@article{ZillmerPikovskij2005, author = {Zillmer, R{\"u}diger and Pikovskij, Arkadij}, title = {Continuous approach for the random-field Ising chain}, year = {2005}, abstract = {We study the random-field Ising chain in the limit of strong exchange coupling. In order to calculate the free energy we apply a continuous Langevin-type approach. This continuous model can be solved exactly, whereupon we are able to locate the crossover between an exponential and a power-law decay of the free energy with increasing coupling strength. In terms of magnetization, this crossover restricts the validity of the linear scaling. The known analytical results for the free energy are recovered in the corresponding limits. The outcomes of numerical computations for the free energy are presented, which confirm the results of the continuous approach. We also discuss the validity of the replica method which we then utilize to investigate the sample-to-sample fluctuations of the finite size free energy}, language = {en} } @article{ZillmerAhlersPikovskij2000, author = {Zillmer, R{\"u}diger and Ahlers, Volker and Pikovskij, Arkadij}, title = {Scaling of Lyapunov exponents of coupled chaotic systems}, year = {2000}, abstract = {We develop a statistical theory of the coupling sensitivity of chaos. The effect was first described by Daido [Prog. Theor. Phys. 72, 853 (1984)]; it appears as a logarithmic singularity in the Lyapunov exponent in coupled chaotic systems at very small couplings. Using a continuous-time stochastic model for the coupled systems we derive a scaling relation for the largest Lyapunov exponent. The singularity is shown to depend on the coupling and the systems' mismatch. Generalizations to the cases of asymmetrical coupling and three interacting oscillators are considered, too. The analytical results are confirmed by numerical simulations.}, language = {en} } @article{ZillmerAhlersPikovskij2000, author = {Zillmer, R{\"u}diger and Ahlers, Volker and Pikovskij, Arkadij}, title = {Stochastic approach to Lapunov exponents in coupled chaotic systems}, isbn = {3-540-41074-0}, year = {2000}, language = {en} } @article{ZhirovPikovskijShepelyansky2011, author = {Zhirov, O. V. and Pikovskij, Arkadij and Shepelyansky, Dima L.}, title = {Quantum vacuum of strongly nonlinear lattices}, series = {Physical review : E, Statistical, nonlinear and soft matter physics}, volume = {83}, journal = {Physical review : E, Statistical, nonlinear and soft matter physics}, number = {1}, publisher = {American Physical Society}, address = {College Park}, issn = {1539-3755}, doi = {10.1103/PhysRevE.83.016202}, pages = {7}, year = {2011}, abstract = {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.}, language = {en} } @article{ZhengToenjesPikovskij2021, author = {Zheng, Chunming and Toenjes, Ralf and Pikovskij, Arkadij}, title = {Transition to synchrony in a three-dimensional swarming model with helical trajectories}, series = {Physical review : E, Statistical, nonlinear and soft matter physics}, volume = {104}, journal = {Physical review : E, Statistical, nonlinear and soft matter physics}, number = {1}, publisher = {American Physical Society}, address = {College Park}, issn = {2470-0045}, doi = {10.1103/PhysRevE.104.014216}, pages = {7}, year = {2021}, abstract = {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.}, language = {en} } @article{ZhengPikovskij2019, author = {Zheng, Chunming and Pikovskij, Arkadij}, title = {Stochastic bursting in unidirectionally delay-coupled noisy excitable systems}, series = {Chaos : an interdisciplinary journal of nonlinear science}, volume = {29}, journal = {Chaos : an interdisciplinary journal of nonlinear science}, number = {4}, publisher = {American Institute of Physics}, address = {Melville}, issn = {1054-1500}, doi = {10.1063/1.5093180}, pages = {9}, year = {2019}, abstract = {We show that "stochastic bursting" is observed in a ring of unidirectional delay-coupled noisy excitable systems, thanks to the combinational action of time-delayed coupling and noise. Under the approximation of timescale separation, i.e., when the time delays in each connection are much larger than the characteristic duration of the spikes, the observed rather coherent spike pattern can be described by an idealized coupled point processwith a leader-follower relationship. We derive analytically the statistics of the spikes in each unit, the pairwise correlations between any two units, and the spectrum of the total output from the network. Theory is in good agreement with the simulations with a network of theta-neurons. Published under license by AIP Publishing.}, language = {en} } @article{ZhengPikovskij2018, author = {Zheng, Chunming and Pikovskij, Arkadij}, title = {Delay-induced stochastic bursting in excitable noisy systems}, series = {Physical review : E, Statistical, nonlinear and soft matter physics}, volume = {98}, journal = {Physical review : E, Statistical, nonlinear and soft matter physics}, number = {4}, publisher = {American Physical Society}, address = {College Park}, issn = {2470-0045}, doi = {10.1103/PhysRevE.98.042148}, pages = {8}, year = {2018}, abstract = {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.}, language = {en} } @article{ZhangPikovskijLiu2017, author = {Zhang, Xiyun and Pikovskij, Arkadij and Liu, Zonghua}, title = {Dynamics of oscillators globally coupled via two mean fields}, series = {Scientific reports}, volume = {7}, journal = {Scientific reports}, publisher = {Nature Publ. Group}, address = {London}, issn = {2045-2322}, doi = {10.1038/s41598-017-02283-1}, pages = {16}, year = {2017}, abstract = {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.}, language = {en} } @article{ZaksPikovskij2017, author = {Zaks, Michael and Pikovskij, Arkadij}, title = {Chimeras and complex cluster states in arrays of spin-torque oscillators}, series = {Scientific reports}, volume = {7}, journal = {Scientific reports}, publisher = {Nature Publ. Group}, address = {London}, issn = {2045-2322}, doi = {10.1038/s41598-017-04918-9}, pages = {10}, year = {2017}, abstract = {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.