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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - INPR A1 - Pikovskij, Arkadij A1 - Zaks, Michael A. A1 - Feudel, Ulrike A1 - Kurths, Jürgen T1 - Singular continuous spectra in dissipative dynamics N2 - We demonstrate the occurrence of regimes with singular continuous (fractal) Fourier spectra in autonomous dissipative dynamical systems. The particular example in an ODE system at the accumulation points of bifurcation sequences associated to the creation of complicated homoclinic orbits. Two different machanisms responsible for the appearance of such spectra are proposed. In the first case when the geometry of the attractor is symbolically represented by the Thue-Morse sequence, both the continuous-time process and its descrete Poincaré map have singular power spectra. The other mechanism owes to the logarithmic divergence of the first return times near the saddle point; here the Poincaré map possesses the discrete spectrum, while the continuous-time process displays the singular one. A method is presented for computing the multifractal characteristics of the singular continuous spectra with the help of the usual Fourier analysis technique. T3 - NLD Preprints - 15 Y1 - 1995 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-13787 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 - INPR A1 - Kurths, Jürgen A1 - Pikovskij, Arkadij A1 - Scheffczyk, Christian T1 - Roughening interfaces in deterministic dynamics N2 - Two deterministic processes leading to roughening interfaces are considered. It is shown that the dynamics of linear perturbations of turbulent regimes in coupled map lattices is governed by a discrete version of the Kardar-Parisi-Zhang equation. The asymptotic scaling behavior of the perturbation field is investigated in the case of large lattices. Secondly, the dynamics of an order-disorder interface is modelled with a simple two-dimensional coupled map lattice, possesing a turbulent and a laminar state. It is demonstrated, that in some range of parameters the spreading of the turbulent state is accompanied by kinetic roughening of the interface. T3 - NLD Preprints - 3 Y1 - 1994 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-13447 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 - 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 - 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 - 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 - 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 -