TY - JOUR A1 - Goldobin, Denis S. A1 - Tyulkina, Irina V. A1 - Klimenko, Lyudmila S. A1 - Pikovskij, Arkadij T1 - Collective mode reductions for populations of coupled noisy oscillators JF - Chaos : an interdisciplinary journal of nonlinear science N2 - 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. Y1 - 2018 U6 - https://doi.org/10.1063/1.5053576 SN - 1054-1500 SN - 1089-7682 VL - 28 IS - 10 PB - American Institute of Physics CY - Melville ER - TY - JOUR A1 - Tyulkina, Irina A1 - Goldobin, Denis S. A1 - Klimenko, Lyudmila S. A1 - Pikovskij, Arkadij T1 - Dynamics of noisy oscillator populations beyond the Ott-Antonsen Ansatz JF - Physical review letters N2 - 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. Y1 - 2018 U6 - https://doi.org/10.1103/PhysRevLett.120.264101 SN - 0031-9007 SN - 1079-7114 VL - 120 IS - 26 PB - American Physical Society CY - College Park ER - TY - JOUR A1 - Rosenblum, Michael A1 - Pikovskij, Arkadij T1 - Efficient determination of synchronization domains from observations of asynchronous dynamics JF - Chaos : an interdisciplinary journal of nonlinear science N2 - 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. Y1 - 2018 U6 - https://doi.org/10.1063/1.5037012 SN - 1054-1500 SN - 1089-7682 VL - 28 IS - 10 PB - American Institute of Physics CY - Melville ER - TY - JOUR A1 - Zheng, Chunming A1 - Pikovskij, Arkadij T1 - Delay-induced stochastic bursting in excitable noisy systems JF - Physical review : E, Statistical, nonlinear and soft matter physics N2 - 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. Y1 - 2018 U6 - https://doi.org/10.1103/PhysRevE.98.042148 SN - 2470-0045 SN - 2470-0053 VL - 98 IS - 4 PB - American Physical Society CY - College Park ER - TY - JOUR A1 - Peter, Franziska A1 - Pikovskij, Arkadij T1 - Transition to collective oscillations in finite Kuramoto ensembles JF - Physical review : E, Statistical, nonlinear and soft matter physics N2 - 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. Y1 - 2018 U6 - https://doi.org/10.1103/PhysRevE.97.032310 SN - 2470-0045 SN - 2470-0053 VL - 97 IS - 3 PB - American Physical Society CY - College Park ER - TY - JOUR A1 - Pikovskij, Arkadij T1 - Reconstruction of a random phase dynamics network from observations JF - Physics letters : A N2 - 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. KW - Phase dynamics KW - Network reconstruction Y1 - 2017 U6 - https://doi.org/10.1016/j.physleta.2017.11.012 SN - 0375-9601 SN - 1873-2429 VL - 382 IS - 4 SP - 147 EP - 152 PB - Elsevier CY - Amsterdam ER - TY - JOUR A1 - Grines, Evgeny A1 - Osipov, Grigory V. A1 - Pikovskij, Arkadij T1 - Describing dynamics of driven multistable oscillators with phase transfer curves JF - Chaos : an interdisciplinary journal of nonlinear science N2 - 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. Y1 - 2018 U6 - https://doi.org/10.1063/1.5037290 SN - 1054-1500 SN - 1089-7682 VL - 28 IS - 10 PB - American Institute of Physics CY - Melville ER - TY - JOUR A1 - Smirnov, Lev A. A1 - Osipov, Grigory V. A1 - Pikovskij, Arkadij T1 - Solitary synchronization waves in distributed oscillator populations JF - Physical review : E, Statistical, nonlinear and soft matter physics N2 - 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. Y1 - 2018 U6 - https://doi.org/10.1103/PhysRevE.98.062222 SN - 2470-0045 SN - 2470-0053 VL - 98 IS - 6 SP - 062222-1 EP - 062222-7 PB - American Physical Society CY - College Park ER - TY - JOUR A1 - Bolotov, Maxim I. A1 - Smirnov, Lev A. A1 - Osipov, Grigory V. A1 - Pikovskij, Arkadij T1 - Simple and complex chimera states in a nonlinearly coupled oscillatory medium JF - Chaos : an interdisciplinary journal of nonlinear science N2 - 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. Y1 - 2018 U6 - https://doi.org/10.1063/1.5011678 SN - 1054-1500 SN - 1089-7682 VL - 28 IS - 4 PB - American Institute of Physics CY - Melville ER - TY - GEN A1 - Bolotov, Maxim A1 - Smirnov, Lev A. A1 - Osipov, Grigory V. A1 - Pikovskij, Arkadij T1 - Complex chimera states in a nonlinearly coupled oscillatory medium T2 - 2018 2nd School on Dynamics of Complex Networks and their Application in Intellectual Robotics (DCNAIR) N2 - 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. KW - phase oscillator KW - nonlocal coupling KW - synchronization KW - chimera state KW - partial synchronization KW - phase lag KW - nonlinear dynamics Y1 - 2018 SN - 978-1-5386-5818-5 U6 - https://doi.org/10.1109/DCNAIR.2018.8589210 SP - 17 EP - 20 PB - IEEE CY - New York ER -