@article{GongZhengToenjesetal.2019,
  author    = {Gong, Chen Chris and Zheng, Chunming and Toenjes, Ralf and Pikovskij, Arkadij},
  title     = {Repulsively coupled Kuramoto-Sakaguchi phase oscillators ensemble subject to common noise},
  series = {Chaos : an interdisciplinary journal of nonlinear science},
  volume    = {29},
  journal   = {Chaos : an interdisciplinary journal of nonlinear science},
  number    = {3},
  publisher = {American Institute of Physics},
  address   = {Melville},
  issn      = {1054-1500},
  doi       = {10.1063/1.5084144},
  pages     = {11},
  year      = {2019},
  abstract  = {We consider the Kuramoto-Sakaguchi model of identical coupled phase oscillators with a common noisy forcing. While common noise always tends to synchronize the oscillators, a strong repulsive coupling prevents the fully synchronous state and leads to a nontrivial distribution of oscillator phases. In previous numerical simulations, the formation of stable multicluster states has been observed in this regime. However, we argue here that because identical phase oscillators in the Kuramoto-Sakaguchi model form a partially integrable system according to the Watanabe-Strogatz theory, the formation of clusters is impossible. Integrating with various time steps reveals that clustering is a numerical artifact, explained by the existence of higher order Fourier terms in the errors of the employed numerical integration schemes. By monitoring the induced change in certain integrals of motion, we quantify these errors. We support these observations by showing, on the basis of the analysis of the corresponding Fokker-Planck equation, that two-cluster states are non-attractive. On the other hand, in ensembles of general limit cycle oscillators, such as Van der Pol oscillators, due to an anharmonic phase response function as well as additional amplitude dynamics, multiclusters can occur naturally. Published under license by AIP Publishing.},
  language  = {en}
}
@article{PeterGongPikovskij2019,
  author    = {Peter, Franziska and Gong, Chen Chris and Pikovskij, Arkadij},
  title     = {Microscopic correlations in the finite-size Kuramoto model of coupled oscillators},
  series = {Physical review : E, Statistical, nonlinear and soft matter physics},
  volume    = {100},
  journal   = {Physical review : E, Statistical, nonlinear and soft matter physics},
  number    = {3},
  publisher = {American Physical Society},
  address   = {College Park},
  issn      = {2470-0045},
  doi       = {10.1103/PhysRevE.100.032210},
  pages     = {6},
  year      = {2019},
  abstract  = {Supercritical Kuramoto oscillators with distributed frequencies can be separated into two disjoint groups: an ordered one locked to the mean field, and a disordered one consisting of effectively decoupled oscillators-at least so in the thermodynamic limit. In finite ensembles, in contrast, such clear separation fails: The mean field fluctuates due to finite-size effects and thereby induces order in the disordered group. This publication demonstrates this effect, similar to noise-induced synchronization, in a purely deterministic system. We start by modeling the situation as a stationary mean field with additional white noise acting on a pair of unlocked Kuramoto oscillators. An analytical expression shows that the cross-correlation between the two increases with decreasing ratio of natural frequency difference and noise intensity. In a deterministic finite Kuramoto model, the strength of the mean-field fluctuations is inextricably linked to the typical natural frequency difference. Therefore, we let a fluctuating mean field, generated by a finite ensemble of active oscillators, act on pairs of passive oscillators with a microscopic natural frequency difference between which we then measure the cross-correlation, at both super- and subcritical coupling.},
  language  = {en}
}
@article{ZaksPikovskij2019,
  author    = {Zaks, Michael A. and Pikovskij, Arkadij},
  title     = {Synchrony breakdown and noise-induced oscillation death in ensembles of serially connected spin-torque oscillators},
  series = {The European physical journal : B, Condensed matter and complex systems},
  volume    = {92},
  journal   = {The European physical journal : B, Condensed matter and complex systems},
  number    = {7},
  publisher = {Springer},
  address   = {New York},
  issn      = {1434-6028},
  doi       = {10.1140/epjb/e2019-100152-2},
  pages     = {12},
  year      = {2019},
  abstract  = {We consider collective dynamics in the ensemble of serially connected spin-torque oscillators governed by the Landau-Lifshitz-Gilbert-Slonczewski magnetization equation. Proximity to homoclinicity hampers synchronization of spin-torque oscillators: when the synchronous ensemble experiences the homoclinic bifurcation, the growth rate per oscillation of small deviations from the ensemble mean diverges. Depending on the configuration of the contour, sufficiently strong common noise, exemplified by stochastic oscillations of the current through the circuit, may suppress precession of the magnetic field for all oscillators. We derive the explicit expression for the threshold amplitude of noise, enabling this suppression.},
  language  = {en}
}
@article{ZhengPikovskij2019,
  author    = {Zheng, Chunming and Pikovskij, Arkadij},
  title     = {Stochastic bursting in unidirectionally delay-coupled noisy excitable systems},
  series = {Chaos : an interdisciplinary journal of nonlinear science},
  volume    = {29},
  journal   = {Chaos : an interdisciplinary journal of nonlinear science},
  number    = {4},
  publisher = {American Institute of Physics},
  address   = {Melville},
  issn      = {1054-1500},
  doi       = {10.1063/1.5093180},
  pages     = {9},
  year      = {2019},
  abstract  = {We show that "stochastic bursting" is observed in a ring of unidirectional delay-coupled noisy excitable systems, thanks to the combinational action of time-delayed coupling and noise. Under the approximation of timescale separation, i.e., when the time delays in each connection are much larger than the characteristic duration of the spikes, the observed rather coherent spike pattern can be described by an idealized coupled point processwith a leader-follower relationship. We derive analytically the statistics of the spikes in each unit, the pairwise correlations between any two units, and the spectrum of the total output from the network. Theory is in good agreement with the simulations with a network of theta-neurons. Published under license by AIP Publishing.},
  language  = {en}
}
@article{RosenblumPikovskij2019,
  author    = {Rosenblum, Michael and Pikovskij, Arkadij},
  title     = {Nonlinear phase coupling functions: a numerical study},
  series = {Philosophical Transactions of the Royal Society of London, Series A : Mathematical, Physical and Engineering Sciences},
  volume    = {377},
  journal   = {Philosophical Transactions of the Royal Society of London, Series A : Mathematical, Physical and Engineering Sciences},