}, language = {en} } @article{ZaksRosenblumPikovskijetal.1997, author = {Zaks, Michael A. and Rosenblum, Michael and Pikovskij, Arkadij and Osipov, Grigory V. and Kurths, J{\"u}rgen}, title = {Phase synchronization of chaotic oscillations in terms of periodic orbits}, issn = {1054-1500}, year = {1997}, language = {en} } @article{ZaksPikovskijKurths1999, author = {Zaks, Michael A. and Pikovskij, Arkadij and Kurths, J{\"u}rgen}, title = {On the generalized dimensions for the fourier spectrum of the thue-morse sequence}, year = {1999}, language = {en} } @article{ZaksPikovskijKurths1998, author = {Zaks, Michael A. and Pikovskij, Arkadij and Kurths, J{\"u}rgen}, title = {Symbolic dynamics behind the singular continuous power spectra of continuous flows}, year = {1998}, language = {en} } @article{ZaksPikovskijKurths1997, author = {Zaks, Michael A. and Pikovskij, Arkadij and Kurths, J{\"u}rgen}, title = {On the correlation dimension of the spectral measure for the Thue-Morse sequence}, year = {1997}, language = {en} } @article{ZaksPikovskij2019, author = {Zaks, Michael A. and Pikovskij, Arkadij}, title = {Synchrony breakdown and noise-induced oscillation death in ensembles of serially connected spin-torque oscillators}, series = {The European physical journal : B, Condensed matter and complex systems}, volume = {92}, journal = {The European physical journal : B, Condensed matter and complex systems}, number = {7}, publisher = {Springer}, address = {New York}, issn = {1434-6028}, doi = {10.1140/epjb/e2019-100152-2}, pages = {12}, year = {2019}, abstract = {We consider collective dynamics in the ensemble of serially connected spin-torque oscillators governed by the Landau-Lifshitz-Gilbert-Slonczewski magnetization equation. Proximity to homoclinicity hampers synchronization of spin-torque oscillators: when the synchronous ensemble experiences the homoclinic bifurcation, the growth rate per oscillation of small deviations from the ensemble mean diverges. Depending on the configuration of the contour, sufficiently strong common noise, exemplified by stochastic oscillations of the current through the circuit, may suppress precession of the magnetic field for all oscillators. We derive the explicit expression for the threshold amplitude of noise, enabling this suppression.}, language = {en} } @misc{ZaksPikovskij2017, author = {Zaks, Michael A. and Pikovskij, Arkadij}, title = {Chimeras and complex cluster states in arrays of spin-torque oscillators}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-402180}, pages = {10}, year = {2017}, abstract = {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.}, language = {en} } @article{ZaksPikovskij2017, author = {Zaks, Michael A. and Pikovskij, Arkadij}, title = {Chimeras and complex cluster states in arrays of spin-torque oscillators}, series = {Scientific reports}, volume = {7}, journal = {Scientific reports}, publisher = {Macmillan Publishers Limited}, address = {London}, issn = {2045-2322}, doi = {10.1038/s41598-017-04918-9}, year = {2017}, abstract = {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.}, language = {en} } @article{YeldesbayPikovskijRosenblum2014, author = {Yeldesbay, Azamat and Pikovskij, Arkadij and Rosenblum, Michael}, title = {Chimeralike states in an ensemble of globally coupled oscillators}, series = {Physical review letters}, volume = {112}, journal = {Physical review letters}, number = {14}, publisher = {American Physical Society}, address = {College Park}, issn = {0031-9007}, doi = {10.1103/PhysRevLett.112.144103}, pages = {5}, year = {2014}, abstract = {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.}, language = {en} } @article{WittKurthsPikovskij1998, author = {Witt, Annette and Kurths, J{\"u}rgen and Pikovskij, Arkadij}, title = {Testing stationarity in time series}, year = {1998}, language = {en} } @article{WittFeudelPikovskij1997, author = {Witt, Annette and Feudel, Ulrike and Pikovskij, Arkadij}, title = {Birth of strange nonchaotic attractors due to interior crisis}, year = {1997}, language = {en} } @article{VlasovRosenblumPikovskij2016, author = {Vlasov, Vladimir and Rosenblum, Michael and Pikovskij, Arkadij}, title = {Dynamics of weakly inhomogeneous oscillator populations: perturbation theory on top of Watanabe-Strogatz integrability}, series = {Journal of physics : A, Mathematical and theoretical}, volume = {49}, journal = {Journal of physics : A, Mathematical and theoretical}, publisher = {IOP Publ. Ltd.}, address = {Bristol}, issn = {1751-8113}, doi = {10.1088/1751-8113/49/31/31LT02}, pages = {8}, year = {2016}, abstract = {As has been shown by Watanabe and Strogatz (WS) (1993 Phys. Rev. Lett. 70 2391), a population of identical phase oscillators, sine-coupled to a common field, is a partially integrable system: for any ensemble size its dynamics reduce to equations for three collective variables. Here we develop a perturbation approach for weakly nonidentical ensembles. We calculate corrections to the WS dynamics for two types of perturbations: those due to a distribution of natural frequencies and of forcing terms, and those due to small white noise. We demonstrate that in both cases, the complex mean field for which the dynamical equations are written is close to the Kuramoto order parameter, up to the leading order in the perturbation. This supports the validity of the dynamical reduction suggested by Ott and Antonsen (2008 Chaos 18 037113) for weakly inhomogeneous populations.}, language = {en} } @article{VlasovPikovskijMacau2015, author = {Vlasov, Vladimir and Pikovskij, Arkadij and Macau, Elbert E. N.}, title = {Star-type oscillatory networks with generic Kuramoto-type coupling: A model for "Japanese drums synchrony"}, series = {Chaos : an interdisciplinary journal of nonlinear science}, volume = {25}, journal = {Chaos : an interdisciplinary journal of nonlinear science}, number = {12}, publisher = {American Institute of Physics}, address = {Melville}, issn = {1054-1500}, doi = {10.1063/1.4938400}, pages = {13}, year = {2015}, abstract = {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.}, language = {en} } @article{VlasovPikovskij2013, author = {Vlasov, Vladimir and Pikovskij, Arkadij}, title = {Synchronization of a Josephson junction array in terms of global variables}, series = {Physical review : E, Statistical, nonlinear and soft matter physics}, volume = {88}, journal = {Physical review : E, Statistical, nonlinear and soft matter physics}, number = {2}, publisher = {American Physical Society}, address = {College Park}, issn = {1539-3755}, doi = {10.1103/PhysRevE.88.022908}, pages = {5}, year = {2013}, abstract = {We consider an array of Josephson junctions with a common LCR load. Application of the Watanabe-Strogatz approach [Physica D 74, 197 (1994)] allows us to formulate the dynamics of the array via the global variables only. For identical junctions this is a finite set of equations, analysis of which reveals the regions of bistability of the synchronous and asynchronous states. For disordered arrays with distributed parameters of the junctions, the problem is formulated as an integro-differential equation for the global variables; here stability of the asynchronous states and the properties of the transition synchrony-asynchrony are established numerically.}, language = {en} } @article{VlasovMacauPikovskij2014, author = {Vlasov, Vladimir and Macau, Elbert E. N. and Pikovskij, Arkadij}, title = {Synchronization of oscillators in a Kuramoto-type model with generic coupling}, series = {Chaos : an interdisciplinary journal of nonlinear science}, volume = {24}, journal = {Chaos : an interdisciplinary journal of nonlinear science}, number = {2}, publisher = {American Institute of Physics}, address = {Melville}, issn = {1054-1500}, doi = {10.1063/1.4880835}, pages = {7}, year = {2014}, abstract = {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.}, language = {en} } @article{VlasovKomarovPikovskij2015, author = {Vlasov, Vladimir and Komarov, Maxim and Pikovskij, Arkadij}, title = {Synchronization transitions in ensembles of noisy oscillators with bi-harmonic coupling}, series = {Journal of physics : A, Mathematical and theoretical}, volume = {48}, journal = {Journal of physics : A, Mathematical and theoretical}, number = {10}, publisher = {IOP Publ. Ltd.