TY - JOUR A1 - Bolotov, Maxim I. A1 - Smirnov, Lev A. A1 - Bubnova, E. S. A1 - Osipov, Grigory V. A1 - Pikovskij, Arkadij T1 - Spatiotemporal regimes in the Kuramoto-Battogtokh system of nonidentical oscillators JF - Journal of experimental and theoretical physics N2 - 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. Y1 - 2021 U6 - https://doi.org/10.1134/S1063776121010106 SN - 1063-7761 SN - 1090-6509 VL - 132 IS - 1 SP - 127 EP - 147 PB - Springer CY - Heidelberg [u.a.] 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 - TY - JOUR A1 - Aranson, Igor S. A1 - Pikovskij, Arkadij T1 - Confinement and collective escape of active particles JF - Physical review letters N2 - 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. Y1 - 2022 U6 - https://doi.org/10.1103/PhysRevLett.128.108001 SN - 0031-9007 SN - 1079-7114 SN - 1092-0145 VL - 128 IS - 10 PB - American Physical Society CY - College Park, Md. ER - TY - JOUR A1 - Gengel, Erik A1 - Pikovskij, Arkadij T1 - Phase reconstruction from oscillatory data with iterated Hilbert transform embeddings BT - benefits and limitations JF - Physica : D, Nonlinear phenomena N2 - 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. KW - Data analysis KW - Phase reconstruction KW - Hilbert transform Y1 - 2021 U6 - https://doi.org/10.1016/j.physd.2021.133070 SN - 0167-2789 SN - 1872-8022 VL - 429 PB - Elsevier CY - Amsterdam ER - TY - JOUR A1 - Pikovskij, Arkadij T1 - Synchronization of oscillators with hyperbolic chaotic phases JF - Izvestija vysšich učebnych zavedenij : naučno-techničeskij žurnal = Izvestiya VUZ. Prikladnaja nelinejnaja dinamika = Applied nonlinear dynamics N2 - 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. N2 - Тема и цель. Синхронизация в популяциях связанных осцилляторов может быть охарактеризована параметрами порядка, описывающими коллективный порядок в ансамблях. Зависимость параметра порядка от коэффициентов связи хорошо известна для связанных периодических осцилляторов. Целью данного исследования является обобщение этого анализа на ансамбли осцилляторов с хаотическими фазами, а именно, с фазами, распределёнными на гиперболическом аттракторе. Модели и методы. В работе исследуются две модели. Первая – абстрактное отображение в дискретном времени, составленное из гиперболического преобразования Бернулли и динамики Курамото. Вторая – это система связанных хаотических осцилляторов в непрерывном времени, где каждый отдельный осциллятор имеет гиперболический аттрактор типа Смейла–Вильямса. Результаты. Модель в дискретном времени изучается с помощью подхода Отта–Антонсена, который, как показано, инвариантен при применении отображения Бернулли. Анализ полученного отображения по параметрам порядка показывает, что асинхронное состояние всегда устойчиво, а синхронное состояние становится устойчивым выше определенной силы связи. Численный анализ модели в непрерывном времени показывает сложную последовательность переходов из асинхронного состояния в полностью синхронный гиперболический хаос с промежуточными стадиями, которые включают режимы с периодическим во времени средним полем, а также со слабо и сильно нерегулярными вариациями среднего поля. Обсуждение. Результаты показывают, что синхронизация систем с гиперболическим фазовым хаосом возможна, хотя требуется довольно сильная связь. Данный подход может быть применен и к другим системам взаимодействующих звеньев с гиперболической хаотической динамикой. T2 - Синхронизация осцилляторов с гиперболическими хаотическими фазами KW - hyperbolic attractor KW - synchronization KW - collective dynamics KW - иперболический аттрактор KW - синхронизация KW - оллективная динамика Y1 - 2021 U6 - https://doi.org/10.18500/0869-6632-2021-29-1-78-87 SN - 0869-6632 SN - 2542-1905 VL - 29 IS - 1 SP - 78 EP - 87 PB - Saratov State University CY - Saratov ER - TY - JOUR A1 - Rosenau, Philip A1 - Pikovskij, Arkadij T1 - Waves in strongly nonlinear Gardner-like equations on a lattice JF - Nonlinearity / the Institute of Physics and the London Mathematical Society N2 - 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. KW - nonlinear lattice KW - solitary wave KW - Gardner equation KW - compacton Y1 - 2021 U6 - https://doi.org/10.1088/1361-6544/ac0f51 SN - 0951-7715 SN - 1361-6544 VL - 34 IS - 8 SP - 5872 EP - 5896 PB - IOP Publ. Ltd. CY - Bristol 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 - Munyaev, Vyacheslav O. A1 - Smirnov, Lev A. A1 - Kostin, Vasily A. A1 - Osipov, Grigory V. A1 - Pikovskij, Arkadij T1 - Analytical approach to synchronous states of globally coupled noisy rotators JF - New journal of physics : the open-access journal for physics N2 - 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. KW - coupled rotators KW - synchronization transition KW - hysteresis KW - Kuramoto KW - model KW - noisy systems Y1 - 2020 U6 - https://doi.org/10.1088/1367-2630/ab6f93 SN - 1367-2630 VL - 22 IS - 2 PB - IOP Publ. Ltd. CY - Bristol ER - TY - JOUR A1 - Rosenau, Philip A1 - Pikovskij, Arkadij T1 - Solitary phase waves in a chain of autonomous oscillators JF - Chaos : an interdisciplinary journal of nonlinear science N2 - 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. Y1 - 2020 U6 - https://doi.org/10.1063/1.5144939 SN - 1054-1500 SN - 1089-7682 VL - 30 IS - 5 PB - American Institute of Physics, AIP CY - Melville, NY ER - TY - JOUR A1 - Pikovskij, Arkadij T1 - Transition to synchrony in chiral active particles JF - Journal of physics. Complexity N2 - 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. KW - active particles KW - chirality KW - synchronization KW - chaos KW - transient chaos Y1 - 2021 U6 - https://doi.org/10.1088/2632-072X/abdadb SN - 2632-072X VL - 2 IS - 2 PB - IOP Publ. Ltd. CY - Bristol ER - TY - JOUR A1 - Pikovskij, Arkadij T1 - Chimeras on a social-type network JF - Mathematical modelling of natural phenomena : MMNP N2 - We consider a social-type network of coupled phase oscillators. Such a network consists of an active core of mutually interacting elements, and of a flock of passive units, which follow the driving from the active elements, but otherwise are not interacting. We consider a ring geometry with a long-range coupling, where active oscillators form a fluctuating chimera pattern. We show that the passive elements are strongly correlated. This is explained by negative transversal Lyapunov exponents. KW - Network KW - Chimera KW - correlations KW - Lyapunov exponent Y1 - 2021 U6 - https://doi.org/10.1051/mmnp/2021012 SN - 0973-5348 SN - 1760-6101 VL - 16 PB - EDP Sciences CY - Les Ulis ER - TY - JOUR A1 - Zheng, Chunming A1 - Toenjes, Ralf A1 - Pikovskij, Arkadij T1 - Transition to synchrony in a three-dimensional swarming model with helical trajectories JF - Physical review : E, Statistical, nonlinear and soft matter physics N2 - 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. Y1 - 2021 U6 - https://doi.org/10.1103/PhysRevE.104.014216 SN - 2470-0045 SN - 2470-0053 VL - 104 IS - 1 PB - American Physical Society CY - College Park ER - TY - JOUR A1 - Smirnov, Lev A. A1 - Osipov, Grigory V. A1 - Pikovskij, Arkadij T1 - Chimera patterns in the Kuramoto-Battogtokh model JF - Journal of physics : A, Mathematical and theoretical N2 - 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. KW - nonlocal coupled oscillators KW - chimera state KW - coarse-grained order parameter KW - Ott-Antonsen reduction KW - perturbation approach KW - linear stability analysis Y1 - 2017 U6 - https://doi.org/10.1088/1751-8121/aa55f1 SN - 1751-8113 SN - 1751-8121 VL - 50 IS - 8 PB - IOP Publ. Ltd. CY - Bristol ER - TY - JOUR A1 - Petereit, Johannes A1 - Pikovskij, Arkadij T1 - Chaos synchronization by nonlinear coupling JF - Communications in nonlinear science & numerical simulation N2 - 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. KW - Chaos synchronization KW - Partial synchrony KW - Intermittency KW - Hypernetwork Y1 - 2016 U6 - https://doi.org/10.1016/j.cnsns.2016.09.002 SN - 1007-5704 SN - 1878-7274 VL - 44 SP - 344 EP - 351 PB - Elsevier CY - Amsterdam ER - TY - JOUR A1 - Popovych, Oleksandr V. A1 - Lysyansky, Borys A1 - Rosenblum, Michael A1 - Pikovskij, Arkadij A1 - Tass, Peter A. T1 - Pulsatile desynchronizing delayed feedback for closed-loop deep brain stimulation JF - PLoS one N2 - 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. Y1 - 2017 U6 - https://doi.org/10.1371/journal.pone.0173363 SN - 1932-6203 VL - 12 PB - PLoS CY - San Fransisco ER - TY - JOUR A1 - Rosenblum, Michael A1 - Pikovskij, Arkadij A1 - Kühn, Andrea A. A1 - Busch, Johannes Leon T1 - Real-time estimation of phase and amplitude with application to neural data JF - Scientific reports N2 - Computation of the instantaneous phase and amplitude via the Hilbert Transform is a powerful tool of data analysis. This approach finds many applications in various science and engineering branches but is not proper for causal estimation because it requires knowledge of the signal’s past and future. However, several problems require real-time estimation of phase and amplitude; an illustrative example is phase-locked or amplitude-dependent stimulation in neuroscience. In this paper, we discuss and compare three causal algorithms that do not rely on the Hilbert Transform but exploit well-known physical phenomena, the synchronization and the resonance. After testing the algorithms on a synthetic data set, we illustrate their performance computing phase and amplitude for the accelerometer tremor measurements and a Parkinsonian patient’s beta-band brain activity. Y1 - 2021 U6 - https://doi.org/10.1038/s41598-021-97560-5 SN - 2045-2322 VL - 11 PB - Springer Nature CY - London 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 - Bolotov, Maxim I. A1 - Smirnov, Lev A. A1 - Osipov, Grigory V. A1 - Pikovskij, Arkadij T1 - Breathing chimera in a system of phase oscillators JF - JETP Letters N2 - 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. Y1 - 2017 U6 - https://doi.org/10.1134/S0021364017180059 SN - 0021-3640 SN - 1090-6487 VL - 106 SP - 393 EP - 399 PB - Pleiades Publ. CY - New York 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 - Sysoev, Ilya V. A1 - Ponomarenko, Vladimir I. A1 - Pikovskij, Arkadij T1 - Reconstruction of coupling architecture of neural field networks from vector time series JF - Communications in nonlinear science & numerical simulation N2 - 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. KW - Network reconstruction KW - Time series KW - Neurooscillators KW - Time delay Y1 - 2017 U6 - https://doi.org/10.1016/j.cnsns.2017.10.006 SN - 1007-5704 SN - 1878-7274 VL - 57 SP - 342 EP - 351 PB - Elsevier CY - Amsterdam 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 - JOUR A1 - Topçu, Çağdaş A1 - Frühwirth, Matthias A1 - Moser, Maximilian A1 - Rosenblum, Michael A1 - Pikovskij, Arkadij T1 - Disentangling respiratory sinus arrhythmia in heart rate variability records JF - Physiological Measurement N2 - Objective: Several different measures of heart rate variability, and particularly of respiratory sinus arrhythmia, are widely used in research and clinical applications. For many purposes it is important to know which features of heart rate variability are directly related to respiration and which are caused by other aspects of cardiac dynamics. Approach: Inspired by ideas from the theory of coupled oscillators, we use simultaneous measurements of respiratory and cardiac activity to perform a nonlinear disentanglement of the heart rate variability into the respiratory-related component and the rest. Main results: The theoretical consideration is illustrated by the analysis of 25 data sets from healthy subjects. In all cases we show how the disentanglement is manifested in the different measures of heart rate variability. Significance: The suggested technique can be exploited as a universal preprocessing tool, both for the analysis of respiratory influence on the heart rate and in cases when effects of other factors on the heart rate variability are in focus. KW - respiratory sinus arrhythmia KW - heart rate variability KW - coupled oscillators model KW - phase dynamics KW - data analysis Y1 - 2018 U6 - https://doi.org/10.1088/1361-6579/aabea4 SN - 0967-3334 SN - 1361-6579 VL - 39 IS - 5 PB - IOP Publ. Ltd. CY - Bristol 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 - Munyaev, Vyacheslav A1 - Smirnov, Lev A. A1 - Kostin, Vasily A1 - Osipov, Grigory V. A1 - Pikovskij, Arkadij T1 - Analytical approach to synchronous states of globally coupled noisy rotators JF - New Journal of Physics N2 - 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. KW - coupled rotators KW - synchronization transition KW - hysteresis KW - Kuramoto model KW - noisy systems Y1 - 2019 VL - 22 IS - 2 PB - Springer Science CY - New York 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 - 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 - 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 - 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 - 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 - 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 - Pikovskij, Arkadij A1 - Dolmatova, A. A1 - Goldobin, Denis S. T1 - Correlations of the States of Non-Entrained Oscillators in the Kuramoto Ensemble with Noise in the Mean Field JF - Radiophysics and Quantum Electronics N2 - 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. Y1 - 2019 U6 - https://doi.org/10.1007/s11141-019-09927-4 SN - 0033-8443 SN - 1573-9120 VL - 61 IS - 8-9 SP - 672 EP - 680 PB - Springer CY - New York ER - TY - JOUR A1 - Tyulkina, Irina V. A1 - Goldobin, Denis S. A1 - Klimenko, Lyudmila S. A1 - Pikovskij, Arkadij T1 - Two-Bunch Solutions for the Dynamics of Ott–Antonsen Phase Ensembles JF - Radiophysics and Quantum Electronics N2 - 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). Y1 - 2019 U6 - https://doi.org/10.1007/s11141-019-09924-7 SN - 0033-8443 SN - 1573-9120 VL - 61 IS - 8-9 SP - 640 EP - 649 PB - Springer CY - New York ER - TY - JOUR A1 - Gong, Chen Chris A1 - Zheng, Chunming A1 - Toenjes, Ralf A1 - Pikovskij, Arkadij T1 - Repulsively coupled Kuramoto-Sakaguchi phase oscillators ensemble subject to common noise JF - Chaos : an interdisciplinary journal of nonlinear science N2 - 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. Y1 - 2019 U6 - https://doi.org/10.1063/1.5084144 SN - 1054-1500 SN - 1089-7682 VL - 29 IS - 3 PB - American Institute of Physics CY - Melville ER - TY - JOUR A1 - Zheng, Chunming A1 - Pikovskij, Arkadij T1 - Stochastic bursting in unidirectionally delay-coupled noisy excitable systems JF - Chaos : an interdisciplinary journal of nonlinear science N2 - 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. Y1 - 2019 U6 - https://doi.org/10.1063/1.5093180 SN - 1054-1500 SN - 1089-7682 VL - 29 IS - 4 PB - American Institute of Physics CY - Melville ER - TY - JOUR A1 - Zaks, Michael A. A1 - Pikovskij, Arkadij T1 - Synchrony breakdown and noise-induced oscillation death in ensembles of serially connected spin-torque oscillators JF - The European physical journal : B, Condensed matter and complex systems N2 - 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. KW - Statistical and Nonlinear Physics Y1 - 2019 U6 - https://doi.org/10.1140/epjb/e2019-100152-2 SN - 1434-6028 SN - 1434-6036 VL - 92 IS - 7 PB - Springer CY - New York ER - TY - JOUR A1 - Goldschmidt, Richard Janis A1 - Pikovskij, Arkadij A1 - Politi, Antonio T1 - Blinking chimeras in globally coupled rotators JF - Chaos : an interdisciplinary journal of nonlinear science N2 - 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. Y1 - 2019 U6 - https://doi.org/10.1063/1.5105367 SN - 1054-1500 SN - 1089-7682 VL - 29 IS - 7 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 - Bolotov, Dmitry A1 - Bolotov, Maxim I. A1 - Smirnov, Lev A. A1 - Osipov, Grigory V. A1 - Pikovskij, Arkadij T1 - Twisted States in a System of Nonlinearly Coupled Phase Oscillators JF - Regular and chaotic dynamics : international scientific journal N2 - 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. KW - twisted state KW - phase oscillators KW - nonlocal coupling KW - Ott - Antonsen reduction KW - stability analysis Y1 - 2019 U6 - https://doi.org/10.1134/S1560354719060091 SN - 1560-3547 SN - 1468-4845 VL - 24 IS - 6 SP - 717 EP - 724 PB - Pleiades publishing inc CY - Moscow ER - 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 - 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 - Rosenblum, Michael A1 - Frühwirth, Martha A1 - Moser, Maximilian A1 - Pikovskij, Arkadij T1 - Dynamical disentanglement in an analysis of oscillatory systems: an application to respiratory sinus arrhythmia JF - Philosophical Transactions of the Royal Society of London, Series A : Mathematical, Physical and Engineering Sciences N2 - We develop a technique for the multivariate data analysis of perturbed self-sustained oscillators. The approach is based on the reconstruction of the phase dynamics model from observations and on a subsequent exploration of this model. For the system, driven by several inputs, we suggest a dynamical disentanglement procedure, allowing us to reconstruct the variability of the system's output that is due to a particular observed input, or, alternatively, to reconstruct the variability which is caused by all the inputs except for the observed one. We focus on the application of the method to the vagal component of the heart rate variability caused by a respiratory influence. We develop an algorithm that extracts purely respiratory-related variability, using a respiratory trace and times of R-peaks in the electrocardiogram. The algorithm can be applied to other systems where the observed bivariate data can be represented as a point process and a slow continuous signal, e.g. for the analysis of neuronal spiking. This article is part of the theme issue 'Coupling functions: dynamical interaction mechanisms in the physical, biological and social sciences'. KW - phase dynamics KW - point process KW - vagal sympathetic activity KW - autonomic nervous system Y1 - 2019 U6 - https://doi.org/10.1098/rsta.2019.0045 SN - 1364-503X SN - 1471-2962 VL - 377 IS - 2160 PB - Royal Society CY - London ER - TY - JOUR A1 - Pawlik, Andreas H. A1 - Pikovskij, Arkadij T1 - Control of oscillators coherence by multiple delayed feedback JF - Modern physics letters : A, Particles and fields, gravitation, cosmology, nuclear physics N2 - 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. KW - phase diffusion KW - delayed feedback KW - control Y1 - 2006 U6 - https://doi.org/10.1016/j.physleta.2006.05.013 SN - 0375-9601 VL - 358 IS - 3 SP - 181 EP - 185 PB - American Institute of Physics CY - Amsterdam ER - TY - JOUR A1 - Zhang, Xiyun A1 - Pikovskij, Arkadij A1 - Liu, Zonghua T1 - Dynamics of oscillators globally coupled via two mean fields JF - Scientific reports N2 - Many studies of synchronization properties of coupled oscillators, based on the classical Kuramoto approach, focus on ensembles coupled via a mean field. Here we introduce a setup of Kuramoto-type