  number    = {2160},
  publisher = {Royal Society},
  address   = {London},
  issn      = {1364-503X},
  doi       = {10.1098/rsta.2019.0093},
  pages     = {12},
  year      = {2019},
  abstract  = {Phase reduction is a general tool widely used to describe forced and interacting self-sustained oscillators. Here, we explore the phase coupling functions beyond the usual first-order approximation in the strength of the force. Taking the periodically forced Stuart-Landau oscillator as the paradigmatic model, we determine and numerically analyse the coupling functions up to the fourth order in the force strength. We show that the found nonlinear phase coupling functions can be used for predicting synchronization regions of the forced oscillator.},
  language  = {en}
}
@article{GengelPikovskij2019,
  author    = {Gengel, Erik and Pikovskij, Arkadij},
  title     = {Phase demodulation with iterative Hilbert transform embeddings},
  series = {Signal processing},
  volume    = {165},
  journal   = {Signal processing},
  publisher = {Elsevier},
  address   = {Amsterdam},
  issn      = {0165-1684},
  doi       = {10.1016/j.sigpro.2019.07.005},
  pages     = {115 -- 127},
  year      = {2019},
  abstract  = {We propose an efficient method for demodulation of phase modulated signals via iterated Hilbert transform embeddings. We show that while a usual approach based on one application of the Hilbert transform provides only an approximation to a proper phase, with iterations the accuracy is essentially improved, up to precision limited mainly by discretization effects. We demonstrate that the method is applicable to arbitrarily complex waveforms, and to modulations fast compared to the basic frequency. Furthermore, we develop a perturbative theory applicable to a simple cosine waveform, showing convergence of the technique.},
  language  = {en}
}
@article{PawlikPikovskij2006,
  author    = {Pawlik, Andreas H. and Pikovskij, Arkadij},
  title     = {Control of oscillators coherence by multiple delayed feedback},
  series = {Modern physics letters : A, Particles and fields, gravitation, cosmology, nuclear physics},
  volume    = {358},
  journal   = {Modern physics letters : A, Particles and fields, gravitation, cosmology, nuclear physics},
  number    = {3},
  publisher = {American Institute of Physics},
  address   = {Amsterdam},
  issn      = {0375-9601},
  doi       = {10.1016/j.physleta.2006.05.013},
  pages     = {181 -- 185},
  year      = {2006},
  abstract  = {We demonstrate that a multiple delayed feedback is a powerful tool to control coherence properties of autonomous self-sustained oscillators. We derive the equation for the phase dynamics in presence of noise and delay, and analyze it analytically. In Gaussian approximation a closed set of equations for the frequency and the diffusion constant is obtained. Solutions of these equations are in good agreement with direct numerical simulations.},
  language  = {en}
}
@misc{MunyaevSmirnovKostinetal.2020,
  author    = {Munyaev, Vyacheslav and Smirnov, Lev A. and Kostin, Vasily and Osipov, Grigory V. and Pikovskij, Arkadij},
  title     = {Analytical approach to synchronous states of globally coupled noisy rotators},
  series = {Postprints der Universit{\"a}t Potsdam : Mathematisch-Naturwissenschaftliche Reihe},
  journal   = {Postprints der Universit{\"a}t Potsdam : Mathematisch-Naturwissenschaftliche Reihe},
  number    = {2},
  issn      = {1866-8372},
  doi       = {10.25932/publishup-52426},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-524261},
  pages     = {17},
  year      = {2020},
  abstract  = {We study populations of globally coupled noisy rotators (oscillators with inertia) allowing a nonequilibrium transition from a desynchronized state to a synchronous one (with the nonvanishing order parameter). The newly developed analytical approaches resulted in solutions describing the synchronous state with constant order parameter for weakly inertial rotators, including the case of zero inertia, when the model is reduced to the Kuramoto model of coupled noise oscillators. These approaches provide also analytical criteria distinguishing supercritical and subcritical transitions to the desynchronized state and indicate the universality of such transitions in rotator ensembles. All the obtained analytical results are confirmed by the numerical ones, both by direct simulations of the large ensembles and by solution of the associated Fokker-Planck equation. We also propose generalizations of the developed approaches for setups where different rotators parameters (natural frequencies, masses, noise intensities, strengths and phase shifts in coupling) are dispersed.},
  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}
}
@article{GongPikovskij2019,
  author    = {Gong, Chen Chris and Pikovskij, Arkadij},
  title     = {Low-dimensional dynamics for higher-order harmonic, globally coupled phase-oscillator ensembles},
  series = {Physical review : E, Statistical, nonlinear and soft matter physics},
  volume    = {100},
  journal   = {Physical review : E, Statistical, nonlinear and soft matter physics},
  number    = {6},
  publisher = {American Physical Society},
  address   = {College Park},
  issn      = {2470-0045},
  doi       = {10.1103/PhysRevE.100.062210},
  pages     = {10},
  year      = {2019},
  abstract  = {The Kuramoto model, despite its popularity as a mean-field theory for many synchronization phenomenon of oscillatory systems, is limited to a first-order harmonic coupling of phases. For higher-order coupling, there only exists a low-dimensional theory in the thermodynamic limit. In this paper, we extend the formulation used by Watanabe and Strogatz to obtain a low-dimensional description of a system of arbitrary size of identical oscillators coupled all-to-all via their higher-order modes. To demonstrate an application of the formulation, we use a second harmonic globally coupled model, with a mean-field equal to the square of the Kuramoto mean-field. This model is known to exhibit asymmetrical clustering in previous numerical studies. We try to explain the phenomenon of asymmetrical clustering using the analytical theory developed here, as well as discuss certain phenomena not observed at the level of first-order harmonic coupling.},
  language  = {en}
}
@article{RosenblumPikovskij2019,
  author    = {Rosenblum, Michael and Pikovskij, Arkadij},
  title     = {Numerical phase reduction beyond the first order approximation},
  series = {Chaos : an interdisciplinary journal of nonlinear science},
  volume    = {29},
  journal   = {Chaos : an interdisciplinary journal of nonlinear science},
  number    = {1},
  publisher = {American Institute of Physics},
  address   = {Melville},
  issn      = {1054-1500},
  doi       = {10.1063/1.5079617},
  pages     = {6},
  year      = {2019},