}, address = {Bristol}, issn = {1751-8113}, doi = {10.1088/1751-8113/48/10/105101}, pages = {16}, year = {2015}, abstract = {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.}, language = {en} } @article{TyulkinaGoldobinKlimenkoetal.2019, author = {Tyulkina, Irina V. and Goldobin, Denis S. and Klimenko, Lyudmila S. and Pikovskij, Arkadij}, title = {Two-Bunch Solutions for the Dynamics of Ott-Antonsen Phase Ensembles}, series = {Radiophysics and Quantum Electronics}, volume = {61}, journal = {Radiophysics and Quantum Electronics}, number = {8-9}, publisher = {Springer}, address = {New York}, issn = {0033-8443}, doi = {10.1007/s11141-019-09924-7}, pages = {640 -- 649}, year = {2019}, abstract = {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).}, language = {en} } @article{TyulkinaGoldobinKlimenkoetal.2018, author = {Tyulkina, Irina and Goldobin, Denis S. and Klimenko, Lyudmila S. and Pikovskij, Arkadij}, title = {Dynamics of noisy oscillator populations beyond the Ott-Antonsen Ansatz}, series = {Physical review letters}, volume = {120}, journal = {Physical review letters}, number = {26}, publisher = {American Physical Society}, address = {College Park}, issn = {0031-9007}, doi = {10.1103/PhysRevLett.120.264101}, pages = {6}, year = {2018}, abstract = {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.}, language = {en} } @article{TurukinaPikovskij2011, author = {Turukina, L. V. and Pikovskij, Arkadij}, title = {Hyperbolic chaos in a system of resonantly coupled weakly nonlinear oscillators}, series = {Modern physics letters : A, Particles and fields, gravitation, cosmology, nuclear physics}, volume = {375}, journal = {Modern physics letters : A, Particles and fields, gravitation, cosmology, nuclear physics}, number = {11}, publisher = {Elsevier}, address = {Amsterdam}, issn = {0375-9601}, doi = {10.1016/j.physleta.2011.02.017}, pages = {1407 -- 1411}, year = {2011}, abstract = {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.}, language = {en} } @article{TsimringPikovskij2001, author = {Tsimring, L. S. and Pikovskij, Arkadij}, title = {Noise-Induced Dynamics in Bistable Systems with Delay}, year = {2001}, abstract = {Noise-induced dynamics of a prototypical bistable system with delayed feedback is studied theoretically and numerically. For small noise and magnitude of the feedback, the problem is reduced to the analysis of the two-state model with transition rates depending on the earlier state of the system. Analytical solutions for the autocorrelation function and the power spectrum have been found. The power spectrum has a peak at the frequency corresponding to the inverse delay time, whose amplitude has a maximum at a certain noise level, thus demonstrating coherence resonance. The linear response to the external periodic force also has maxima at the frequencies corresponding to the inverse delay time and its harmonics.}, language = {en} } @misc{TopcuFruehwirthMoseretal.2018, author = {Top{\c{c}}u, {\c{C}}ağda{\c{s}} and Fr{\"u}hwirth, Matthias and Moser, Maximilian and Rosenblum, Michael and Pikovskij, Arkadij}, title = {Disentangling respiratory sinus arrhythmia in heart rate variability records}, series = {Postprints der Universit{\"a}t Potsdam : Mathematisch-Naturwissenschaftliche Reihe}, journal = {Postprints der Universit{\"a}t Potsdam : Mathematisch-Naturwissenschaftliche Reihe}, number = {913}, issn = {1866-8372}, doi = {10.25932/publishup-43631}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-436315}, pages = {15}, year = {2018}, abstract = {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.}, language = {en} } @article{TopcuFruehwirthMoseretal.2018, author = {Top{\c{c}}u, {\c{C}}ağda{\c{s}} and Fr{\"u}hwirth, Matthias and Moser, Maximilian and Rosenblum, Michael and Pikovskij, Arkadij}, title = {Disentangling respiratory sinus arrhythmia in heart rate variability records}, series = {Physiological Measurement}, volume = {39}, journal = {Physiological Measurement}, number = {5}, publisher = {IOP Publ. Ltd.}, address = {Bristol}, issn = {0967-3334}, doi = {10.1088/1361-6579/aabea4}, pages = {12}, year = {2018}, abstract = {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.}, language = {en} } @article{TopajKyePikovskij2001, author = {Topaj, Dmitri and Kye, W.-H and Pikovskij, Arkadij}, title = {Transition to Coherence in Populations of Coupled Chaotic Oscillators: A Linear Response Approach}, year = {2001}, abstract = {We consider the collective dynamics in an ensemble of globally coupled chaotic maps. The transition to the coherent state with a macroscopic mean field is analyzed in the framework of the linear response theory. The linear response function for the chaotic system is obtained using the perturbation approach to the Frobenius-Perron operator. The transition point is defined from this function by virtue of the self-excitation condition for the feedback loop. Analytical results for the coupled Bernoulli maps are confirmed by the numerics.}, language = {en} } @article{TassRosenblumWeuleetal.1998, author = {Tass, Peter and Rosenblum, Michael and Weule, J. and Kurths, J{\"u}rgen and Pikovskij, Arkadij and Volkmann, J. and Schnitzler, A. and Freund, H.-J.}, title = {Detection of n:m phase locking from noisy data : application to magnetoencephalography}, year = {1998}, abstract = {We use the concept of phase synchronization for the analysis of noisy nonstationary bivariate data. Phase synchronization is understood in a statistical sense as an existence of preferred values of the phase difference, and two techniques are proposed for a reliable detection of synchronous epochs. These methods are applied to magnetoencephalograms and records of muscle activity of a Parkinsonian patient. We reveal that}, language = {en} } @article{SysoevPonomarenkoPikovskij2017, author = {Sysoev, Ilya V. and Ponomarenko, Vladimir I. and Pikovskij, Arkadij}, title = {Reconstruction of coupling architecture of neural field networks from vector time series}, series = {Communications in nonlinear science \& numerical simulation}, volume = {57}, journal = {Communications in nonlinear science \& numerical simulation}, publisher = {Elsevier}, address = {Amsterdam}, issn = {1007-5704}, doi = {10.1016/j.cnsns.2017.10.006}, pages = {342 -- 351}, year = {2017}, abstract = {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.}, language = {en} } @article{StraubePikovskij2011, author = {Straube, Arthur V. and Pikovskij, Arkadij}, title = {Pattern formation induced by time-dependent advection}, series = {Mathematical modelling of natural phenomena}, volume = {6}, journal = {Mathematical modelling of natural phenomena}, number = {1}, publisher = {EDP Sciences}, address = {Les Ulis}, issn = {0973-5348}, doi = {10.1051/mmnp/20116107}, pages = {138 -- 148}, year = {2011}, abstract = {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.