@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} } @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} }