  abstract  = {We develop a numerical approach to reconstruct the phase dynamics of driven or coupled self-sustained oscillators. Employing a simple algorithm for computation of the phase of a perturbed system, we construct numerically the equation for the evolution of the phase. Our simulations demonstrate that the description of the dynamics solely by phase variables can be valid for rather strong coupling strengths and large deviations from the limit cycle. Coupling functions depend crucially on the coupling and are generally non-decomposable in phase response and forcing terms. We also discuss the limitations of the approach. Published under license by AIP Publishing.},
  language  = {en}
}
@article{RosenblumPikovskij2018,
  author    = {Rosenblum, Michael and Pikovskij, Arkadij},
  title     = {Efficient determination of synchronization domains from observations of asynchronous dynamics},
  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       = {10.1063/1.5037012},
  pages     = {8},
  year      = {2018},
  abstract  = {We develop an approach for a fast experimental inference of synchronization properties of an oscillator. While the standard technique for determination of synchronization domains implies that the oscillator under study is forced with many different frequencies and amplitudes, our approach requires only several observations of a driven system. Reconstructing the phase dynamics from data, we successfully determine synchronization domains of noisy and chaotic oscillators. Our technique is especially important for experiments with living systems where an external action can be harmful and shall be minimized. Published by AIP Publishing.},
  language  = {en}
}
@article{ZhengPikovskij2018,
  author    = {Zheng, Chunming and Pikovskij, Arkadij},
  title     = {Delay-induced stochastic bursting in excitable noisy systems},
  series = {Physical review : E, Statistical, nonlinear and soft matter physics},
  volume    = {98},
  journal   = {Physical review : E, Statistical, nonlinear and soft matter physics},
  number    = {4},
  publisher = {American Physical Society},
  address   = {College Park},
  issn      = {2470-0045},
  doi       = {10.1103/PhysRevE.98.042148},
  pages     = {8},
  year      = {2018},
  abstract  = {We show that a combined action of noise and delayed feedback on an excitable theta-neuron leads to rather coherent stochastic bursting. An idealized point process, valid if the characteristic timescales in the problem are well separated, is used to describe statistical properties such as the power spectral density and the interspike interval distribution. We show how the main parameters of the point process, the spontaneous excitation rate, and the probability to induce a spike during the delay action can be calculated from the solutions of a stationary and a forced Fokker-Planck equation.},
  language  = {en}
}
@article{RosenauPikovskij2020,
  author    = {Rosenau, Philip and Pikovskij, Arkadij},
  title     = {Solitary phase waves in a chain of autonomous oscillators},
  series = {Chaos : an interdisciplinary journal of nonlinear science},
  volume    = {30},
  journal   = {Chaos : an interdisciplinary journal of nonlinear science},
  number    = {5},
  publisher = {American Institute of Physics, AIP},
  address   = {Melville, NY},
  issn      = {1054-1500},
  doi       = {10.1063/1.5144939},
  pages     = {8},
  year      = {2020},
  abstract  = {In the present paper, we study phase waves of self-sustained oscillators with a nearest-neighbor dispersive coupling on an infinite lattice. To analyze the underlying dynamics, we approximate the lattice with a quasi-continuum (QC). The resulting partial differential model is then further reduced to the Gardner equation, which predicts many properties of the underlying solitary structures. Using an iterative procedure on the original lattice equations, we determine the shapes of solitary waves, kinks, and the flat-like solitons that we refer to as flatons. Direct numerical experiments reveal that the interaction of solitons and flatons on the lattice is notably clean. All in all, we find that both the QC and the Gardner equation predict remarkably well the discrete patterns and their dynamics.},
  language  = {en}
}
@article{Pikovskij2021,
  author    = {Pikovskij, Arkadij},
  title     = {Transition to synchrony in chiral active particles},
  series = {Journal of physics. Complexity},
  volume    = {2},
  journal   = {Journal of physics. Complexity},
  number    = {2},
  publisher = {IOP Publ. Ltd.},
  address   = {Bristol},
  issn      = {2632-072X},
  doi       = {10.1088/2632-072X/abdadb},
  pages     = {8},
  year      = {2021},
  abstract  = {I study deterministic dynamics of chiral active particles in two dimensions. Particles are considered as discs interacting with elastic repulsive forces. An ensemble of particles, started from random initial conditions, demonstrates chaotic collisions resulting in their normal diffusion. This chaos is transient, as rather abruptly a synchronous collisionless state establishes. The life time of chaos grows exponentially with the number of particles. External forcing (periodic or chaotic) is shown to facilitate the synchronization transition.},
  language  = {en}
}
@article{ZhengToenjesPikovskij2021,
  author    = {Zheng, Chunming and Toenjes, Ralf and Pikovskij, Arkadij},
  title     = {Transition to synchrony in a three-dimensional swarming model with helical trajectories},
  series = {Physical review : E, Statistical, nonlinear and soft matter physics},
  volume    = {104},
  journal   = {Physical review : E, Statistical, nonlinear and soft matter physics},
  number    = {1},
  publisher = {American Physical Society},
  address   = {College Park},
  issn      = {2470-0045},
  doi       = {10.1103/PhysRevE.104.014216},
  pages     = {7},
  year      = {2021},
  abstract  = {We investigate the transition from incoherence to global collective motion in a three-dimensional swarming model of agents with helical trajectories, subject to noise and global coupling. Without noise this model was recently proposed as a generalization of the Kuramoto model and it was found that alignment of the velocities occurs discontinuously for arbitrarily small attractive coupling. Adding noise to the system resolves this singular limit and leads to a continuous transition, either to a directed collective motion or to center-of-mass rotations.},
  language  = {en}
}
@article{CestnikPikovskij2022,
  author    = {Cestnik, Rok and Pikovskij, Arkadij},
  title     = {Exact finite-dimensional reduction for a population of noisy oscillators and its link to Ott-Antonsen and Watanabe-Strogatz theories},
  series = {Chaos : an interdisciplinary journal of nonlinear science},
  volume    = {32},
  journal   = {Chaos : an interdisciplinary journal of nonlinear science},
  number    = {11},
  publisher = {AIP},
  address   = {Melville},
  issn      = {1054-1500},
  doi       = {10.1063/5.0106171},
  pages     = {15},