}, language = {en} } @misc{StraubePikovskij2011, author = {Straube, Arthur V. and Pikovskij, Arkadij}, title = {Pattern formation induced by time-dependent advection}, series = {Postprints der Universit{\"a}t Potsdam : Mathematisch Naturwissenschaftliche Reihe}, journal = {Postprints der Universit{\"a}t Potsdam : Mathematisch Naturwissenschaftliche Reihe}, number = {575}, issn = {1866-8372}, doi = {10.25932/publishup-41314}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-413140}, pages = {138-147}, year = {2011}, abstract = {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.}, language = {en} } @article{StraubeAbelPikovskij2004, author = {Straube, Arthur V. and Abel, Markus and Pikovskij, Arkadij}, title = {Temporal chaos versus spatial mixing in reaction-advection-diffusion systems}, issn = {0031-9007}, year = {2004}, abstract = {We develop a theory describing the transition to a spatially homogeneous regime in a mixing flow with a chaotic in time reaction. The transverse Lyapunov exponent governing the stability of the homogeneous state can be represented as a combination of Lyapunov exponents for spatial mixing and temporal chaos. This representation, being exact for time- independent flows and equal Peclet numbers of different components, is demonstrated to work accurately for time- dependent flows and different Peclet numbers}, language = {en} } @article{StarkFeudelGlendinningetal.2002, author = {Stark, J. and Feudel, Ulrike and Glendinning, P. A. and Pikovskij, Arkadij}, title = {Rotation numbers for quasi-periodically forced monotone circle maps}, issn = {1468-9367}, year = {2002}, language = {en} } @article{SmirnovOsipovPikovskij2018, author = {Smirnov, Lev A. and Osipov, Grigory V. and Pikovskij, Arkadij}, title = {Solitary synchronization waves in distributed oscillator populations}, series = {Physical review : E, Statistical, nonlinear and soft matter physics}, volume = {98}, journal = {Physical review : E, Statistical, nonlinear and soft matter physics}, number = {6}, publisher = {American Physical Society}, address = {College Park}, issn = {2470-0045}, doi = {10.1103/PhysRevE.98.062222}, pages = {062222-1 -- 062222-7}, year = {2018}, abstract = {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.}, language = {en} } @article{SmirnovOsipovPikovskij2017, author = {Smirnov, Lev A. and Osipov, Grigory V. and Pikovskij, Arkadij}, title = {Chimera patterns in the Kuramoto-Battogtokh model}, series = {Journal of physics : A, Mathematical and theoretical}, volume = {50}, journal = {Journal of physics : A, Mathematical and theoretical}, number = {8}, publisher = {IOP Publ. Ltd.}, address = {Bristol}, issn = {1751-8113}, doi = {10.1088/1751-8121/aa55f1}, pages = {10}, year = {2017}, abstract = {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.}, language = {en} } @article{SmirnovBolotovOsipovetal.2021, author = {Smirnov, Lev A. and Bolotov, Maxim I. and Osipov, Grigorij V. and Pikovskij, Arkadij}, title = {Disorder fosters chimera in an array of motile particles}, series = {Physical review : E, Statistical, nonlinear and soft matter physics}, volume = {104}, journal = {Physical review : E, Statistical, nonlinear and soft matter physics}, number = {3}, publisher = {American Physical Society}, address = {Melville, NY}, issn = {2470-0045}, doi = {10.1103/PhysRevE.104.034205}, pages = {8}, year = {2021}, abstract = {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.}, language = {en} } @article{ShepelyanskyPikovskijSchmidtetal.2009, author = {Shepelyansky, Dima L. and Pikovskij, Arkadij and Schmidt, J{\"u}rgen and Spahn, Frank}, title = {Synchronization mechanism of sharp edges in rings of Saturn}, issn = {0035-8711}, doi = {10.1111/j.1365-2966.2009.14719.x}, year = {2009}, abstract = {We propose a new mechanism which explains the existence of enormously sharp edges in the rings of Saturn. This mechanism is based on the synchronization phenomenon due to which the epicycle rotational phases of particles in the ring, under certain conditions, become synchronized with the phase of external satellite, e. g. with the phase of Mimas in the case of the outer B ring edge. This synchronization eliminates collisions between particles and suppresses the diffusion induced by collisions by orders of magnitude. The minimum of the diffusion is reached at the centre of the synchronization regime corresponding to the ratio 2:1 between the orbital frequency at the edge of B ring and the orbital frequency of Mimas. The synchronization theory gives the sharpness of the edge in a few tens of meters that is in agreement with available observations.}, language = {en} } @article{SchwabedalPikovskijKralemannetal.2012, author = {Schwabedal, Justus T. C. and Pikovskij, Arkadij and Kralemann, Bj{\"o}rn and Rosenblum, Michael}, title = {Optimal phase description of chaotic oscillators}, series = {Physical review : E, Statistical, nonlinear and soft matter physics}, volume = {85}, journal = {Physical review : E, Statistical, nonlinear and soft matter physics}, number = {2}, publisher = {American Physical Society}, address = {College Park}, issn = {1539-3755}, doi = {10.1103/PhysRevE.85.026216}, pages = {9}, year = {2012}, abstract = {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.}, language = {en} } @article{SchwabedalPikovskij2010, author = {Schwabedal, Justus T. C. and Pikovskij, Arkadij}, title = {Effective phase description of noise-perturbed and noise-induced oscillations}, issn = {1951-6355}, doi = {10.1140/epjst/e2010-01271-6}, year = {2010}, abstract = {An effective dynamical description of a general class of stochastic phase oscillators is presented. For this, the effective phase velocity is defined either by the stochastic phase oscillators invariant probability density or its first passage times. Using the first approach the effective phase exhibits the correct frequency and invariant distribution density, whereas the second approach models the proper phase resetting curve. The discrepancy of the effective models is most pronounced for noise-induced oscillations and is related to non-monotonicity of the stochastic phase variable due to fluctuations.}, language = {en} } @article{SchwabedalPikovskij2010, author = {Schwabedal, Justus T. C. and Pikovskij, Arkadij}, title = {Effective phase dynamics of noise-induced oscillations in excitable systems}, issn = {1539-3755}, doi = {10.1103/Physreve.81.046218}, year = {2010}, abstract = {We develop an effective description of noise-induced oscillations based on deterministic phase dynamics. The phase equation is constructed to exhibit correct frequency and distribution density of noise-induced oscillations. In the simplest one-dimensional case the effective phase equation is obtained analytically, whereas for more complex situations a simple method of data processing is suggested. As an application an effective coupling function is constructed that quantitatively describes periodically forced noise-induced oscillations.