phase oscillators coupled via two mean fields. We derive stability properties of the incoherent state and find traveling wave solutions with different locking patterns; stability properties of these waves are found numerically. Mostly nontrivial states appear when the two fields compete, i.e. one tends to synchronize oscillators while the other one desynchronizes them. Here we identify normal branches which bifurcate from the incoherent state in a usual way, and anomalous branches, appearance of which cannot be described as a bifurcation. Furthermore, hybrid branches combining properties of both are described. In the situations where no stable traveling wave exists, modulated quasiperiodic in time dynamics is observed. Our results indicate that a competition between two coupling channels can lead to a complex system behavior, providing a potential generalized framework for understanding of complex phenomena in natural oscillatory systems. Y1 - 2017 U6 - https://doi.org/10.1038/s41598-017-02283-1 SN - 2045-2322 VL - 7 PB - Nature Publ. Group CY - London ER - TY - JOUR A1 - Goldobin, Denis S. A1 - Pimenova, Anastasiya V. A1 - Rosenblum, Michael A1 - Pikovskij, Arkadij T1 - Competing influence of common noise and desynchronizing coupling on synchronization in the Kuramoto-Sakaguchi ensemble JF - European physical journal special topics N2 - We describe analytically synchronization and desynchronization effects in an ensemble of phase oscillators driven by common noise and by global coupling. Adopting the Ott-Antonsen ansatz, we reduce the dynamics to closed stochastic equations for the order parameters, and study these equations for the cases of populations of identical and nonidentical oscillators. For nonidentical oscillators we demonstrate a counterintuitive effect of divergence of individual frequencies for moderate repulsive coupling, while the order parameter remains large. Y1 - 2017 U6 - https://doi.org/10.1140/epjst/e2017-70039-y SN - 1951-6355 SN - 1951-6401 VL - 226 SP - 1921 EP - 1937 PB - Springer CY - Heidelberg ER - TY - JOUR A1 - Politi, Antonio A1 - Pikovskij, Arkadij A1 - Ullner, Ekkehard T1 - Chaotic macroscopic phases in one-dimensional oscillators JF - European physical journal special topics N2 - The connection between the macroscopic description of collective chaos and the underlying microscopic dynamics is thoroughly analysed in mean-field models of one-dimensional oscillators. We investigate to what extent infinitesimal perturbations of the microscopic configurations can provide information also on the stability of the corresponding macroscopic phase. In ensembles of identical one-dimensional dynamical units, it is possible to represent the microscopic configurations so as to make transparent their connection with the macroscopic world. As a result, we find evidence of an intermediate, mesoscopic, range of distances, over which the instability is neither controlled by the microscopic equations nor by the macroscopic ones. We examine a whole series of indicators, ranging from the usual microscopic Lyapunov exponents, to the collective ones, including finite-amplitude exponents. A system of pulse-coupled oscillators is also briefly reviewed as an example of non-identical phase oscillators where collective chaos spontaneously emerges. Y1 - 2017 U6 - https://doi.org/10.1140/epjst/e2017-70056-4 SN - 1951-6355 SN - 1951-6401 VL - 226 SP - 1791 EP - 1810 PB - Springer CY - Heidelberg ER - TY - JOUR A1 - Zaks, Michael A1 - Pikovskij, Arkadij T1 - Chimeras and complex cluster states in arrays of spin-torque oscillators JF - Scientific reports N2 - 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. Y1 - 2017 U6 - https://doi.org/10.1038/s41598-017-04918-9 SN - 2045-2322 VL - 7 PB - Nature Publ. Group CY - London ER - TY - JOUR A1 - Pikovskij, Arkadij T1 - Reconstruction of a scalar voltage-based neural field network from observed time series JF - epl : a letters journal exploring the frontiers of physics Y1 - 2017 U6 - https://doi.org/10.1209/0295-5075/119/30004 SN - 0295-5075 SN - 1286-4854 VL - 119 PB - EDP Sciences CY - Mulhouse ER - TY - JOUR A1 - Dolmatova, Anastasiya V. A1 - Goldobin, Denis S. A1 - Pikovskij, Arkadij T1 - Synchronization of coupled active rotators by common noise JF - Physical review : E, Statistical, nonlinear and soft matter physics N2 - We study the effect of common noise on coupled active rotators. While such a noise always facilitates synchrony, coupling may be attractive (synchronizing) or repulsive (desynchronizing). We develop an analytical approach based on a transformation to approximate angle-action variables and averaging over fast rotations. For identical rotators, we describe a transition from full to partial synchrony at a critical value of repulsive coupling. For nonidentical rotators, the most nontrivial effect occurs at moderate repulsive coupling, where a juxtaposition of phase locking with frequency repulsion (anti-entrainment) is observed. We show that the frequency repulsion obeys a nontrivial power law. Y1 - 2017 U6 - https://doi.org/10.1103/PhysRevE.96.062204 SN - 2470-0045 SN - 2470-0053 VL - 96 SP - E10648 EP - E10657 PB - American Physical Society CY - College Park 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, M. I. A1 - Osipov, Grigory V. A1 - Pikovskij, Arkadij T1 - Marginal chimera state at cross-frequency locking of pulse-coupled neural networks JF - Physical review : E, Statistical, nonlinear and soft matter physics N2 - We consider two coupled populations of leaky integrate-and-fire neurons. Depending on the coupling strength, mean fields generated by these populations can have incommensurate frequencies or become frequency locked. In the observed 2:1 locking state of the mean fields, individual neurons in one population are asynchronous with the mean fields, while in another population they have the same frequency as the mean field. These synchronous neurons form a chimera state, where part of them build a fully synchronized cluster, while other remain scattered. We explain this chimera as a marginal one, caused by a self-organized neutral dynamics of the effective circle map. Y1 - 2016 U6 - https://doi.org/10.1103/PhysRevE.93.032202 SN - 2470-0045 SN - 2470-0053 VL - 93 PB - American Physical Society CY - College Park ER - TY - JOUR A1 - Pikovskij, Arkadij T1 - Reconstruction of a neural network from a time series of firing rates JF - Physical review : E, Statistical, nonlinear and soft matter physics N2 - Randomly coupled neural fields demonstrate irregular variation of firing rates, if the coupling is strong enough, as has been shown by Sompolinsky et al. [Phys. Rev. Lett. 61, 259 (1988)]. We present a method for reconstruction of the coupling matrix from a time series of irregular firing rates. The approach is based on the particular property of the nonlinearity in the coupling, as the latter is determined by a sigmoidal gain function. We demonstrate that for a large enough data set and a small measurement noise, the method gives an accurate estimation of the coupling matrix and of other parameters of the system, including the gain function. Y1 - 2016 U6 - https://doi.org/10.1103/PhysRevE.93.062313 SN - 2470-0045 SN - 2470-0053 VL - 93 PB - American Physical Society CY - College Park ER - TY - JOUR A1 - Nagornov, Roman A1 - Osipoy, Grigory A1 - Komarov, Maxim A1 - Pikovskij, Arkadij A1 - Shilnikov, Andrey T1 - Mixed-mode synchronization between two inhibitory neurons with post-inhibitory rebound JF - Communications in nonlinear science & numerical simulation N2 - We study an array of activity rhythms generated by a half-center oscillator (HCO), represented by a pair of reciprocally coupled neurons with post-inhibitory rebounds (PIR). Such coupling induced bursting possesses two time scales, one for fast spiking and another for slow quiescent periods, is shown to exhibit an array of synchronization properties. We discuss several HCO configurations constituted by two endogenous bursters, by tonic-spiking and quiescent neurons, as well as mixed-mode configurations composed of neurons of different type. We demonstrate that burst synchronization can be accompanied by complex, often chaotic, interactions of fast spikes within synchronized bursts. (C) 2015 Elsevier B.V. All rights reserved. KW - Synchronization KW - Hodgkin-Huxley model KW - Half-center oscillator KW - Post-inhibitory rebound Y1 - 2016 U6 - https://doi.org/10.1016/j.cnsns.2015.11.024 SN - 1007-5704 SN - 1878-7274 VL - 36 SP - 175 EP - 191 PB - Elsevier CY - Amsterdam ER - TY - JOUR A1 - Vlasov, Vladimir A1 - Rosenblum, Michael A1 - Pikovskij, Arkadij T1 - Dynamics of weakly inhomogeneous oscillator populations: perturbation theory on top of Watanabe-Strogatz integrability JF - Journal of physics : A, Mathematical and theoretical N2 - 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. KW - Kuramoto model KW - oscillator populations KW - integrability KW - perturbation theory Y1 - 2016 U6 - https://doi.org/10.1088/1751-8113/49/31/31LT02 SN - 1751-8113 SN - 1751-8121 VL - 49 PB - IOP Publ. Ltd. CY - Bristol ER - TY - JOUR 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 JF - Scientific reports 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. Y1 - 2016 U6 - https://doi.org/10.1038/srep38518 SN - 2045-2322 VL - 6 PB - Nature Publ. Group CY - London ER - TY - JOUR A1 - Zaks, Michael A. A1 - Pikovskij, Arkadij T1 - Chimeras and complex cluster states in arrays of spin-torque oscillators JF - Scientific