  year      = {2022},
  abstract  = {Populations of globally coupled phase oscillators are described in the thermodynamic limit by kinetic equations for the distribution densities or, equivalently, by infinite hierarchies of equations for the order parameters. Ott and Antonsen [Chaos 18, 037113 (2008)] have found an invariant finite-dimensional subspace on which the dynamics is described by one complex variable per population. For oscillators with Cauchy distributed frequencies or for those driven by Cauchy white noise, this subspace is weakly stable and, thus, describes the asymptotic dynamics. Here, we report on an exact finite-dimensional reduction of the dynamics outside of the Ott-Antonsen subspace. We show that the evolution from generic initial states can be reduced to that of three complex variables, plus a constant function. For identical noise-free oscillators, this reduction corresponds to the Watanabe-Strogatz system of equations [Watanabe and Strogatz, Phys. Rev. Lett. 70, 2391 (1993)]. We discuss how the reduced system can be used to explore the transient dynamics of perturbed ensembles. Published under an exclusive license by AIP Publishing.},
  language  = {en}
}
@article{PikovskijDolmatovaGoldobin2019,
  author    = {Pikovskij, Arkadij and Dolmatova, A. and Goldobin, Denis S.},
  title     = {Correlations of the States of Non-Entrained Oscillators in the Kuramoto Ensemble with Noise in the Mean Field},
  series = {Radiophysics and Quantum Electronics},
  volume    = {61},
  journal   = {Radiophysics and Quantum Electronics},
  number    = {8-9},
  publisher = {Springer},
  address   = {New York},
  issn      = {0033-8443},
  doi       = {10.1007/s11141-019-09927-4},
  pages     = {672 -- 680},
  year      = {2019},
  abstract  = {We consider the dynamics of the Kuramoto ensemble oscillators not included in a common synchronized cluster, where the mean field is subject to fluctuations. The fluctuations can be either related to the finite size of the ensemble or superimposed on the mean field in the form of common noise due to the constructive features of the system. It is shown that the states of such oscillators with close natural frequencies appear correlated with each other, since the mean-field fluctuations act as common noise. We quantify the effect with the synchronization index of two oscillators, which is calculated numerically and analytically as a function of the frequency difference and noise intensity. The results are rigorous for large ensembles with additional noise superimposed on the mean field and are qualitatively true for the systems where the mean-field fluctuations are due to the finite size of the ensemble. In the latter case, the effect is found to be independent of the number of oscillators in the ensemble.},
  language  = {en}
}
@article{TyulkinaGoldobinKlimenkoetal.2019,
  author    = {Tyulkina, Irina V. and Goldobin, Denis S. and Klimenko, Lyudmila S. and Pikovskij, Arkadij},
  title     = {Two-Bunch Solutions for the Dynamics of Ott-Antonsen Phase Ensembles},
  series = {Radiophysics and Quantum Electronics},
  volume    = {61},
  journal   = {Radiophysics and Quantum Electronics},
  number    = {8-9},
  publisher = {Springer},
  address   = {New York},
  issn      = {0033-8443},
  doi       = {10.1007/s11141-019-09924-7},
  pages     = {640 -- 649},
  year      = {2019},
  abstract  = {We have developed a method for deriving systems of closed equations for the dynamics of order parameters in the ensembles of phase oscillators. The Ott-Antonsen equation for the complex order parameter is a particular case of such equations. The simplest nontrivial extension of the Ott-Antonsen equation corresponds to two-bunch states of the ensemble. Based on the equations obtained, we study the dynamics of multi-bunch chimera states in coupled Kuramoto-Sakaguchi ensembles. We show an increase in the dimensionality of the system dynamics for two-bunch chimeras in the case of identical phase elements and a transition to one-bunch "Abrams chimeras" for imperfect identity (in the latter case, the one-bunch chimeras become attractive).},
  language  = {en}
}
@misc{PimenovaGoldobinRosenblumetal.2016,
  author    = {Pimenova, Anastasiya V. and Goldobin, Denis S. and Rosenblum, Michael and Pikovskij, Arkadij},
  title     = {Interplay of coupling and common noise at the transition to synchrony in oscillator populations},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-103471},
  pages     = {7},
  year      = {2016},
  abstract  = {There are two ways to synchronize oscillators: by coupling and by common forcing, which can be pure noise. By virtue of the Ott-Antonsen ansatz for sine-coupled phase oscillators, we obtain analytically tractable equations for the case where both coupling and common noise are present. While noise always tends to synchronize the phase oscillators, the repulsive coupling can act against synchrony, and we focus on this nontrivial situation. For identical oscillators, the fully synchronous state remains stable for small repulsive coupling; moreover it is an absorbing state which always wins over the asynchronous regime. For oscillators with a distribution of natural frequencies, we report on a counter-intuitive effect of dispersion (instead of usual convergence) of the oscillators frequencies at synchrony; the latter effect disappears if noise vanishes.},
  language  = {en}
}
@article{GoldschmidtPikovskijPoliti2019,
  author    = {Goldschmidt, Richard Janis and Pikovskij, Arkadij and Politi, Antonio},
  title     = {Blinking chimeras in globally coupled rotators},
  series = {Chaos : an interdisciplinary journal of nonlinear science},
  volume    = {29},
  journal   = {Chaos : an interdisciplinary journal of nonlinear science},
  number    = {7},
  publisher = {American Institute of Physics},
  address   = {Melville},
  issn      = {1054-1500},
  doi       = {10.1063/1.5105367},
  pages     = {7},
  year      = {2019},
  abstract  = {In globally coupled ensembles of identical oscillators so-called chimera states can be observed. The chimera state is a symmetry-broken regime, where a subset of oscillators forms a cluster, a synchronized population, while the rest of the system remains a collection of nonsynchronized, scattered units. We describe here a blinking chimera regime in an ensemble of seven globally coupled rotators (Kuramoto oscillators with inertia). It is characterized by a death-birth process, where a long-term stable cluster of four oscillators suddenly dissolves and is very quickly reborn with a new reshuffled configuration. We identify three different kinds of rare blinking events and give a quantitative characterization by applying stability analysis to the long-lived chaotic state and to the short-lived regular regimes that arise when the cluster dissolves.},
  language  = {en}
}
@article{SysoevPonomarenkoPikovskij2017,
  author    = {Sysoev, Ilya V. and Ponomarenko, Vladimir I. and Pikovskij, Arkadij},