}, language = {en} } @article{SchwabedalPikovskij2013, author = {Schwabedal, Justus T. C. and Pikovskij, Arkadij}, title = {Phase description of stochastic oscillations}, series = {Physical review letters}, volume = {110}, journal = {Physical review letters}, number = {20}, publisher = {American Physical Society}, address = {College Park}, issn = {0031-9007}, doi = {10.1103/PhysRevLett.110.204102}, pages = {5}, year = {2013}, abstract = {We introduce an invariant phase description of stochastic oscillations by generalizing the concept of standard isophases. The average isophases are constructed as sections in the state space, having a constant mean first return time. The approach allows us to obtain a global phase variable of noisy oscillations, even in the cases where the phase is ill defined in the deterministic limit. A simple numerical method for finding the isophases is illustrated for noise-induced switching between two coexisting limit cycles, and for noise-induced oscillation in an excitable system. We also discuss how to determine isophases of observed irregular oscillations, providing a basis for a refined phase description in data analysis.}, language = {en} } @article{RuzickScheffczykPikovskijetal.1997, author = {Ruzick, Oliver and Scheffczyk, Christian and Pikovskij, Arkadij and Kurths, J{\"u}rgen}, title = {Dynamics of chaos-order interface in coupled map lattices}, year = {1997}, language = {en} } @article{RuffoPikovskij1999, author = {Ruffo, Stefano and Pikovskij, Arkadij}, title = {Finite-size effects in a population of interacting oscillators}, year = {1999}, abstract = {We consider a large population of globally coupled noisy phase oscillators. In the thermodynamic limit N this system exhibits a nonequilibrium phase transition, at which amacroscopic mean field appears. It is shown that for large but finite system size N the system can be described by the noisy Stuart-Landau equation, yielding scaling behavior of statistical characteristics of the macroscopic mean field with N. The predictions of the theory are checked numerically.}, language = {en} } @article{RoyPikovskij2012, author = {Roy, S. and Pikovskij, Arkadij}, title = {Spreading of energy in the Ding-Dong model}, series = {Chaos : an interdisciplinary journal of nonlinear science}, volume = {22}, journal = {Chaos : an interdisciplinary journal of nonlinear science}, number = {2}, publisher = {American Institute of Physics}, address = {Melville}, issn = {1054-1500}, doi = {10.1063/1.3695369}, pages = {7}, year = {2012}, abstract = {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.}, language = {en} } @misc{RosenblumPikovskijKuehnetal.2021, author = {Rosenblum, Michael and Pikovskij, Arkadij and K{\"u}hn, Andrea A. and Busch, Johannes Leon}, title = {Real-time estimation of phase and amplitude with application to neural data}, series = {Zweitver{\"o}ffentlichungen der Universit{\"a}t Potsdam : Mathematisch-Naturwissenschaftliche Reihe}, journal = {Zweitver{\"o}ffentlichungen der Universit{\"a}t Potsdam : Mathematisch-Naturwissenschaftliche Reihe}, publisher = {Universit{\"a}tsverlag Potsdam}, address = {Potsdam}, issn = {1866-8372}, doi = {10.25932/publishup-54963}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-549630}, pages = {11}, year = {2021}, abstract = {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.}, language = {en} } @article{RosenblumPikovskijKuehnetal.2021, author = {Rosenblum, Michael and Pikovskij, Arkadij and K{\"u}hn, Andrea A. and Busch, Johannes Leon}, title = {Real-time estimation of phase and amplitude with application to neural data}, series = {Scientific reports}, volume = {11}, journal = {Scientific reports}, publisher = {Springer Nature}, address = {London}, issn = {2045-2322}, doi = {10.1038/s41598-021-97560-5}, pages = {11}, year = {2021}, abstract = {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.}, language = {en} } @article{RosenblumPikovskijKurthsetal.2002, author = {Rosenblum, Michael and Pikovskij, Arkadij and Kurths, J{\"u}rgen and Osipov, Grigory V. and Kiss, Istvan Z. and Hudson, J. L.}, title = {Locking-based frequency measurement and synchronization of chaotic oscillators with complex dynamics}, year = {2002}, language = {en} } @article{RosenblumPikovskijKurths2004, author = {Rosenblum, Michael and Pikovskij, Arkadij and Kurths, J{\"u}rgen}, title = {Synchronization approach to analysis of biological systems}, issn = {0219-4775}, year = {2004}, abstract = {In this article we review the application of the synchronization theory to the analysis of multivariate biological signals. We address the problem of phase estimation from data and detection and quantification of weak interaction, as well as quantification of the direction of coupling. We discuss the potentials as well as limitations and misinterpretations of the approach}, language = {en} } @article{RosenblumPikovskijKurths1997, author = {Rosenblum, Michael and Pikovskij, Arkadij and Kurths, J{\"u}rgen}, title = {From Phase to Lag Synchronization in Coupled Chaotic Oscillators}, year = {1997}, abstract = {We study synchronization transitions in a system of two coupled self-sustained chaotic oscillators. We demonstrate that with the increase of coupling strength the system first undergoes the transition to phase synchronization. With a further increase of coupling, a new synchronous regime is observed, where the states of two oscillators are nearly identical, but one system lags in time to the other. We describe thisregime as a state with correlated amplitudes and a constant phase shift. These transitions are traced in the Lyapunov spectrum.}, language = {en} } @article{RosenblumPikovskijKurths1997, author = {Rosenblum, Michael and Pikovskij, Arkadij and Kurths, J{\"u}rgen}, title = {Phase synchronization in noisy and chaotic oscillators}, year = {1997}, language = {en} } @article{RosenblumPikovskijKurths1997, author = {Rosenblum, Michael and Pikovskij, Arkadij and Kurths, J{\"u}rgen}, title = {Phase synchronization in driven and coupled chaotic oscillators}, year = {1997}, language = {en} } @article{RosenblumPikovskijKurths1997, author = {Rosenblum, Michael and Pikovskij, Arkadij and Kurths, J{\"u}rgen}, title = {Effect of phase synchronization in driven chaotic oscillators}, year = {1997}, language = {en} } @article{RosenblumPikovskij2019, author = {Rosenblum, Michael and Pikovskij, Arkadij}, title = {Nonlinear phase coupling functions: a numerical study}, series = {Philosophical Transactions of the Royal Society of London, Series A : Mathematical, Physical and Engineering Sciences}, volume = {377}, journal = {Philosophical Transactions of the Royal Society of London, Series A : Mathematical, Physical and Engineering Sciences}, number = {2160}, publisher = {Royal Society}, address = {London}, issn = {1364-503X}, doi = {10.1098/rsta.2019.0093}, pages = {12}, year = {2019}, abstract = {Phase reduction is a general tool widely used to describe forced and interacting self-sustained oscillators. Here, we explore the phase coupling functions beyond the usual first-order approximation in the strength of the force. Taking the periodically forced Stuart-Landau oscillator as the paradigmatic model, we determine and numerically analyse the coupling functions up to the fourth order in the force strength. We show that the found nonlinear phase coupling functions can be used for predicting synchronization regions of the forced oscillator.