TY - JOUR A1 - Gengel, Erik A1 - Pikovskij, Arkadij T1 - Phase demodulation with iterative Hilbert transform embeddings JF - Signal processing N2 - 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. KW - Phase modulation KW - Hilbert transform KW - Embedding Y1 - 2019 U6 - https://doi.org/10.1016/j.sigpro.2019.07.005 SN - 0165-1684 SN - 1872-7557 VL - 165 SP - 115 EP - 127 PB - Elsevier CY - Amsterdam ER - TY - JOUR A1 - Rosenblum, Michael A1 - Pikovskij, Arkadij T1 - Numerical phase reduction beyond the first order approximation JF - Chaos : an interdisciplinary journal of nonlinear science N2 - 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. Y1 - 2019 U6 - https://doi.org/10.1063/1.5079617 SN - 1054-1500 SN - 1089-7682 VL - 29 IS - 1 PB - American Institute of Physics CY - Melville ER - TY - JOUR A1 - Rosenblum, Michael A1 - Pikovskij, Arkadij T1 - Nonlinear phase coupling functions: a numerical study JF - Philosophical Transactions of the Royal Society of London, Series A : Mathematical, Physical and Engineering Sciences N2 - 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. KW - phase approximation KW - coupling function KW - phase response curve Y1 - 2019 U6 - https://doi.org/10.1098/rsta.2019.0093 SN - 1364-503X SN - 1471-2962 VL - 377 IS - 2160 PB - Royal Society CY - London ER - TY - JOUR A1 - Chigarev, Vladimir A1 - Kazakov, Alexey A1 - Pikovskij, Arkadij T1 - Mutual singularities of overlapping attractor and repeller JF - Chaos : an interdisciplinary journal of nonlinear science N2 - 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. Y1 - 2021 U6 - https://doi.org/10.1063/5.0056891 SN - 1054-1500 SN - 1089-7682 VL - 31 IS - 8 PB - American Institute of Physics CY - Melville ER - TY - JOUR A1 - Peter, Franziska A1 - Gong, Chen Chris A1 - Pikovskij, Arkadij T1 - Microscopic correlations in the finite-size Kuramoto model of coupled oscillators JF - Physical review : E, Statistical, nonlinear and soft matter physics N2 - 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. Y1 - 2019 U6 - https://doi.org/10.1103/PhysRevE.100.032210 SN - 2470-0045 SN - 2470-0053 VL - 100 IS - 3 PB - American Physical Society CY - College Park ER - TY - JOUR A1 - Gong, Chen Chris A1 - Pikovskij, Arkadij T1 - Low-dimensional dynamics for higher-order harmonic, globally coupled phase-oscillator ensembles JF - Physical review : E, Statistical, nonlinear and soft matter physics N2 - 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. Y1 - 2019 U6 - https://doi.org/10.1103/PhysRevE.100.062210 SN - 2470-0045 SN - 2470-0053 VL - 100 IS - 6 PB - American Physical Society CY - College Park ER - TY - GEN A1 - Pimenova, Anastasiya V. A1 - Goldobin, Denis S. A1 - Rosenblum, Michael A1 - Pikovskij, Arkadij T1 - Interplay of coupling and common noise at the transition to synchrony in oscillator populations N2 - 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. T3 - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - 310 Y1 - 2016 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-103471 ER - TY - JOUR A1 - Cestnik, Rok A1 - Pikovskij, Arkadij T1 - Exact finite-dimensional reduction for a population of noisy oscillators and its link to Ott-Antonsen and Watanabe-Strogatz theories JF - Chaos : an interdisciplinary journal of nonlinear science N2 - 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. Y1 - 2022 U6 - https://doi.org/10.1063/5.0106171 SN - 1054-1500 SN - 1089-7682 VL - 32 IS - 11 PB - AIP CY - Melville 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 - Smirnov, Lev A. A1 - Bolotov, Maxim I. A1 - Osipov, Grigorij V. A1 - Pikovskij, Arkadij T1 - Disorder fosters chimera in an array of motile particles JF - Physical review : E, Statistical, nonlinear and soft matter physics N2 - 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. Y1 - 2021 U6 - https://doi.org/10.1103/PhysRevE.104.034205 SN - 2470-0045 SN - 2470-0053 VL - 104 IS - 3 PB - American Physical Society CY - Melville, NY ER -