reports N2 - 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. Y1 - 2017 U6 - https://doi.org/10.1038/s41598-017-04918-9 SN - 2045-2322 VL - 7 PB - Macmillan Publishers Limited CY - London ER - TY - JOUR A1 - Vlasov, Vladimir A1 - Komarov, Maxim A1 - Pikovskij, Arkadij T1 - Synchronization transitions in ensembles of noisy oscillators with bi-harmonic coupling JF - Journal of physics : A, Mathematical and theoretical N2 - We describe synchronization transitions in an ensemble of globally coupled phase oscillators with a bi-harmonic coupling function, and two sources of disorder-diversity of the intrinsic oscillators' frequencies, and external independent noise forces. Based on the self-consistent formulation, we derive analytic solutions for different synchronous states. We report on various non-trivial transitions from incoherence to synchrony, with the following possible scenarios: simple supercritical transition (similar to classical Kuramoto model); subcritical transition with large area of bistability of incoherent and synchronous solutions; appearance of a symmetric two-cluster solution which can coexist with the regular synchronous state. We show that the interplay between relatively small white noise and finite-size fluctuations can lead to metastability of the asynchronous solution. KW - synchronization KW - bi-harmonic coupling KW - noise Y1 - 2015 U6 - https://doi.org/10.1088/1751-8113/48/10/105101 SN - 1751-8113 SN - 1751-8121 VL - 48 IS - 10 PB - IOP Publ. Ltd. CY - Bristol ER - TY - JOUR A1 - Freitas, Celso A1 - Macau, Elbert A1 - Pikovskij, Arkadij T1 - Partial synchronization in networks of non-linearly coupled oscillators: The Deserter Hubs Model JF - Chaos : an interdisciplinary journal of nonlinear science N2 - We study the Deserter Hubs Model: a Kuramoto-like model of coupled identical phase oscillators on a network, where attractive and repulsive couplings are balanced dynamically due to nonlinearity of interactions. Under weak force, an oscillator tends to follow the phase of its neighbors, but if an oscillator is compelled to follow its peers by a sufficient large number of cohesive neighbors, then it actually starts to act in the opposite manner, i.e., in anti-phase with the majority. Analytic results yield that if the repulsion parameter is small enough in comparison with the degree of the maximum hub, then the full synchronization state is locally stable. Numerical experiments are performed to explore the model beyond this threshold, where the overall cohesion is lost. We report in detail partially synchronous dynamical regimes, like stationary phase-locking, multistability, periodic and chaotic states. Via statistical analysis of different network organizations like tree, scale-free, and random ones, we found a measure allowing one to predict relative abundance of partially synchronous stationary states in comparison to time-dependent ones. (C) 2015 AIP Publishing LLC. Y1 - 2015 U6 - https://doi.org/10.1063/1.4919246 SN - 1054-1500 SN - 1089-7682 VL - 25 IS - 4 PB - American Institute of Physics CY - Melville ER - TY - JOUR A1 - Komarov, Maxim A1 - Pikovskij, Arkadij T1 - Intercommunity resonances in multifrequency ensembles of coupled oscillators JF - Physical review : E, Statistical, nonlinear and soft matter physics N2 - We generalize the Kuramoto model of globally coupled oscillators to multifrequency communities. A situation when mean frequencies of two subpopulations are close to the resonance 2 : 1 is considered in detail. We construct uniformly rotating solutions describing synchronization inside communities and between them. Remarkably, cross coupling across the frequencies can promote synchrony even when ensembles are separately asynchronous. We also show that the transition to synchrony due to the cross coupling is accompanied by a huge multiplicity of distinct synchronous solutions, which is directly related to a multibranch entrainment. On the other hand, for synchronous populations, the cross-frequency coupling can destroy phase locking and lead to chaos of mean fields. Y1 - 2015 U6 - https://doi.org/10.1103/PhysRevE.92.012906 SN - 1539-3755 SN - 1550-2376 VL - 92 IS - 1 PB - American Physical Society CY - College Park ER - TY - JOUR A1 - Rosenblum, Michael A1 - Pikovskij, Arkadij T1 - Two types of quasiperiodic partial synchrony in oscillator ensembles JF - Physical review : E, Statistical, nonlinear and soft matter physics N2 - We analyze quasiperiodic partially synchronous states in an ensemble of Stuart-Landau oscillators with global nonlinear coupling. We reveal two types of such dynamics: in the first case the time-averaged frequencies of oscillators and of the mean field differ, while in the second case they are equal, but the motion of oscillators is additionally modulated. We describe transitions from the synchronous state to both types of quasiperiodic dynamics, and a transition between two different quasiperiodic states. We present an example of a bifurcation diagram, where we show the borderlines for all these transitions, as well as domain of bistability. Y1 - 2015 U6 - https://doi.org/10.1103/PhysRevE.92.012919 SN - 1539-3755 SN - 1550-2376 VL - 92 IS - 1 PB - American Physical Society CY - College Park ER - TY - JOUR A1 - Pikovskij, Arkadij T1 - First and second sound in disordered strongly nonlinear lattices: numerical study JF - Journal of statistical mechanics: theory and experiment N2 - We study numerically secondary modes on top of a chaotic state in disordered nonlinear lattices. Two basic models are considered, with or without a local on-site potential. By performing periodic spatial modulation of displacement and kinetic energy, and following the temporal evolution of the corresponding spatial profiles, we reveal different modes which can be interpreted as first and second sound. KW - disordered systems (theory) KW - connections between chaos and statistical physics Y1 - 2015 U6 - https://doi.org/10.1088/1742-5468/2015/08/P08007 SN - 1742-5468 PB - IOP Publ. Ltd. CY - Bristol ER - TY - JOUR A1 - Komarov, Maxim A1 - Pikovskij, Arkadij T1 - Finite-size-induced transitions to synchrony in oscillator ensembles with nonlinear global coupling JF - Physical review : E, Statistical, nonlinear and soft matter physics N2 - We report on finite-sized-induced transitions to synchrony in a population of phase oscillators coupled via a nonlinear mean field, which microscopically is equivalent to a hypernetwork organization of interactions. Using a self-consistent approach and direct numerical simulations, we argue that a transition to synchrony occurs only for finite-size ensembles and disappears in the thermodynamic limit. For all considered setups, which include purely deterministic oscillators with or without heterogeneity in natural oscillatory frequencies, and an ensemble of noise-driven identical oscillators, we establish scaling relations describing the order parameter as a function of the coupling constant and the system size. Y1 - 2015 U6 - https://doi.org/10.1103/PhysRevE.92.020901 SN - 1539-3755 SN - 1550-2376 VL - 92 IS - 2 PB - American Physical Society CY - College Park ER - TY - JOUR A1 - Pikovskij, Arkadij T1 - Maximizing Coherence of Oscillations by External Locking JF - Physical review letters N2 - We study how coherence of noisy oscillations can be optimally enhanced by external locking. Based on the condition of minimizing the phase diffusion constant, we find the optimal forcing explicitly in the limits of small and large noise, in dependence of the phase sensitivity of the oscillator. We show analytically that the form of the optimal force bifurcates with the noise intensity; this is confirmed by the analysis of an optimal locking forcing for an experimentally obtained phase sensitivity of a neural cell. In the limit of small noise, the results are compared with purely deterministic conditions of optimal locking. Y1 - 2015 U6 - https://doi.org/10.1103/PhysRevLett.115.070602 SN - 0031-9007 SN - 1079-7114 VL - 115 IS - 7 PB - American Physical Society CY - College Park ER - TY - JOUR A1 - Pikovskij, Arkadij A1 - Rosenblum, Michael T1 - Dynamics of globally coupled oscillators: Progress and perspectives JF - Chaos : an interdisciplinary journal of nonlinear science N2 - In this paper, we discuss recent progress in research of ensembles of mean field coupled oscillators. Without an ambition to present a comprehensive review, we outline most interesting from our viewpoint results and surprises, as well as interrelations between different approaches. (c) 2015 AIP Publishing LLC. Y1 - 2015 U6 - https://doi.org/10.1063/1.4922971 SN - 1054-1500 SN - 1089-7682 VL - 25 IS - 9 PB - American Institute of Physics CY - Melville ER - TY - JOUR A1 - Vlasov, Vladimir A1 - Pikovskij, Arkadij A1 - Macau, Elbert E. N. T1 - Star-type oscillatory networks with generic Kuramoto-type coupling: A model for "Japanese drums synchrony" JF - Chaos : an interdisciplinary journal of nonlinear science N2 - We analyze star-type networks of phase oscillators by virtue of two methods. For identical oscillators we adopt the Watanabe-Strogatz approach, which gives full analytical description of states, rotating with constant frequency. For nonidentical oscillators, such states can be obtained by virtue of the self-consistent approach in a parametric form. In this case stability analysis cannot be performed, however with the help of direct numerical simulations we show which solutions are stable and which not. We consider this