}, language = {en} } @article{RosenblumPikovskij2019, author = {Rosenblum, Michael and Pikovskij, Arkadij}, title = {Numerical phase reduction beyond the first order approximation}, series = {Chaos : an interdisciplinary journal of nonlinear science}, volume = {29}, journal = {Chaos : an interdisciplinary journal of nonlinear science}, number = {1}, publisher = {American Institute of Physics}, address = {Melville}, issn = {1054-1500}, doi = {10.1063/1.5079617}, pages = {6}, year = {2019}, abstract = {We develop a numerical approach to reconstruct the phase dynamics of driven or coupled self-sustained oscillators. Employing a simple algorithm for computation of the phase of a perturbed system, we construct numerically the equation for the evolution of the phase. Our simulations demonstrate that the description of the dynamics solely by phase variables can be valid for rather strong coupling strengths and large deviations from the limit cycle. Coupling functions depend crucially on the coupling and are generally non-decomposable in phase response and forcing terms. We also discuss the limitations of the approach. Published under license by AIP Publishing.}, language = {en} } @article{RosenblumPikovskij2018, author = {Rosenblum, Michael and Pikovskij, Arkadij}, title = {Efficient determination of synchronization domains from observations of asynchronous dynamics}, series = {Chaos : an interdisciplinary journal of nonlinear science}, volume = {28}, journal = {Chaos : an interdisciplinary journal of nonlinear science}, number = {10}, publisher = {American Institute of Physics}, address = {Melville}, issn = {1054-1500}, doi = {10.1063/1.5037012}, pages = {8}, year = {2018}, abstract = {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.}, language = {en} } @article{RosenblumPikovskij2004, author = {Rosenblum, Michael and Pikovskij, Arkadij}, title = {Delayed feedback control of collective synchrony : an approach to suppression of pathological brain rhythms}, issn = {1063-651X}, year = {2004}, abstract = {We suggest a method for suppression of synchrony in a globally coupled oscillator network, based on the time- delayed feedback via the mean field. Having in mind possible applications for suppression of pathological rhythms in neural ensembles, we present numerical results for different models of coupled bursting neurons. A theory is developed based on the consideration of the synchronization transition as a Hopf bifurcation}, language = {en} } @article{RosenblumPikovskij2004, author = {Rosenblum, Michael and Pikovskij, Arkadij}, title = {Controlling synchronization in an ensemble of globally coupled oscillators}, issn = {0031-9007}, year = {2004}, abstract = {We propose a technique to control coherent collective oscillations in ensembles of globally coupled units (self- sustained oscillators or maps). We demonstrate numerically and theoretically that a time delayed feedback in the mean field can, depending on the parameters, enhance or suppress the self-synchronization in the population. We discuss possible applications of the technique}, language = {en} } @article{RosenblumPikovskij2015, author = {Rosenblum, Michael and Pikovskij, Arkadij}, title = {Two types of quasiperiodic partial synchrony in oscillator ensembles}, series = {Physical review : E, Statistical, nonlinear and soft matter physics}, volume = {92}, journal = {Physical review : E, Statistical, nonlinear and soft matter physics}, number = {1}, publisher = {American Physical Society}, address = {College Park}, issn = {1539-3755}, doi = {10.1103/PhysRevE.92.012919}, pages = {8}, year = {2015}, abstract = {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.}, language = {en} } @article{RosenblumOsipovPikovskijetal.1997, author = {Rosenblum, Michael and Osipov, Grigory V. and Pikovskij, Arkadij and Kurths, J{\"u}rgen}, title = {Phase synchronization of chaotic oscillators by external driving}, year = {1997}, language = {en} } @article{RosenblumKurthsPikovskijetal.1998, author = {Rosenblum, Michael and Kurths, J{\"u}rgen and Pikovskij, Arkadij and Schafer, C. and Tass, Peter and Abel, Hans-Henning}, title = {Synchronization in Noisy Systems and Cardiorespiratory Interaction}, year = {1998}, language = {en} } @article{RosenblumKurthsPikovskij2001, author = {Rosenblum, Michael and Kurths, J{\"u}rgen and Pikovskij, Arkadij}, title = {Comment on "Phase synchronization in discrete chaotic systems"}, year = {2001}, abstract = {Chen et al. [Phys. Rev. E 61, 2559 (2000)] recently proposed an extension of the concept of phase for discrete chaotic systems. Using the newly introduced definition of phase they studied the dynamics of coupled map lattices and compared these dynamics with phase synchronization of coupled continuous-time chaotic systems. In this paper we illustrate by two simple counterexamples that the angle variable introduced by Chen et al. fails to satisfy the basic requirements to the proper phase. Furthermore, we argue that an extension of the notion of phase synchronization to generic discrete maps is doubtful.}, language = {en} } @article{RosenblumFruehwirthMoseretal.2019, author = {Rosenblum, Michael and Fr{\"u}hwirth, Martha and Moser, Maximilian and Pikovskij, Arkadij}, title = {Dynamical disentanglement in an analysis of oscillatory systems: an application to respiratory sinus arrhythmia}, series = {Philosophical Transactions of the Royal Society of London, Series A : Mathematical, Physical and Engineering Sciences}, volume = {377}, journal = {Philosophical Transactions of the Royal Society of London, Series A : Mathematical, Physical and Engineering Sciences}, number = {2160}, publisher = {Royal Society}, address = {London}, issn = {1364-503X}, doi = {10.1098/rsta.2019.0045}, pages = {14}, year = {2019}, abstract = {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'.}, language = {en} } @article{RosenauPikovskij2020, author = {Rosenau, Philip and Pikovskij, Arkadij}, title = {Solitary phase waves in a chain of autonomous oscillators}, series = {Chaos : an interdisciplinary journal of nonlinear science}, volume = {30}, journal = {Chaos : an interdisciplinary journal of nonlinear science}, number = {5}, publisher = {American Institute of Physics, AIP}, address = {Melville, NY}, issn = {1054-1500}, doi = {10.1063/1.5144939}, pages = {8}, year = {2020}, abstract = {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.}, language = {en} } @article{RosenauPikovskij2005, author = {Rosenau, Philip and Pikovskij, Arkadij}, title = {Phase compactons in chains of dispersively coupled oscillators}, issn = {0031-9007}, year = {2005}, abstract = {We study the phase dynamics of a chain of autonomous oscillators with a dispersive coupling. In the quasicontinuum limit the basic discrete model reduces to a Korteveg-de Vries-like equation, but with a nonlinear dispersion. The system supports compactons: solitary waves with a compact support and kovatons which are compact formations of glued together kink-antikink pairs that may assume an arbitrary width. These robust objects seem to collide elastically and, together with wave trains, are the building blocks of the dynamics for typical initial conditions. Numerical studies of the complex Ginzburg-Landau and Van der Pol lattices show that the presence of a nondispersive coupling does not affect kovatons, but causes a damping and deceleration or growth and acceleration of compactons}, language = {en} } @article{RosenauPikovskij2014, author = {Rosenau, Philip and Pikovskij, Arkadij}, title = {Breathers in strongly anharmonic lattices}, series = {Physical review : E, Statistical, nonlinear and soft matter physics}, volume = {89}, journal = {Physical review : E, Statistical, nonlinear and soft matter physics}, number = {2}, publisher = {American Physical Society}, address = {College Park}, issn = {1539-3755}, doi = {10.1103/PhysRevE.89.022924}, pages = {9}, year = {2014}, abstract = {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.