instability leads to a breathing chimera, for which the synchronous domain splits into subdomains with different mean frequencies. Further development of instability leads to turbulent chimeras. Published by AIP Publishing.},
  language  = {en}
}
@article{BolotovSmirnovOsipovetal.2017,
  author    = {Bolotov, Maxim I. and Smirnov, Lev A. and Osipov, Grigory V. and Pikovskij, Arkadij},
  title     = {Breathing chimera in a system of phase oscillators},
  series = {JETP Letters},
  volume    = {106},
  journal   = {JETP Letters},
  publisher = {Pleiades Publ.},
  address   = {New York},
  issn      = {0021-3640},
  doi       = {10.1134/S0021364017180059},
  pages     = {393 -- 399},
  year      = {2017},
  abstract  = {Chimera states consisting of synchronous and asynchronous domains in a medium of nonlinearly coupled phase oscillators have been considered. Stationary inhomogeneous solutions of the Ott-Antonsen equation for a complex order parameter that correspond to fundamental chimeras have been constructed. The direct numerical simulation has shown that these structures under certain conditions are transformed to oscillatory (breathing) chimera regimes because of the development of instability.},
  language  = {en}
}
@misc{BolotovSmirnovOsipovetal.2018,
  author    = {Bolotov, Maxim and Smirnov, Lev A. and Osipov, Grigory V. and Pikovskij, Arkadij},
  title     = {Complex chimera states in a nonlinearly coupled oscillatory medium},
  series = {2018 2nd School on Dynamics of Complex Networks and their Application in Intellectual Robotics (DCNAIR)},
  journal   = {2018 2nd School on Dynamics of Complex Networks and their Application in Intellectual Robotics (DCNAIR)},
  publisher = {IEEE},
  address   = {New York},
  isbn      = {978-1-5386-5818-5},
  doi       = {10.1109/DCNAIR.2018.8589210},
  pages     = {17 -- 20},
  year      = {2018},
  abstract  = {We consider chimera states in a one-dimensional medium of nonlinear nonlocally coupled phase oscillators. Stationary inhomogeneous solutions of the Ott-Antonsen equation for a complex order parameter that correspond to fundamental chimeras have been constructed. Stability calculations reveal that only some of these states are stable. The direct numerical simulation has shown that these structures under certain conditions are transformed to breathing chimera regimes because of the development of instability. Further development of instability leads to turbulent chimeras.},
  language  = {en}
}
@article{GrinesOsipovPikovskij2018,
  author    = {Grines, Evgeny and Osipov, Grigory V. and Pikovskij, Arkadij},
  title     = {Describing dynamics of driven multistable oscillators with phase transfer curves},
  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       = {10.1063/1.5037290},
  pages     = {6},
  year      = {2018},
  abstract  = {Phase response curve is an important tool in the studies of stable self-sustained oscillations; it describes a phase shift under action of an external perturbation. We consider multistable oscillators with several stable limit cycles. Under a perturbation, transitions from one oscillating mode to another one may occur. We define phase transfer curves to describe the phase shifts at such transitions. This allows for a construction of one-dimensional maps that characterize periodically kicked multistable oscillators. We show that these maps are good approximations of the full dynamics for large periods of forcing. Published by AIP Publishing.},
  language  = {en}
}
@article{IvanchenkoOsipovShalfeevetal.2004,
  author    = {Ivanchenko, Mikhail V. and Osipov, Grigory V. and Shalfeev, V. D. and Kurths, J{\"u}rgen},
  title     = {Phase synchronization in ensembles of bursting oscillators},
  issn      = {0031-9007},
  year      = {2004},
  abstract  = {We study the effects of mutual and external chaotic phase synchronization in ensembles of bursting oscillators. These oscillators (used for modeling neuronal dynamics) are essentially multiple time scale systems. We show that a transition to mutual phase synchronization takes place on the bursting time scale of globally coupled oscillators, while on the spiking time scale, they behave asynchronously. We also demonstrate the effect of the onset of external chaotic phase synchronization of the bursting behavior in the studied ensemble by a periodic driving applied to one arbitrarily taken neuron. We also propose an explanation of the mechanism behind this effect. We infer that the demonstrated phenomenon can be used efficiently for controlling bursting activity in neural ensembles},
  language  = {en}
}