  title     = {Reconstruction of coupling architecture of neural field networks from vector time series},
  series = {Communications in nonlinear science \& numerical simulation},
  volume    = {57},
  journal   = {Communications in nonlinear science \& numerical simulation},
  publisher = {Elsevier},
  address   = {Amsterdam},
  issn      = {1007-5704},
  doi       = {10.1016/j.cnsns.2017.10.006},
  pages     = {342 -- 351},
  year      = {2017},
  abstract  = {We propose a method of reconstruction of the network coupling matrix for a basic voltage-model of the neural field dynamics. Assuming that the multivariate time series of observations from all nodes are available, we describe a technique to find coupling constants which is unbiased in the limit of long observations. Furthermore, the method is generalized for reconstruction of networks with time-delayed coupling, including the reconstruction of unknown time delays. The approach is compared with other recently proposed techniques.},
  language  = {en}
}
@unpublished{PikovskijZaksFeudeletal.1995,
  author    = {Pikovskij, Arkadij and Zaks, Michael A. and Feudel, Ulrike and Kurths, J{\"u}rgen},
  title     = {Singular continuous spectra in dissipative dynamics},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-13787},
  year      = {1995},
  abstract  = {We demonstrate the occurrence of regimes with singular continuous (fractal) Fourier spectra in autonomous dissipative dynamical systems. The particular example in an ODE system at the accumulation points of bifurcation sequences associated to the creation of complicated homoclinic orbits. Two different machanisms responsible for the appearance of such spectra are proposed. In the first case when the geometry of the attractor is symbolically represented by the Thue-Morse sequence, both the continuous-time process and its descrete Poincar{\´e} map have singular power spectra. The other mechanism owes to the logarithmic divergence of the first return times near the saddle point; here the Poincar{\´e} map possesses the discrete spectrum, while the continuous-time process displays the singular one. A method is presented for computing the multifractal characteristics of the singular continuous spectra with the help of the usual Fourier analysis technique.},
  language  = {en}
}
@article{PetereitPikovskij2017,
  author    = {Petereit, Johannes and Pikovskij, Arkadij},
  title     = {Chaos synchronization by nonlinear coupling},
  series = {Communications in nonlinear science \& numerical simulation},
  volume    = {44},
  journal   = {Communications in nonlinear science \& numerical simulation},
  publisher = {Elsevier},
  address   = {Amsterdam},
  issn      = {1007-5704},
  doi       = {10.1016/j.cnsns.2016.09.002},
  pages     = {344 -- 351},
  year      = {2017},
  abstract  = {We study synchronization properties of three nonlinearly coupled chaotic maps. Coupling is introduced in such a way, that it cannot be reduced to pairwise terms, but includes combined action of all interacting units. For two models of nonlinear coupling we characterize the transition to complete synchrony, as well as partially synchronized states. Relation to hypernetworks of chaotic units is also discussed.},
  language  = {en}
}
@unpublished{PikovskijFeudel1994,
  author    = {Pikovskij, Arkadij and Feudel, Ulrike},
  title     = {Characterizing strange nonchaotic attractors},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-13405},
  year      = {1994},
  abstract  = {Strange nonchaotic attractors typically appear in quasiperiodically driven nonlinear systems. Two methods of their characterization are proposed. The first one is based on the bifurcation analysis of the systems, resulting from periodic approximations of the quasiperiodic forcing. Secondly, we propose th characterize their strangeness by calculating a phase sensitivity exponent, that measures the sensitivity with respect to changes of the phase of the external force. It is shown, that phase sensitivity appears if there is a non-zero probability for positive local Lyapunov exponents to occur.},
  language  = {en}
}
@article{ZaksPikovskij2017,
  author    = {Zaks, Michael A. and Pikovskij, Arkadij},
  title     = {Chimeras and complex cluster states in arrays of spin-torque oscillators},
  series = {Scientific reports},
  volume    = {7},
  journal   = {Scientific reports},
  publisher = {Macmillan Publishers Limited},
  address   = {London},
  issn      = {2045-2322},
  doi       = {10.1038/s41598-017-04918-9},
  year      = {2017},
  abstract  = {We consider synchronization properties of arrays of spin-torque nano-oscillators coupled via an RC load. We show that while the fully synchronized state of identical oscillators may be locally stable in some parameter range, this synchrony is not globally attracting. Instead, regimes of different levels of compositional complexity are observed. These include chimera states (a part of the array forms a cluster while other units are desynchronized), clustered chimeras (several clusters plus desynchronized oscillators), cluster state (all oscillators form several clusters), and partial synchronization (no clusters but a nonvanishing mean field). Dynamically, these states are also complex, demonstrating irregular and close to quasiperiodic modulation. Remarkably, when heterogeneity of spin-torque oscillators is taken into account, dynamical complexity even increases: close to the onset of a macroscopic mean field, the dynamics of this field is rather irregular.},
  language  = {en}
}
@article{PopovychLysyanskyRosenblumetal.2017,
  author    = {Popovych, Oleksandr V. and Lysyansky, Borys and Rosenblum, Michael and Pikovskij, Arkadij and Tass, Peter A.},
  title     = {Pulsatile desynchronizing delayed feedback for closed-loop deep brain stimulation},
  series = {PLoS one},
  volume    = {12},
  journal   = {PLoS one},
  publisher = {PLoS},
  address   = {San Fransisco},
  issn      = {1932-6203},
  doi       = {10.1371/journal.pone.0173363},
  pages     = {29},
  year      = {2017},
  abstract  = {High-frequency (HF) deep brain stimulation (DBS) is the gold standard for the treatment of medically refractory movement disorders like Parkinson's disease, essential tremor, and dystonia, with a significant potential for application to other neurological diseases. The standard setup of HF DBS utilizes an open-loop stimulation protocol, where a permanent HF electrical pulse train is administered to the brain target areas irrespectively of the ongoing neuronal dynamics. Recent experimental and clinical studies demonstrate that a closed-loop, adaptive DBS might be superior to the open-loop setup. We here combine the notion of the adaptive high-frequency stimulation approach, that aims at delivering stimulation adapted to the extent of appropriately detected biomarkers, with specifically desynchronizing stimulation protocols. To this end, we extend the delayed feedback stimulation methods, which are intrinsically closed-loop techniques and specifically designed to desynchronize abnormal neuronal synchronization, to pulsatile electrical brain stimulation. We show that permanent pulsatile high-frequency stimulation subjected to an amplitude modulation by linear or nonlinear delayed feedback methods can effectively and robustly desynchronize a STN-GPe network of model neurons and suggest this approach for desynchronizing closed-loop DBS.},
