@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{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} } @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{LueckPikovskij2011, author = {Lueck, S. and Pikovskij, Arkadij}, title = {Dynamics of multi-frequency oscillator ensembles with resonant coupling}, 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 = {28-29}, publisher = {Elsevier}, address = {Amsterdam}, issn = {0375-9601}, doi = {10.1016/j.physleta.2011.06.016}, pages = {2714 -- 2719}, year = {2011}, abstract = {We study dynamics of populations of resonantly coupled oscillators having different frequencies. Starting from the coupled van der Pol equations we derive the Kuramoto-type phase model for the situation, where the natural frequencies of two interacting subpopulations are in relation 2 : 1. Depending on the parameter of coupling, ensembles can demonstrate fully synchronous clusters, partial synchrony (only one subpopulation synchronizes), or asynchrony in both subpopulations. Theoretical description of the dynamics based on the Watanabe-Strogatz approach is developed.}, language = {en} } @article{VlasovRosenblumPikovskij2016, author = {Vlasov, Vladimir and Rosenblum, Michael and Pikovskij, Arkadij}, title = {Dynamics of weakly inhomogeneous oscillator populations: perturbation theory on top of Watanabe-Strogatz integrability}, series = {Journal of physics : A, Mathematical and theoretical}, volume = {49}, journal = {Journal of physics : A, Mathematical and theoretical}, publisher = {IOP Publ. Ltd.}, address = {Bristol}, issn = {1751-8113}, doi = {10.1088/1751-8113/49/31/31LT02}, pages = {8}, year = {2016}, abstract = {As has been shown by Watanabe and Strogatz (WS) (1993 Phys. Rev. Lett. 70 2391), a population of identical phase oscillators, sine-coupled to a common field, is a partially integrable system: for any ensemble size its dynamics reduce to equations for three collective variables. Here we develop a perturbation approach for weakly nonidentical ensembles. We calculate corrections to the WS dynamics for two types of perturbations: those due to a distribution of natural frequencies and of forcing terms, and those due to small white noise. We demonstrate that in both cases, the complex mean field for which the dynamical equations are written is close to the Kuramoto order parameter, up to the leading order in the perturbation. This supports the validity of the dynamical reduction suggested by Ott and Antonsen (2008 Chaos 18 037113) for weakly inhomogeneous populations.}, language = {en} } @article{OcampoEspindolaOmel'chenkoKiss2021, author = {Ocampo-Espindola, Jorge Luis and Omel'chenko, Oleh and Kiss, Istvan Z.}, title = {Non-monotonic transients to synchrony in Kuramoto networks and electrochemical oscillators}, series = {Journal of physics. Complexity}, volume = {2}, journal = {Journal of physics. Complexity}, number = {1}, publisher = {IOP Publ. Ltd.}, address = {Bristol}, issn = {2632-072X}, doi = {10.1088/2632-072X/abe109}, pages = {15}, year = {2021}, abstract = {We performed numerical simulations with the Kuramoto model and experiments with oscillatory nickel electrodissolution to explore the dynamical features of the transients from random initial conditions to a fully synchronized (one-cluster) state. The numerical simulations revealed that certain networks (e.g., globally coupled or dense Erdos-Renyi random networks) showed relatively simple behavior with monotonic increase of the Kuramoto order parameter from the random initial condition to the fully synchronized state and that the transient times exhibited a unimodal distribution. However, some modular networks with bridge elements were identified which exhibited non-monotonic variation of the order parameter with local maximum and/or minimum. In these networks, the histogram of the transients times became bimodal and the mean transient time scaled well with inverse of the magnitude of the second largest eigenvalue of the network Laplacian matrix. The non-monotonic transients increase the relative standard deviations from about 0.3 to 0.5, i.e., the transient times became more diverse. The non-monotonic transients are related to generation of phase patterns where the modules are synchronized but approximately anti-phase to each other. The predictions of the numerical simulations were demonstrated in a population of coupled oscillatory electrochemical reactions in global, modular, and irregular tree networks. The findings clarify the role of network structure in generation of complex transients that can, for example, play a role in intermittent desynchronization of the circadian clock due to external cues or in deep brain stimulations where long transients are required after a desynchronization stimulus.}, language = {en} } @article{KomarovPikovskij2014, author = {Komarov, Maxim and Pikovskij, Arkadij}, title = {The Kuramoto model of coupled oscillators with a bi-harmonic coupling function}, series = {Physica : D, Nonlinear phenomena}, volume = {289}, journal = {Physica : D, Nonlinear phenomena}, publisher = {Elsevier}, address = {Amsterdam}, issn = {0167-2789}, doi = {10.1016/j.physd.2014.09.002}, pages = {18 -- 31}, year = {2014}, abstract = {We study synchronization in a Kuramoto model of globally coupled phase oscillators with a bi-harmonic coupling function, in the thermodynamic limit of large populations. We develop a method for an analytic solution of self-consistent equations describing uniformly rotating complex order parameters, both for single-branch (one possible state of locked oscillators) and multi-branch (two possible values of locked phases) entrainment. We show that synchronous states coexist with the neutrally linearly stable asynchronous regime. The latter has a finite life time for finite ensembles, this time grows with the ensemble size as a power law. (C) 2014 Elsevier B.V. All rights reserved.}, language = {en} }