system as a model for a drum orchestra, where we assume that the drummers follow the signal of the leader without listening to each other and the coupling parameters are determined by a geometrical organization of the orchestra. (C) 2015 AIP Publishing LLC. Y1 - 2015 U6 - https://doi.org/10.1063/1.4938400 SN - 1054-1500 SN - 1089-7682 VL - 25 IS - 12 PB - American Institute of Physics CY - Melville ER - TY - JOUR A1 - Rosenau, Philip A1 - Pikovskij, Arkadij T1 - Breathers in strongly anharmonic lattices JF - Physical review : E, Statistical, nonlinear and soft matter physics N2 - We present and study a family of finite amplitude breathers on a genuinely anharmonic Klein-Gordon lattice embedded in a nonlinear site potential. The direct numerical simulations are supported by a quasilinear Schrodinger equation (QLS) derived by averaging out the fast oscillations assuming small, albeit finite, amplitude vibrations. The genuinely anharmonic interlattice forces induce breathers which are strongly localized with tails evanescing at a doubly exponential rate and are either close to a continuum, with discrete effects being suppressed, or close to an anticontinuum state, with discrete effects being enhanced. Whereas the D-QLS breathers appear to be always stable, in general there is a stability threshold which improves with spareness of the lattice. Y1 - 2014 U6 - https://doi.org/10.1103/PhysRevE.89.022924 SN - 1539-3755 SN - 1550-2376 VL - 89 IS - 2 PB - American Physical Society CY - College Park ER - TY - JOUR A1 - Pollatos, Olga A1 - Yeldesbay, Azamat A1 - Pikovskij, Arkadij A1 - Rosenblum, Michael T1 - How much time has passed? Ask your heart JF - Frontiers in neurorobotics N2 - Internal signals like one's heartbeats are centrally processed via specific pathways and both their neural representations as well as their conscious perception (interoception) provide key information for many cognitive processes. Recent empirical findings propose that neural processes in the insular cortex, which are related to bodily signals, might constitute a neurophysiological mechanism for the encoding of duration. Nevertheless, the exact nature of such a proposed relationship remains unclear. We aimed to address this question by searching for the effects of cardiac rhythm on time perception by the use of a duration reproduction paradigm. Time intervals used were of 0.5, 2, 3, 7, 10, 14, 25, and 40s length. In a framework of synchronization hypothesis, measures of phase locking between the cardiac cycle and start/stop signals of the reproduction task were calculated to quantify this relationship. The main result is that marginally significant synchronization indices (Sls) between the heart cycle and the time reproduction responses for the time intervals of 2, 3, 10, 14, and 25s length were obtained, while results were not significant for durations of 0.5, 7, and 40s length. On the single participant level, several subjects exhibited some synchrony between the heart cycle and the time reproduction responses, most pronounced for the time interval of 25s (8 out of 23 participants for 20% quantile). Better time reproduction accuracy was not related with larger degree of phase locking, but with greater vagal control of the heart. A higher interoceptive sensitivity (IS) was associated with a higher synchronization index (SI) for the 2s time interval only. We conclude that information obtained from the cardiac cycle is relevant for the encoding and reproduction of time in the time span of 2-25s. Sympathovagal tone as well as interoceptive processes mediate the accuracy of time estimation. KW - time interval reproduction KW - synchronization KW - heart cycle KW - interoception KW - interoceptive sensitivity Y1 - 2014 U6 - https://doi.org/10.3389/fnbot.2014.00015 SN - 1662-5218 VL - 8 SP - 1 EP - 9 PB - Frontiers Research Foundation CY - Lausanne ER - TY - JOUR A1 - Yeldesbay, Azamat A1 - Pikovskij, Arkadij A1 - Rosenblum, Michael T1 - Chimeralike states in an ensemble of globally coupled oscillators JF - Physical review letters N2 - We demonstrate the emergence of a complex state in a homogeneous ensemble of globally coupled identical oscillators, reminiscent of chimera states in nonlocally coupled oscillator lattices. In this regime some part of the ensemble forms a regularly evolving cluster, while all other units irregularly oscillate and remain asynchronous. We argue that the chimera emerges because of effective bistability, which dynamically appears in the originally monostable system due to internal delayed feedback in individual units. Additionally, we present two examples of chimeras in bistable systems with frequency-dependent phase shift in the global coupling. Y1 - 2014 U6 - https://doi.org/10.1103/PhysRevLett.112.144103 SN - 0031-9007 SN - 1079-7114 VL - 112 IS - 14 PB - American Physical Society CY - College Park ER - TY - JOUR A1 - Komarov, Maxim A1 - Gupta, Shamik A1 - Pikovskij, Arkadij T1 - Synchronization transitions in globally coupled rotors in the presence of noise and inertia: Exact results JF - epl : a letters journal exploring the frontiers of physics N2 - We study a generic model of globally coupled rotors that includes the effects of noise, phase shift in the coupling, and distributions of moments of inertia and natural frequencies of oscillation. As particular cases, the setup includes previously studied Sakaguchi-Kuramoto, Hamiltonian and Brownian mean-field, and Tanaka-Lichtenberg-Oishi and Acebron-Bonilla-Spigler models. We derive an exact solution of the self-consistent equations for the order parameter in the stationary state, valid for arbitrary parameters in the dynamics, and demonstrate nontrivial phase transitions to synchrony that include reentrant synchronous regimes. Copyright (C) EPLA, 2014 Y1 - 2014 U6 - https://doi.org/10.1209/0295-5075/106/40003 SN - 0295-5075 SN - 1286-4854 VL - 106 IS - 4 PB - EDP Sciences CY - Mulhouse ER - TY - JOUR A1 - Levnajic, Zoran A1 - Pikovskij, Arkadij T1 - Untangling complex dynamical systems via derivative-variable correlations JF - Scientific reports N2 - Inferring the internal interaction patterns of a complex dynamical system is a challenging problem. Traditional methods often rely on examining the correlations among the dynamical units. However, in systems such as transcription networks, one unit's variable is also correlated with the rate of change of another unit's variable. Inspired by this, we introduce the concept of derivative-variable correlation, and use it to design a new method of reconstructing complex systems (networks) from dynamical time series. Using a tunable observable as a parameter, the reconstruction of any system with known interaction functions is formulated via a simple matrix equation. We suggest a procedure aimed at optimizing the reconstruction from the time series of length comparable to the characteristic dynamical time scale. Our method also provides a reliable precision estimate. We illustrate the method's implementation via elementary dynamical models, and demonstrate its robustness to both model error and observation error. Y1 - 2014 U6 - https://doi.org/10.1038/srep05030 SN - 2045-2322 VL - 4 PB - Nature Publ. Group CY - London ER - TY - JOUR A1 - Vlasov, Vladimir A1 - Macau, Elbert E. N. A1 - Pikovskij, Arkadij T1 - Synchronization of oscillators in a Kuramoto-type model with generic coupling JF - Chaos : an interdisciplinary journal of nonlinear science N2 - We study synchronization properties of coupled oscillators on networks that allow description in terms of global mean field coupling. These models generalize the standard Kuramoto-Sakaguchi model, allowing for different contributions of oscillators to the mean field and to different forces from the mean field on oscillators. We present the explicit solutions of self-consistency equations for the amplitude and frequency of the mean field in a parametric form, valid for noise-free and noise-driven oscillators. As an example, we consider spatially spreaded oscillators for which the coupling properties are determined by finite velocity of signal propagation. (C) 2014 AIP Publishing LLC. Y1 - 2014 U6 - https://doi.org/10.1063/1.4880835 SN - 1054-1500 SN - 1089-7682 VL - 24 IS - 2 PB - American Institute of Physics CY - Melville ER - TY - JOUR A1 - Kruglov, Vyacheslav P. A1 - Kuznetsov, Sergey P. A1 - Pikovskij, Arkadij T1 - Attractor of Smale - Williams type in an autonomous distributed system JF - Regular and chaotic dynamics : international scientific journal N2 - We consider an autonomous system of partial differential equations for a one-dimensional distributed medium with periodic boundary conditions. Dynamics in time consists of alternating birth and death of patterns with spatial phases transformed from one stage of activity to another by the doubly expanding circle map. So, the attractor in the Poincar, section is uniformly hyperbolic, a kind of Smale - Williams solenoid. Finite-dimensional models are derived as ordinary differential equations for amplitudes of spatial Fourier modes (the 5D and 7D models). Correspondence of the reduced models to the original system is demonstrated numerically. Computational verification of the hyperbolicity criterion is performed for the reduced models: the distribution of angles of intersection for stable and unstable manifolds on the attractor is separated from zero, i.e., the touches are excluded. The example considered gives a partial justification for the old hopes that the chaotic