  language  = {en}
}
@article{PeterPikovskij2018,
  author    = {Peter, Franziska and Pikovskij, Arkadij},
  title     = {Transition to collective oscillations in finite Kuramoto ensembles},
  series = {Physical review : E, Statistical, nonlinear and soft matter physics},
  volume    = {97},
  journal   = {Physical review : E, Statistical, nonlinear and soft matter physics},
  number    = {3},
  publisher = {American Physical Society},
  address   = {College Park},
  issn      = {2470-0045},
  doi       = {10.1103/PhysRevE.97.032310},
  pages     = {10},
  year      = {2018},
  abstract  = {We present an alternative approach to finite-size effects around the synchronization transition in the standard Kuramoto model. Our main focus lies on the conditions under which a collective oscillatory mode is well defined. For this purpose, the minimal value of the amplitude of the complex Kuramoto order parameter appears as a proper indicator. The dependence of this minimum on coupling strength varies due to sampling variations and correlates with the sample kurtosis of the natural frequency distribution. The skewness of the frequency sample determines the frequency of the resulting collective mode. The effects of kurtosis and skewness hold in the thermodynamic limit of infinite ensembles. We prove this by integrating a self-consistency equation for the complex Kuramoto order parameter for two families of distributions with controlled kurtosis and skewness, respectively.},
  language  = {en}
}
@article{Pikovskij2018,
  author    = {Pikovskij, Arkadij},
  title     = {Reconstruction of a random phase dynamics network from observations},
  series = {Physics letters : A},
  volume    = {382},
  journal   = {Physics letters : A},
  number    = {4},
  publisher = {Elsevier},
  address   = {Amsterdam},
  issn      = {0375-9601},
  doi       = {10.1016/j.physleta.2017.11.012},
  pages     = {147 -- 152},
  year      = {2018},
  abstract  = {We consider networks of coupled phase oscillators of different complexity: Kuramoto-Daido-type networks, generalized Winfree networks, and hypernetworks with triple interactions. For these setups an inverse problem of reconstruction of the network connections and of the coupling function from the observations of the phase dynamics is addressed. We show how a reconstruction based on the minimization of the squared error can be implemented in all these cases. Examples include random networks with full disorder both in the connections and in the coupling functions, as well as networks where the coupling functions are taken from experimental data of electrochemical oscillators. The method can be directly applied to asynchronous dynamics of units, while in the case of synchrony, additional phase resettings are necessary for reconstruction.},
  language  = {en}
}
@unpublished{KurthsPikovskijScheffczyk1994,
  author    = {Kurths, J{\"u}rgen and Pikovskij, Arkadij and Scheffczyk, Christian},
  title     = {Roughening interfaces in deterministic dynamics},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-13447},
  year      = {1994},
  abstract  = {Two deterministic processes leading to roughening interfaces are considered. It is shown that the dynamics of linear perturbations of turbulent regimes in coupled map lattices is governed by a discrete version of the Kardar-Parisi-Zhang equation. The asymptotic scaling behavior of the perturbation field is investigated in the case of large lattices. Secondly, the dynamics of an order-disorder interface is modelled with a simple two-dimensional coupled map lattice, possesing a turbulent and a laminar state. It is demonstrated, that in some range of parameters the spreading of the turbulent state is accompanied by kinetic roughening of the interface.},
  language  = {en}
}
@article{AransonPikovskij2022,
  author    = {Aranson, Igor S. and Pikovskij, Arkadij},
  title     = {Confinement and collective escape of active particles},
  series = {Physical review letters},
  volume    = {128},
  journal   = {Physical review letters},
  number    = {10},
  publisher = {American Physical Society},
  address   = {College Park, Md.},
  issn      = {0031-9007},
  doi       = {10.1103/PhysRevLett.128.108001},
  pages     = {6},
  year      = {2022},
  abstract  = {Active matter broadly covers the dynamics of self-propelled particles. While the onset of collective behavior in homogenous active systems is relatively well understood, the effect of inhomogeneities such as obstacles and traps lacks overall clarity. Here, we study how interacting, self-propelled particles become trapped and released from a trap. We have found that captured particles aggregate into an orbiting condensate with a crystalline structure. As more particles are added, the trapped condensates escape as a whole. Our results shed light on the effects of confinement and quenched disorder in active matter.},
  language  = {en}
}
@article{GengelPikovskij2022,
  author    = {Gengel, Erik and Pikovskij, Arkadij},
  title     = {Phase reconstruction from oscillatory data with iterated Hilbert transform embeddings},
  series = {Physica : D, Nonlinear phenomena},
  volume    = {429},
  journal   = {Physica : D, Nonlinear phenomena},
  publisher = {Elsevier},
  address   = {Amsterdam},
  issn      = {0167-2789},
  doi       = {10.1016/j.physd.2021.133070},
  pages     = {9},
  year      = {2022},
  abstract  = {In the data analysis of oscillatory systems, methods based on phase reconstruction are widely used to characterize phase-locking properties and inferring the phase dynamics. The main component in these studies is an extraction of the phase from a time series of an oscillating scalar observable. We discuss a practical procedure of phase reconstruction by virtue of a recently proposed method termed iterated Hilbert transform embeddings. We exemplify the potential benefits and limitations of the approach by applying it to a generic observable of a forced Stuart-Landau oscillator. Although in many cases, unavoidable amplitude modulation of the observed signal does not allow for perfect phase reconstruction, in cases of strong stability of oscillations and a high frequency of the forcing, iterated Hilbert transform embeddings significantly improve the quality of the reconstructed phase. We also demonstrate that for significant amplitude modulation, iterated embeddings do not provide any improvement.},
  language  = {en}
}
@article{Pikovskij2021,
  author    = {Pikovskij, Arkadij},
  title     = {Synchronization of oscillators with hyperbolic chaotic phases},