}, language = {en} } @article{RosenauPikovskij2021, author = {Rosenau, Philip and Pikovskij, Arkadij}, title = {Waves in strongly nonlinear Gardner-like equations on a lattice}, series = {Nonlinearity / the Institute of Physics and the London Mathematical Society}, volume = {34}, journal = {Nonlinearity / the Institute of Physics and the London Mathematical Society}, number = {8}, publisher = {IOP Publ. Ltd.}, address = {Bristol}, issn = {0951-7715}, doi = {10.1088/1361-6544/ac0f51}, pages = {5872 -- 5896}, year = {2021}, abstract = {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.}, language = {en} } @article{PopovychMaistrenkoMosekildeetal.2001, author = {Popovych, Orest and Maistrenko, Yu and Mosekilde, Erik and Pikovskij, Arkadij and Kurths, J{\"u}rgen}, title = {Transcritical riddling in a system of coupled maps}, year = {2001}, abstract = {The transition from fully synchronized behavior to two-cluster dynamics is investigated for a system of N globally coupled chaotic oscillators by means of a model of two coupled logistic maps. An uneven distribution of oscillators between the two clusters causes an asymmetry to arise in the coupling of the model system. While the transverse period-doubling bifurcation remains essentially unaffected by this asymmetry, the transverse pitchfork bifurcation is turned into a saddle-node bifurcation followed by a transcritical riddling bifurcation in which a periodic orbit embedded in the synchronized chaotic state loses its transverse stability. We show that the transcritical riddling transition is always hard. For this, we study the sequence of bifurcations that the asynchronous point cycles produced in the saddle-node bifurcation undergo, and show how the manifolds of these cycles control the magnitude of asynchronous bursts. In the case where the system involves two subpopulations of oscillators with a small mismatch of the parameters, the transcritical riddling will be replaced by two subsequent saddle-node bifurcations, or the saddle cycle involved in the transverse destabilization of the synchronized chaotic state may smoothly shift away from the synchronization manifold. In this way, the transcritical riddling bifurcation is substituted by a symmetry-breaking bifurcation, which is accompanied by the destruction of a thin invariant region around the symmetrical chaotic state.}, language = {en} } @article{PopovychMaistrenkoMosekildeetal.2000, author = {Popovych, Orest and Maistrenko, Yu and Mosekilde, Erik and Pikovskij, Arkadij and Kurths, J{\"u}rgen}, title = {Transcritical loss of synchronization in coupled chaotic systems}, year = {2000}, language = {en} } @article{PopovychLysyanskyRosenblumetal.2017, author = {Popovych, Oleksandr V. and Lysyansky, Borys and Rosenblum, Michael and Pikovskij, Arkadij and Tass, Peter A.}, title = {Pulsatile desynchronizing delayed feedback for closed-loop deep brain stimulation}, series = {PLoS one}, volume = {12}, journal = {PLoS one}, publisher = {PLoS}, address = {San Fransisco}, issn = {1932-6203}, doi = {10.1371/journal.pone.0173363}, pages = {29}, year = {2017}, abstract = {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.}, language = {en} } @article{PollatosYeldesbayPikovskijetal.2014, author = {Pollatos, Olga and Yeldesbay, Azamat and Pikovskij, Arkadij and Rosenblum, Michael}, title = {How much time has passed? Ask your heart}, series = {Frontiers in neurorobotics}, volume = {8}, journal = {Frontiers in neurorobotics}, publisher = {Frontiers Research Foundation}, address = {Lausanne}, issn = {1662-5218}, doi = {10.3389/fnbot.2014.00015}, pages = {1 -- 9}, year = {2014}, abstract = {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.}, language = {en} } @article{PolitiPikovskijUllner2017, author = {Politi, Antonio and Pikovskij, Arkadij and Ullner, Ekkehard}, title = {Chaotic macroscopic phases in one-dimensional oscillators}, series = {European physical journal special topics}, volume = {226}, journal = {European physical journal special topics}, publisher = {Springer}, address = {Heidelberg}, issn = {1951-6355}, doi = {10.1140/epjst/e2017-70056-4}, pages = {1791 -- 1810}, year = {2017}, abstract = {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.}, language = {en} } @misc{PolitiPikovskijUllner2017, author = {Politi, Antonio and Pikovskij, Arkadij and Ullner, Ekkehard}, title = {Chaotic macroscopic phases in one-dimensional oscillators}, series = {Postprints der Universit{\"a}t Potsdam Mathematisch-Naturwissenschaftliche Reihe}, journal = {Postprints der Universit{\"a}t Potsdam Mathematisch-Naturwissenschaftliche Reihe}, number = {721}, issn = {1866-8372}, doi = {10.25932/publishup-42979}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-429790}, pages = {20}, year = {2017}, abstract = {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.}, language = {en} } @misc{PimenovaGoldobinRosenblumetal.2016, author = {Pimenova, Anastasiya V. and Goldobin, Denis S. and Rosenblum, Michael and Pikovskij, Arkadij}, title = {Interplay of coupling and common noise at the transition to synchrony in oscillator populations}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-103471}, pages = {7}, year = {2016}, abstract = {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.}, language = {en} } @article{PimenovaGoldobinRosenblumetal.2016, author = {Pimenova, Anastasiya V. and Goldobin, Denis S. and Rosenblum, Michael and Pikovskij, Arkadij}, title = {Interplay of coupling and common noise at the transition to synchrony in oscillator populations}, series = {Scientific reports}, volume = {6}, journal = {Scientific reports}, publisher = {Nature Publ. Group}, address = {London}, issn = {2045-2322}, doi = {10.1038/srep38518}, pages = {7}, year = {2016}, abstract = {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.}, language = {en} } @unpublished{PikovskijZaksFeudeletal.1995, author = {Pikovskij, Arkadij and Zaks, Michael A. and Feudel, Ulrike and Kurths, J{\"u}rgen}, title = {Singular continuous spectra in dissipative dynamics}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-13787}, year = {1995}, abstract = {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 Poincar{\´e} map have singular power spectra. The other mechanism owes to the logarithmic divergence of the first return times near the saddle point; here the Poincar{\´e} 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.}, language = {en} } @article{PikovskijZaikindelaCasa2002, author = {Pikovskij, Arkadij and Zaikin, Alexei A. and de la Casa, M. A.