behavior of autonomous distributed systems may be associated with uniformly hyperbolic attractors. KW - Smale - Williams solenoid KW - hyperbolic attractor KW - chaos KW - Swift - Hohenberg equation KW - Lyapunov exponent Y1 - 2014 U6 - https://doi.org/10.1134/S1560354714040042 SN - 1560-3547 SN - 1468-4845 VL - 19 IS - 4 SP - 483 EP - 494 PB - Pleiades Publ. CY - New York ER - TY - JOUR A1 - Levanova, T. A. A1 - Osipov, Grigory V. A1 - Pikovskij, Arkadij T1 - Coherence properties of cycling chaos JF - Communications in nonlinear science & numerical simulation N2 - Cycling chaos is a heteroclinic connection between several chaotic attractors, at which switchings between the chaotic sets occur at growing time intervals. Here we characterize the coherence properties of these switchings, considering nearly periodic regimes that appear close to the cycling chaos due to imperfections or to instability. Using numerical simulations of coupled Lorenz, Roessler, and logistic map models, we show that the coherence is high in the case of imperfection (so that asymptotically the cycling chaos is very regular), while it is low close to instability of the cycling chaos. (C) 2014 Elsevier B. V. All rights reserved. KW - Heteroclinic cycle KW - Chaos KW - Coherence Y1 - 2014 U6 - https://doi.org/10.1016/j.cnsns.2014.01.011 SN - 1007-5704 SN - 1878-7274 VL - 19 IS - 8 SP - 2734 EP - 2739 PB - Elsevier CY - Amsterdam ER - TY - JOUR A1 - Kralemann, Bjoern A1 - Pikovskij, Arkadij A1 - Rosenblum, Michael T1 - Reconstructing effective phase connectivity of oscillator networks from observations JF - New journal of physics : the open-access journal for physics N2 - We present a novel approach for recovery of the directional connectivity of a small oscillator network by means of the phase dynamics reconstruction from multivariate time series data. The main idea is to use a triplet analysis instead of the traditional pairwise one. Our technique reveals an effective phase connectivity which is generally not equivalent to a structural one. We demonstrate that by comparing the coupling functions from all possible triplets of oscillators, we are able to achieve in the reconstruction a good separation between existing and non-existing connections, and thus reliably reproduce the network structure. KW - network reconstruction KW - coupled oscillators KW - connectivity KW - data analysis Y1 - 2014 U6 - https://doi.org/10.1088/1367-2630/16/8/085013 SN - 1367-2630 VL - 16 PB - IOP Publ. Ltd. CY - Bristol ER - TY - JOUR A1 - Kralemann, Bjoern A1 - Pikovskij, Arkadij A1 - Rosenblum, Michael T1 - Reconstructing connectivity of oscillator networks from multimodal observations JF - Biomedizinische Technik = Biomedical engineering Y1 - 2014 U6 - https://doi.org/10.1515/bmt-2014-4089 SN - 0013-5585 SN - 1862-278X VL - 59 SP - S220 EP - S220 PB - De Gruyter CY - Berlin ER - TY - JOUR A1 - Lepri, Stefano A1 - Pikovskij, Arkadij T1 - Nonreciprocal wave scattering on nonlinear string-coupled oscillators JF - Chaos : an interdisciplinary journal of nonlinear science N2 - We study scattering of a periodic wave in a string on two lumped oscillators attached to it. The equations can be represented as a driven (by the incident wave) dissipative (due to radiation losses) system of delay differential equations of neutral type. Nonlinearity of oscillators makes the scattering non-reciprocal: The same wave is transmitted differently in two directions. Periodic regimes of scattering are analyzed approximately, using amplitude equation approach. We show that this setup can act as a nonreciprocal modulator via Hopf bifurcations of the steady solutions. Numerical simulations of the full system reveal nontrivial regimes of quasiperiodic and chaotic scattering. Moreover, a regime of a "chaotic diode," where transmission is periodic in one direction and chaotic in the opposite one, is reported. (C) 2014 AIP Publishing LLC. Y1 - 2014 U6 - https://doi.org/10.1063/1.4899205 SN - 1054-1500 SN - 1089-7682 VL - 24 IS - 4 PB - American Institute of Physics CY - Melville ER - TY - JOUR A1 - Komarov, Maxim A1 - Pikovskij, Arkadij T1 - The Kuramoto model of coupled oscillators with a bi-harmonic coupling function JF - Physica : D, Nonlinear phenomena N2 - 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. KW - Kuramoto model KW - Bi-harmonic coupling function KW - Multi-branch entrainment KW - Synchronization Y1 - 2014 U6 - https://doi.org/10.1016/j.physd.2014.09.002 SN - 0167-2789 SN - 1872-8022 VL - 289 SP - 18 EP - 31 PB - Elsevier CY - Amsterdam ER - TY - JOUR A1 - Pikovskij, Arkadij A1 - Gupta, Shamik A1 - Teles, Tarcisio N. A1 - Benetti, Fernanda P. C. A1 - Pakter, Renato A1 - Levin, Yan A1 - Ruffo, Stefano T1 - Ensemble inequivalence in a mean-field XY model with ferromagnetic and nematic couplings JF - Physical review : E, Statistical, nonlinear and soft matter physics N2 - We explore ensemble inequivalence in long-range interacting systems by studying an XY model of classical spinswith ferromagnetic and nematic coupling. We demonstrate the inequivalence bymapping themicrocanonical phase diagram onto the canonical one, and also by doing the inverse mapping. We show that the equilibrium phase diagrams within the two ensembles strongly disagree within the regions of first-order transitions, exhibiting interesting features like temperature jumps. In particular, we discuss the coexistence and forbidden regions of different macroscopic states in both the phase diagrams. Y1 - 2014 U6 - https://doi.org/10.1103/PhysRevE.90.062141 SN - 1539-3755 SN - 1550-2376 VL - 90 IS - 6 PB - American Physical Society CY - College Park ER - TY - JOUR A1 - Straube, Arthur V. A1 - Pikovskij, Arkadij T1 - Pattern formation induced by time-dependent advection JF - Mathematical modelling of natural phenomena N2 - We study pattern-forming instabilities in reaction-advection-diffusion systems. We develop an approach based on Lyapunov-Bloch exponents to figure out the impact of a spatially periodic mixing flow on the stability of a spatially homogeneous state. We deal with the flows periodic in space that may have arbitrary time dependence. We propose a discrete in time model, where reaction, advection, and diffusion act as successive operators, and show that a mixing advection can lead to a pattern-forming instability in a two-component system where only one of the species is advected. Physically, this can be explained as crossing a threshold of Turing instability due to effective increase of one of the diffusion constants. KW - pattern formation KW - reaction-advection-diffusion equation Y1 - 2011 U6 - https://doi.org/10.1051/mmnp/20116107 SN - 0973-5348 VL - 6 IS - 1 SP - 138 EP - 148 PB - EDP Sciences CY - Les Ulis ER - TY - JOUR A1 - Zhirov, O. V. A1 - Pikovskij, Arkadij A1 - Shepelyansky, Dima L. T1 - Quantum vacuum of strongly nonlinear lattices JF - Physical review : E, Statistical, nonlinear and soft matter physics N2 - We study the properties of classical and quantum strongly nonlinear chains by means of extensive numerical simulations. Due to strong nonlinearity, the classical dynamics of such chains remains chaotic at arbitrarily low energies. We show that the collective excitations of classical chains are described by sound waves whose decay rate scales algebraically with the wave number with a generic exponent value. The properties of the quantum chains are studied by the quantum Monte Carlo method and it is found that the low-energy excitations are well described by effective phonon modes with the sound velocity dependent on an effective Planck constant. Our results show that at low energies the quantum effects lead to a suppression of chaos and drive the system to a quasi-integrable regime of effective phonon modes. Y1 - 2011 U6 - https://doi.org/10.1103/PhysRevE.83.016202 SN - 1539-3755 VL - 83 IS - 1 PB - American Physical Society CY - College Park ER - TY - JOUR A1 - Mulansky, Mario A1 - Ahnert, Karsten A1 - Pikovskij, Arkadij T1 - Scaling of energy spreading in strongly nonlinear disordered lattices JF - Physical review : E, Statistical, nonlinear and soft matter physics N2 - To characterize a destruction of Anderson localization by nonlinearity, we study the spreading behavior of initially localized states in disordered, strongly nonlinear lattices. Due to chaotic nonlinear interaction of localized linear or nonlinear modes, energy spreads nearly subdiffusively. Based on a phenomenological description by virtue of a nonlinear diffusion equation, we establish a one-parameter scaling relation between the velocity of spreading and the density, which is confirmed numerically. From this scaling it follows that for very low densities the spreading slows down compared to the pure power law. Y1 - 2011 U6 - https://doi.org/10.1103/PhysRevE.83.026205 SN - 1539-3755 VL - 83 IS - 2 PB - American Physical Society CY - College Park ER - TY - JOUR A1 - Pikovskij, Arkadij A1 - Fishman, Shmuel T1 - Scaling properties of weak chaos in nonlinear disordered lattices JF - Physical review : E, Statistical, nonlinear and soft matter physics N2 - We study the discrete nonlinear Schrodinger equation with a random potential in one dimension. It is characterized by the length, the strength of the random potential, and the field density that determines the effect of nonlinearity. Following the time evolution of the field and calculating the largest Lyapunov exponent, the probability of the system to be regular is established numerically and found to be a scaling