  series = {Izvestija vysšich učebnych zavedenij : naučno-techničeskij žurnal = Izvestiya VUZ. Prikladnaja nelinejnaja dinamika = Applied nonlinear dynamics},
  volume    = {29},
  journal   = {Izvestija vysšich učebnych zavedenij : naučno-techničeskij žurnal = Izvestiya VUZ. Prikladnaja nelinejnaja dinamika = Applied nonlinear dynamics},
  number    = {1},
  publisher = {Saratov State University},
  address   = {Saratov},
  issn      = {0869-6632},
  doi       = {10.18500/0869-6632-2021-29-1-78-87},
  pages     = {78 -- 87},
  year      = {2021},
  abstract  = {Topic and aim. Synchronization in populations of coupled oscillators can be characterized with order parameters that describe collective order in ensembles. A dependence of the order parameter on the coupling constants is well-known for coupled periodic oscillators. The goal of the study is to extend this analysis to ensembles of oscillators with chaotic phases, moreover with phases possessing hyperbolic chaos. Models and methods. Two models are studied in the paper. One is an abstract discrete-time map, composed with a hyperbolic Bernoulli transformation and with Kuramoto dynamics. Another model is a system of coupled continuous-time chaotic oscillators, where each individual oscillator has a hyperbolic attractor of Smale-Williams type. Results. The discrete-time model is studied with the Ott-Antonsen ansatz, which is shown to be invariant under the application of the Bernoulli map. The analysis of the resulting map for the order parameter shows, that the asynchronouis state is always stable, but the synchronous one becomes stable above a certain coupling strength. Numerical analysis of the continuous-time model reveals a complex sequence of transitions from an asynchronous state to a completely synchronous hyperbolic chaos, with intermediate stages that include regimes with periodic in time mean field, as well as with weakly and strongly irregular mean field variations. Discussion. Results demonstrate that synchronization of systems with hyperbolic chaos of phases is possible, although a rather strong coupling is required. The approach can be applied to other systems of interacting units with hyperbolic chaotic dynamics.},
  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{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{SmirnovOsipovPikovskij2018,
  author    = {Smirnov, Lev A. and Osipov, Grigory V. and Pikovskij, Arkadij},
  title     = {Solitary synchronization waves in distributed oscillator populations},
  series = {Physical review : E, Statistical, nonlinear and soft matter physics},
  volume    = {98},
  journal   = {Physical review : E, Statistical, nonlinear and soft matter physics},
  number    = {6},
  publisher = {American Physical Society},
  address   = {College Park},
  issn      = {2470-0045},
  doi       = {10.1103/PhysRevE.98.062222},
  pages     = {062222-1 -- 062222-7},
  year      = {2018},
  abstract  = {We demonstrate the existence of solitary waves of synchrony in one-dimensional arrays of oscillator populations with Laplacian coupling. Characterizing each community with its complex order parameter, we obtain lattice equations similar to those of the discrete nonlinear Schrodinger system. Close to full synchrony, we find solitary waves for the order parameter perturbatively, starting from the known phase compactons and kovatons; these solutions are extended numerically to the full domain of possible synchrony levels. For nonidentical oscillators, the existence of dissipative solitons is shown.},
  language  = {en}
}
@article{MunyaevSmirnovKostinetal.2020,
  author    = {Munyaev, Vyacheslav O. and Smirnov, Lev A. and Kostin, Vasily A. and Osipov, Grigory V. and Pikovskij, Arkadij},
  title     = {Analytical approach to synchronous states of globally coupled noisy rotators},
  series = {New journal of physics : the open-access journal for physics},
  volume    = {22},
  journal   = {New journal of physics : the open-access journal for physics},
  number    = {2},
  publisher = {IOP Publ. Ltd.},
  address   = {Bristol},
  issn      = {1367-2630},
  doi       = {10.1088/1367-2630/ab6f93},
  pages     = {14},
  year      = {2020},
  abstract  = {We study populations of globally coupled noisy rotators (oscillators with inertia) allowing a nonequilibrium transition from a desynchronized state to a synchronous one (with the nonvanishing order parameter). The newly developed analytical approaches resulted in solutions describing the synchronous state with constant order parameter for weakly inertial rotators, including the case of zero inertia, when the model is reduced to the Kuramoto model of coupled noise oscillators. These approaches provide also analytical criteria distinguishing supercritical and subcritical transitions to the desynchronized state and indicate the universality of such transitions in rotator ensembles. All the obtained analytical results are confirmed by the numerical ones, both by direct simulations of the large ensembles and by solution of the associated Fokker-Planck equation. We also propose generalizations of the developed approaches for setups where different rotators parameters (natural frequencies, masses, noise intensities, strengths and phase shifts in coupling) are dispersed.},
  language  = {en}
}
@article{SmirnovOsipovPikovskij2017,
  author    = {Smirnov, Lev A. and Osipov, Grigory V. and Pikovskij, Arkadij},
  title     = {Chimera patterns in the Kuramoto-Battogtokh model},
  series = {Journal of physics : A, Mathematical and theoretical},
  volume    = {50},
  journal   = {Journal of physics : A, Mathematical and theoretical},
  number    = {8},
  publisher = {IOP Publ. Ltd.},
  address   = {Bristol},
  issn      = {1751-8113},
  doi       = {10.1088/1751-8121/aa55f1},
  pages     = {10},
  year      = {2017},
  abstract  = {Kuramoto and Battogtokh (2002 Nonlinear Phenom. Complex Syst. 5 380) discovered chimera states represented by stable coexisting synchrony and asynchrony domains in a lattice of coupled oscillators. After a reformulation in terms of a local order parameter, the problem can be reduced to partial differential equations. We find uniformly rotating, spatially periodic chimera patterns as solutions of a reversible ordinary differential equation, and demonstrate a plethora of such states. In the limit of neutral coupling they reduce to analytical solutions in the form of one-and two-point chimera patterns as well as localized chimera solitons. Patterns at weakly attracting coupling are characterized by virtue of a perturbative approach. Stability analysis reveals that only the simplest chimeras with one synchronous region are stable.},
  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}
}
@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{MunyaevSmirnovKostinetal.2020,
  author    = {Munyaev, Vyacheslav and Smirnov, Lev A. and Kostin, Vasily and Osipov, Grigory V. and Pikovskij, Arkadij},
  title     = {Analytical approach to synchronous states of globally coupled noisy rotators},
  series = {New Journal of Physics},
  volume    = {22},
  journal   = {New Journal of Physics},
  number    = {2},
  publisher = {Springer Science},
  address   = {New York},
  pages     = {15},
  year      = {2020},
  abstract  = {We study populations of globally coupled noisy rotators (oscillators with inertia) allowing a nonequilibrium transition from a desynchronized state to a synchronous one (with the nonvanishing order parameter). The newly developed analytical approaches resulted in solutions describing the synchronous state with constant order parameter for weakly inertial rotators, including the case of zero inertia, when the model is reduced to the Kuramoto model of coupled noise oscillators. These approaches provide also analytical criteria distinguishing supercritical and subcritical transitions to the desynchronized state and indicate the universality of such transitions in rotator ensembles. All the obtained analytical results are confirmed by the numerical ones, both by direct simulations of the large ensembles and by solution of the associated Fokker-Planck equation. We also propose generalizations of the developed approaches for setups where different rotators parameters (natural frequencies, masses, noise intensities, strengths and phase shifts in coupling) are dispersed.},
  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}
}
@article{RosenauPikovskij2021,
  author    = {Rosenau, Philip and Pikovskij, Arkadij},
  title     = {Waves in strongly nonlinear Gardner-like equations on a lattice},
  series = {Nonlinearity / the Institute of Physics and the London Mathematical Society},
  volume    = {34},
  journal   = {Nonlinearity / the Institute of Physics and the London Mathematical Society},
  number    = {8},
  publisher = {IOP Publ. Ltd.},
  address   = {Bristol},
  issn      = {0951-7715},
  doi       = {10.1088/1361-6544/ac0f51},
  pages     = {5872 -- 5896},
  year      = {2021},
  abstract  = {We introduce and study a family of lattice equations which may be viewed either as a strongly nonlinear discrete extension of the Gardner equation, or a non-convex variant of the Lotka-Volterra chain. Their deceptively simple form supports a very rich family of complex solitary patterns. Some of these patterns are also found in the quasi-continuum rendition, but the more intriguing ones, like interlaced pairs of solitary waves, or waves which may reverse their direction either spontaneously or due a collision, are an intrinsic feature of the discrete realm.},
  language  = {en}
}
@article{BolotovSmirnovBubnovaetal.2021,
  author    = {Bolotov, Maxim I. and Smirnov, Lev A. and Bubnova, E. S. and Osipov, Grigory V. and Pikovskij, Arkadij},
  title     = {Spatiotemporal regimes in the Kuramoto-Battogtokh system of nonidentical oscillators},
  series = {Journal of experimental and theoretical physics},
  volume    = {132},
  journal   = {Journal of experimental and theoretical physics},
  number    = {1},
  publisher = {Springer},
  address   = {Heidelberg [u.a.]},
  issn      = {1063-7761},
  doi       = {10.1134/S1063776121010106},
  pages     = {127 -- 147},
  year      = {2021},
  abstract  = {We consider the spatiotemporal states of an ensemble of nonlocally coupled nonidentical phase oscillators, which correspond to different regimes of the long-term evolution of such a system. We have obtained homogeneous, twisted, and nonhomogeneous stationary solutions to the Ott-Antonsen equations corresponding to key variants of the realized collective rotational motion of elements of the medium in question with nonzero mesoscopic characteristics determining the degree of coherence of the dynamics of neighboring particles. We have described the procedures of the search for the class of nonhomogeneous solutions as stationary points of the auxiliary point map and of determining the stability based on analysis of the eigenvalue spectrum of the composite operator. Static and breather cluster regimes have been demonstrated and described, as well as the regimes with an irregular behavior of averaged complex fields including, in particular, the local order parameter.},
  language  = {en}
}
@article{SmirnovBolotovOsipovetal.2021,
  author    = {Smirnov, Lev A. and Bolotov, Maxim I. and Osipov, Grigorij V. and Pikovskij, Arkadij},
  title     = {Disorder fosters chimera in an array of motile particles},
  series = {Physical review : E, Statistical, nonlinear and soft matter physics},
  volume    = {104},
  journal   = {Physical review : E, Statistical, nonlinear and soft matter physics},
  number    = {3},
  publisher = {American Physical Society},
  address   = {Melville, NY},
  issn      = {2470-0045},
  doi       = {10.1103/PhysRevE.104.034205},
  pages     = {8},
  year      = {2021},
  abstract  = {We consider an array of nonlocally coupled oscillators on a ring, which for equally spaced units possesses a Kuramoto-Battogtokh chimera regime and a synchronous state. We demonstrate that disorder in oscillators positions leads to a transition from the synchronous to the chimera state. For a static (quenched) disorder we find that the probability of synchrony survival depends on the number of particles, from nearly zero at small populations to one in the thermodynamic limit. Furthermore, we demonstrate how the synchrony gets destroyed for randomly (ballistically or diffusively) moving oscillators. We show that, depending on the number of oscillators, there are different scalings of the transition time with this number and the velocity of the units.},
  language  = {en}
}
@article{ChigarevKazakovPikovskij2021,
  author    = {Chigarev, Vladimir and Kazakov, Alexey and Pikovskij, Arkadij},
  title     = {Mutual singularities of overlapping attractor and repeller},
  series = {Chaos : an interdisciplinary journal of nonlinear science},
  volume    = {31},
  journal   = {Chaos : an interdisciplinary journal of nonlinear science},
  number    = {8},
  publisher = {American Institute of Physics},
  address   = {Melville},
  issn      = {1054-1500},
  doi       = {10.1063/5.0056891},
  pages     = {10},
  year      = {2021},
  abstract  = {We apply the concepts of relative dimensions and mutual singularities to characterize the fractal properties of overlapping attractor and repeller in chaotic dynamical systems. We consider one analytically solvable example (a generalized baker's map); two other examples, the Anosov-Mobius and the Chirikov-Mobius maps, which possess fractal attractor and repeller on a two-dimensional torus, are explored numerically. We demonstrate that although for these maps the stable and unstable directions are not orthogonal to each other, the relative Renyi and Kullback-Leibler dimensions as well as the mutual singularity spectra for the attractor and repeller can be well approximated under orthogonality assumption of two fractals.},
  language  = {en}
}