function of the parameters. This property is used to calculate the asymptotic properties of the system in regimes beyond our computational power. Y1 - 2011 U6 - https://doi.org/10.1103/PhysRevE.83.025201 SN - 1539-3755 SN - 1550-2376 VL - 83 IS - 2 PB - American Physical Society CY - College Park ER - TY - JOUR A1 - Turukina, L. V. A1 - Pikovskij, Arkadij T1 - Hyperbolic chaos in a system of resonantly coupled weakly nonlinear oscillators JF - Modern physics letters : A, Particles and fields, gravitation, cosmology, nuclear physics N2 - We show that a hyperbolic chaos can be observed in resonantly coupled oscillators near a Hopf bifurcation, described by normal-form-type equations for complex amplitudes. The simplest example consists of four oscillators, comprising two alternatively activated, due to an external periodic modulation, pairs. In terms of the stroboscopic Poincare map, the phase differences change according to an expanding Bernoulli map that depends on the coupling type. Several examples of hyperbolic chaos for different types of coupling are illustrated numerically. KW - Coupled oscillators KW - Hyperbolic chaos Y1 - 2011 U6 - https://doi.org/10.1016/j.physleta.2011.02.017 SN - 0375-9601 VL - 375 IS - 11 SP - 1407 EP - 1411 PB - Elsevier CY - Amsterdam ER - TY - JOUR A1 - Pikovskij, Arkadij A1 - Rosenblum, Michael T1 - Dynamics of heterogeneous oscillator ensembles in terms of collective variables JF - Physica :D, Nonlinear phenomena N2 - 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. KW - Coupled oscillators KW - Oscillator ensembles KW - Kuramoto model KW - Nonlinear coupling KW - Watanabe-Strogatz theory KW - Ott-Antonsen theory Y1 - 2011 U6 - https://doi.org/10.1016/j.physd.2011.01.002 SN - 0167-2789 VL - 240 IS - 9-10 SP - 872 EP - 881 PB - Elsevier CY - Amsterdam ER - TY - JOUR A1 - Kralemann, Björn A1 - Pikovskij, Arkadij A1 - Rosenblum, Michael T1 - Reconstructing phase dynamics of oscillator networks JF - Chaos : an interdisciplinary journal of nonlinear science N2 - We generalize our recent approach to the reconstruction of phase dynamics of coupled oscillators from data [B. Kralemann et al., Phys. Rev. E 77, 066205 (2008)] to cover the case of small networks of coupled periodic units. Starting from a multivariate time series, we first reconstruct genuine phases and then obtain the coupling functions in terms of these phases. Partial norms of these coupling functions quantify directed coupling between oscillators. We illustrate the method by different network motifs for three coupled oscillators and for random networks of five and nine units. We also discuss nonlinear effects in coupling. Y1 - 2011 U6 - https://doi.org/10.1063/1.3597647 SN - 1054-1500 VL - 21 IS - 2 PB - American Institute of Physics CY - Melville ER - TY - JOUR A1 - Lueck, S. A1 - Pikovskij, Arkadij T1 - Dynamics of multi-frequency oscillator ensembles with resonant coupling JF - Modern physics letters : A, Particles and fields, gravitation, cosmology, nuclear physics N2 - 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. KW - Oscillator populations KW - Kuramoto model KW - Resonant interaction Y1 - 2011 U6 - https://doi.org/10.1016/j.physleta.2011.06.016 SN - 0375-9601 VL - 375 IS - 28-29 SP - 2714 EP - 2719 PB - Elsevier CY - Amsterdam ER - TY - JOUR A1 - Levnajic, Zoran A1 - Pikovskij, Arkadij T1 - Network reconstruction from random phase resetting JF - Physical review letters N2 - We propose a novel method of reconstructing the topology and interaction functions for a general oscillator network. An ensemble of initial phases and the corresponding instantaneous frequencies is constructed by repeating random phase resets of the system dynamics. The desired details of network structure are then revealed by appropriately averaging over the ensemble. The method is applicable for a wide class of networks with arbitrary emergent dynamics, including full synchrony. Y1 - 2011 U6 - https://doi.org/10.1103/PhysRevLett.107.034101 SN - 0031-9007 VL - 107 IS - 3 PB - American Physical Society CY - College Park ER - TY - JOUR A1 - Komarov, Maxim A1 - Pikovskij, Arkadij T1 - Effects of nonresonant interaction in ensembles of phase oscillators JF - Physical review : E, Statistical, nonlinear and soft matter physics N2 - We consider general properties of groups of interacting oscillators, for which the natural frequencies are not in resonance. Such groups interact via nonoscillating collective variables like the amplitudes of the order parameters defined for each group. We treat the phase dynamics of the groups using the Ott-Antonsen ansatz and reduce it to a system of coupled equations for the order parameters. We describe different regimes of cosynchrony in the groups. For a large number of groups, heteroclinic cycles, corresponding to a sequential synchronous activity of groups and chaotic states where the order parameters oscillate irregularly, are possible. Y1 - 2011 U6 - https://doi.org/10.1103/PhysRevE.84.016210 SN - 1539-3755 VL - 84 IS - 1 PB - American Physical Society CY - College Park ER - TY - JOUR A1 - Burylko, Oleksandr A1 - Pikovskij, Arkadij T1 - Desynchronization transitions in nonlinearly coupled phase oscillators JF - Physica :D, Nonlinear phenomena N2 - 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. KW - Coupled oscillators KW - Oscillator ensembles KW - Kuramoto model KW - Nonlinear coupling KW - Bifurcations Y1 - 2011 U6 - https://doi.org/10.1016/j.physd.2011.05.016 SN - 0167-2789 VL - 240 IS - 17 SP - 1352 EP - 1361 PB - Elsevier CY - Amsterdam ER - TY - JOUR A1 - Engbert, Ralf A1 - Mergenthaler, Konstantin A1 - Sinn, Petra A1 - Pikovskij, Arkadij T1 - An integrated model of fixational eye movements and microsaccades JF - Proceedings of the National Academy of Sciences of the United States of America N2 - When we fixate a stationary target, our eyes generate miniature (or fixational) eye movements involuntarily. These fixational eye movements are classified as slow components (physiological drift, tremor) and microsaccades, which represent rapid, small-amplitude movements. Here we propose an integrated mathematical model for the generation of slow fixational eye movements and microsaccades. The model is based on the concept of self-avoiding random walks in a potential, a process driven by a self-generated activation field. The self-avoiding walk generates persistent movements on a short timescale, whereas, on a longer timescale, the potential produces antipersistent motions that keep the eye close to an intended fixation position. We introduce microsaccades as fast movements triggered by critical activation values. As a consequence, both slow movements and microsaccades follow the same law of motion; i.e., movements are driven by the self-generated activation field. Thus, the model contributes a unified explanation of why it has been a long-standing problem to separate slow movements and microsaccades with respect to their motion-generating principles. We conclude that the concept of a self-avoiding random walk captures fundamental properties of fixational eye movements and provides a coherent theoretical framework for two physiologically distinct movement types. Y1 - 2011 U6 - https://doi.org/10.1073/pnas.1102730108 SN - 0027-8424 VL - 108 IS - 39 SP - E765 EP - E770 PB - National Acad. of Sciences CY - Washington ER - TY - JOUR A1 - Blaha, Karen A. A1 - Pikovskij, Arkadij A1 - Rosenblum, Michael A1 - Clark, Matthew T. A1 - Rusin, Craig G. A1 - Hudson, John L. T1 - Reconstruction of two-dimensional phase dynamics from experiments on coupled oscillators JF - Physical review : E, Statistical, nonlinear and soft matter physics N2 - Phase models are a powerful method to quantify the coupled dynamics of nonlinear oscillators from measured data. We use two phase modeling methods to quantify the dynamics of pairs of coupled electrochemical oscillators, based on the phases of the two oscillators independently and the phase difference, respectively. We discuss the benefits of the two-dimensional approach relative to the one-dimensional approach using phase difference. We quantify the dependence of the coupling functions on the coupling magnitude and coupling time delay. We show differences in synchronization predictions of the two models using a toy model. We show that the two-dimensional approach reveals behavior not detected by the one-dimensional model in a driven experimental oscillator. This approach is broadly applicable to quantify interactions between nonlinear oscillators, especially where intrinsic oscillator sensitivity and coupling evolve with time. Y1 - 2011 U6 - https://doi.org/10.1103/PhysRevE.84.046201 SN - 1539-3755 VL - 84 IS - 4 PB - American Physical Society CY - College Park ER - TY - JOUR A1 - Mulansky, Mario A1 - Ahnert, Karsten A1 - Pikovskij, Arkadij A1 - Shepelyansky, Dima L. T1 - Strong and weak chaos in weakly nonintegrable many-body hamiltonian systems JF - Journal of statistical physics N2 - We study properties of chaos in generic one-dimensional nonlinear Hamiltonian lattices comprised of weakly coupled nonlinear oscillators by numerical simulations of continuous-time systems and symplectic maps. For small coupling, the measure of chaos is found to be proportional to the coupling strength and lattice length, with the typical maximal Lyapunov exponent being proportional to the square root of coupling. This strong chaos appears as a result of triplet resonances between nearby modes. In addition to strong chaos we observe a weakly chaotic component having much smaller Lyapunov exponent, the measure of which drops approximately as a square of the coupling strength down to smallest couplings we were able to reach. We argue that this weak chaos is linked to the regime of fast Arnold diffusion discussed by Chirikov and Vecheslavov. In disordered lattices of large size we find a subdiffusive spreading of initially localized wave packets over larger and larger number of modes. The relations between the exponent of this spreading and the exponent in the dependence of the fast Arnold diffusion on coupling strength are analyzed. We also trace parallels between the slow spreading of chaos and deterministic rheology. KW - Lyapunov exponent KW - Arnold diffusion KW - Chaos spreading Y1 - 2011 U6 - https://doi.org/10.1007/s10955-011-0335-3 SN - 0022-4715 VL - 145 IS - 5 SP - 1256 EP - 1274 PB - Springer CY - New York ER - TY - JOUR A1 - Schwabedal, Justus T. C. A1 - Pikovskij, Arkadij A1 - Kralemann, Björn A1 - Rosenblum, Michael T1 - Optimal phase description of chaotic oscillators JF - Physical review : E, Statistical, nonlinear and soft matter physics N2 - We introduce an optimal phase description of chaotic oscillations by generalizing the concept of isochrones. On chaotic attractors possessing a general phase description, we define the optimal isophases as Poincare surfaces showing return times as constant as possible. The dynamics of the resultant optimal phase is maximally decoupled from the amplitude dynamics and provides a proper description of the phase response of chaotic oscillations. The method is illustrated with the Rossler and Lorenz systems. Y1 - 2012 U6 - https://doi.org/10.1103/PhysRevE.85.026216 SN - 1539-3755 VL - 85 IS - 2 PB - American Physical Society CY - College Park ER - TY - JOUR A1 - Kuptsov, Pavel V. A1 - Kuznetsov, Sergey P. A1 - Pikovskij, Arkadij T1 - Hyperbolic chaos of turing patterns JF - Physical review letters N2 - We consider time evolution of Turing patterns in an extended system governed by an equation of the Swift-Hohenberg type, where due to an external periodic parameter modulation longwave and shortwave patterns with length scales related as 1:3 emerge in succession. We show theoretically and demonstrate numerically that the spatial phases of the patterns, being observed stroboscopically, are governed by an expanding circle map, so that the corresponding chaos of Turing patterns is hyperbolic, associated with a strange attractor of the Smale-Williams solenoid type. This chaos is shown to be robust with respect to variations of parameters and boundary conditions. Y1 - 2012 U6 - https://doi.org/10.1103/PhysRevLett.108.194101 SN - 0031-9007 VL - 108 IS - 19 PB - American Physical Society CY - College Park ER - TY - JOUR A1 - Roy, S. A1 - Pikovskij, Arkadij T1 - Spreading of energy in the Ding-Dong model JF - Chaos : an interdisciplinary journal of nonlinear science N2 - We study the properties of energy spreading in a lattice of elastically colliding harmonic oscillators (Ding-Dong model). We demonstrate that in the regular lattice the spreading from a localized initial state is mediated by compactons and chaotic breathers. In a disordered lattice, the compactons do not exist, and the spreading eventually stops, resulting in a finite configuration with a few chaotic spots. Y1 - 2012 U6 - https://doi.org/10.1063/1.3695369 SN - 1054-1500 VL - 22 IS - 2 PB - American Institute of Physics CY - Melville ER - TY - JOUR A1 - Braun, W. A1 - Pikovskij, Arkadij A1 - Matias, M. A. A1 - Colet, P. T1 - Global dynamics of oscillator populations under common noise JF - epl : a letters journal exploring the frontiers of physics N2 - Common noise acting on a population of identical oscillators can synchronize them. We develop a description of this process which is not limited to the states close to synchrony, but provides a global picture of the evolution of the ensembles. The theory is based on the Watanabe-Strogatz transformation, allowing us to obtain closed stochastic equations for the global variables. We show that at the initial stage, the order parameter grows linearly in time, while at the later stages the convergence to synchrony is exponentially fast. Furthermore, we extend the theory to nonidentical ensembles with the Lorentzian distribution of natural frequencies and determine the stationary values of the order parameter in dependence on driving noise and mismatch. Y1 - 2012 U6 - https://doi.org/10.1209/0295-5075/99/20006 SN - 0295-5075 VL - 99 IS - 2 PB - EDP Sciences CY - Mulhouse ER - TY - JOUR A1 - Mulansky, Mario A1 - Pikovskij, Arkadij T1 - Scaling properties of energy spreading in nonlinear Hamiltonian two-dimensional lattices JF - Physical review : E, Statistical, nonlinear and soft matter physics N2 - In nonlinear disordered Hamiltonian lattices, where there are no propagating phonons, the spreading of energy is of subdiffusive nature. Recently, the universality class of the subdiffusive spreading according to the nonlinear diffusion equation (NDE) has been suggested and checked for one-dimensional lattices. Here, we apply this approach to two-dimensional strongly nonlinear lattices and find a nice agreement of the scaling predicted from the NDE with the spreading results from extensive numerical studies. Moreover, we show that the scaling works also for regular lattices with strongly nonlinear coupling, for which the scaling exponent is estimated analytically. This shows that the process of chaotic diffusion in such lattices does not require disorder. Y1 - 2012 U6 - https://doi.org/10.1103/PhysRevE.86.056214 SN - 1539-3755 VL - 86 IS - 5 PB - American Physical Society CY - College Park ER - TY - JOUR A1 - Kuptsov, Pavel V. A1 - Kuznetsov, Sergey P. A1 - Pikovskij, Arkadij T1 - Hyperbolic chaos at blinking coupling of noisy oscillators JF - Physical review : E, Statistical, nonlinear and soft matter physics N2 - We study an ensemble of identical noisy phase oscillators with a blinking mean-field coupling, where onecluster and two-cluster synchronous states alternate. In the thermodynamic limit the population is described by a nonlinear Fokker-Planck equation. We show that the dynamics of the order parameters demonstrates hyperbolic chaos. The chaoticity manifests itself in phases of the complex mean field, which obey a strongly chaotic Bernoulli map. Hyperbolicity is confirmed by numerical tests based on the calculations of relevant invariant Lyapunov vectors and Lyapunov exponents. We show how the chaotic dynamics of the phases is slightly smeared by finite-size fluctuations. DOI: 10.1103/PhysRevE.87.032912 Y1 - 2013 U6 - https://doi.org/10.1103/PhysRevE.87.032912 SN - 1539-3755 VL - 87 IS - 3 PB - American Physical Society CY - College Park ER - TY - JOUR A1 - Komarov, Maxim A1 - Pikovskij, Arkadij T1 - Dynamics of multifrequency oscillator communities JF - Physical review letters N2 - We consider a generalization of the Kuramoto model of coupled oscillators to the situation where communities of oscillators having essentially different natural frequencies interact. General equations describing possible resonances between the communities' frequencies are derived. The simplest situation of three resonantly interacting groups is analyzed in detail. We find conditions for the mutual coupling to promote or suppress synchrony in individual populations and present examples where the interaction between communities leads to their synchrony or to a partially asynchronous state or to a chaotic dynamics of order parameters. DOI: 10.1103/PhysRevLett.110.134101 Y1 - 2013 U6 - https://doi.org/10.1103/PhysRevLett.110.134101 SN - 0031-9007 VL - 110 IS - 13 PB - American Physical Society CY - College Park ER - TY - JOUR A1 - Mulansky, Mario A1 - Pikovskij, Arkadij T1 - Energy spreading in strongly nonlinear disordered lattices JF - New journal of physics : the open-access journal for physics N2 - We study the scaling properties of energy spreading in disordered strongly nonlinear Hamiltonian lattices. Such lattices consist of nonlinearly coupled local linear or nonlinear oscillators, and demonstrate a rather slow, subdiffusive spreading of initially localized wave packets. We use a fractional nonlinear diffusion equation as a heuristic model of this process, and confirm that the scaling predictions resulting from a self-similar solution of this equation are indeed applicable to all studied cases. We show that the spreading in nonlinearly coupled linear oscillators slows down compared to a pure power law, while for nonlinear local oscillators a power law is valid in the whole studied range of parameters. Y1 - 2013 U6 - https://doi.org/10.1088/1367-2630/15/5/053015 SN - 1367-2630 VL - 15 IS - 5 PB - IOP Publ. Ltd. CY - Bristol ER - TY - JOUR A1 - Kralemann, Björn A1 - Pikovskij, Arkadij A1 - Rosenblum, Michael T1 - Detecting triplet locking by triplet synchronization indices JF - Physical review : E, Statistical, nonlinear and soft matter physics N2 - We discuss the effect of triplet synchrony in oscillatory networks. In this state the phases and the frequencies of three coupled oscillators fulfill the conditions of a triplet locking, whereas every pair of systems remains asynchronous. We suggest an easy to compute measure, a triplet synchronization index, which can be used to detect such states from experimental data. Y1 - 2013 U6 - https://doi.org/10.1103/PhysRevE.87.052904 SN - 1539-3755 VL - 87 IS - 5 PB - American Physical Society CY - College Park ER -