}, title = {System Size Resonance in Coupled Noisy Systems and in the Ising Model}, year = {2002}, abstract = {We consider an ensemble of coupled nonlinear noisy oscillators demonstrating in the thermodynamic limit an Ising-type transition. In the ordered phase and for finite ensembles stochastic flips of the mean field are observed with the rate depending on the ensemble size. When a small periodic force acts on the ensemble, the linear response of the system has a maximum at a certain system size, similar to the stochastic resonance phenomenon. We demonstrate this effect of system size resonance for different types of noisy oscillators and for different ensembles{\`u}lattices with nearest neighbors coupling and globally coupled populations. The Ising model is also shown to demonstrate the system size resonance.}, language = {en} } @article{PikovskijRosenblumZaksetal.1999, author = {Pikovskij, Arkadij and Rosenblum, Michael and Zaks, Michael A. and Kurths, J{\"u}rgen}, title = {Phase synchronization of regular and chaotic oscillators}, year = {1999}, language = {en} } @article{PikovskijRosenblumOsipovetal.1997, author = {Pikovskij, Arkadij and Rosenblum, Michael and Osipov, Grigory V. and Kurths, J{\"u}rgen}, title = {Phase synchronization effects in a lattice of nonidentical R{\"o}ssler oscillators}, year = {1997}, language = {en} } @article{PikovskijRosenblumKurths2000, author = {Pikovskij, Arkadij and Rosenblum, Michael and Kurths, J{\"u}rgen}, title = {Phase synchronization in regular and chaotic systems}, issn = {0218-1274}, year = {2000}, language = {en} } @book{PikovskijRosenblumKurths2001, author = {Pikovskij, Arkadij and Rosenblum, Michael and Kurths, J{\"u}rgen}, title = {Synchronization : a universal concept in nonlinear sciences}, series = {Cambridge nonlinear science series}, volume = {12}, journal = {Cambridge nonlinear science series}, edition = {1st paperback ed., repr}, publisher = {Cambridge Univ. Press}, address = {Cambridge}, isbn = {0-521-59285-2}, pages = {XIX, 411 S. : Ill., graph. Darst.}, year = {2001}, language = {en} } @article{PikovskijRosenblum2001, author = {Pikovskij, Arkadij and Rosenblum, Michael}, title = {Comment on "Intermittency in chaotic rotations"}, year = {2001}, abstract = {Lai et al. [Phys. Rev. E 62, R29 (2000)] claim that the angular velocity of the phase point moving along the chaotic trajectory in a properly chosen projection (the instantaneous frequency) is intermittent. Using the same examples, namely the R{\"o}ssler and the Lorenz systems, we show the absence of intermittency in the dynamics of the instantaneous frequency.This is confirmed by demonstrating that the phase dynamics exhibits normal diffusion. We argue that the nonintermittent behavior is generic.}, language = {en} } @article{PikovskijRosenblum2001, author = {Pikovskij, Arkadij and Rosenblum, Michael}, title = {Detecting direction of coupling in interacting oscillators}, year = {2001}, abstract = {We propose a method for experimental detection of directionality of weak coupling between two self-sustained oscillators from bivariate data. The technique is applicable to both noisy and chaotic systems that can be nonidentical or even structurally different. We introduce an index that quantifies the asymmetry in coupling.}, language = {en} } @article{PikovskijRosenblum2009, author = {Pikovskij, Arkadij and Rosenblum, Michael}, title = {Self-organized partially synchronous dynamics in populations of nonlinearly coupled oscillators}, issn = {0167-2789}, doi = {10.1016/j.physd.2008.08.018}, year = {2009}, abstract = {We analyze a minimal model of a population of identical oscillators with a nonlinear coupling-a generalization of the popular Kuramoto model. In addition to well-known for the Kuramoto model regimes of full synchrony, full asynchrony, and integrable neutral quasiperiodic states, ensembles of nonlinearly coupled oscillators demonstrate two novel nontrivial types of partially synchronized dynamics: self-organized bunch states and self-organized quasiperiodic dynamics. The analysis based on the Watanabe-Strogatz ansatz allows us to describe the self-organized bunch states in any finite ensemble as a set of equilibria, and the self-organized quasiperiodicity as a two-frequency quasiperiodic regime. An analytic solution in the thermodynamic limit of infinitely many oscillators is also discussed.}, language = {en} } @article{PikovskijRosenblum2011, author = {Pikovskij, Arkadij and Rosenblum, Michael}, title = {Dynamics of heterogeneous oscillator ensembles in terms of collective variables}, series = {Physica :D, Nonlinear phenomena}, volume = {240}, journal = {Physica :D, Nonlinear phenomena}, number = {9-10}, publisher = {Elsevier}, address = {Amsterdam}, issn = {0167-2789}, doi = {10.1016/j.physd.2011.01.002}, pages = {872 -- 881}, year = {2011}, abstract = {We consider general heterogeneous ensembles of phase oscillators, sine coupled to arbitrary external fields. Starting with the infinitely large ensembles, we extend the Watanabe-Strogatz theory, valid for identical oscillators, to cover the case of an arbitrary parameter distribution. The obtained equations yield the description of the ensemble dynamics in terms of collective variables and constants of motion. As a particular case of the general setup we consider hierarchically organized ensembles, consisting of a finite number of subpopulations, whereas the number of elements in a subpopulation can be both finite or infinite. Next, we link the Watanabe-Strogatz and Ott-Antonsen theories and demonstrate that the latter one corresponds to a particular choice of constants of motion. The approach is applied to the standard Kuramoto-Sakaguchi model, to its extension for the case of nonlinear coupling, and to the description of two interacting subpopulations, exhibiting a chimera state. With these examples we illustrate that, although the asymptotic dynamics can be found within the framework of the Ott-Antonsen theory, the transients depend on the constants of motion. The most dramatic effect is the dependence of the basins of attraction of different synchronous regimes on the initial configuration of phases.}, language = {en} } @article{PikovskijRosenblum2015, author = {Pikovskij, Arkadij and Rosenblum, Michael}, title = {Dynamics of globally coupled oscillators: Progress and perspectives}, series = {Chaos : an interdisciplinary journal of nonlinear science}, volume = {25}, journal = {Chaos : an interdisciplinary journal of nonlinear science}, number = {9}, publisher = {American Institute of Physics}, address = {Melville}, issn = {1054-1500}, doi = {10.1063/1.4922971}, pages = {11}, year = {2015}, abstract = {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.}, language = {en} } @article{PikovskijRosenau2006, author = {Pikovskij, Arkadij and Rosenau, Philip}, title = {Phase compactons}, doi = {10.1016/j.physd.2006.04.015}, year = {2006}, abstract = {We study the phase dynamics of a chain of autonomous, self-sustained, dispersively coupled oscillators. In the quasicontinuum limit the basic discrete model reduces to a Korteveg-de Vries-like equation, but with a nonlinear dispersion. The system supports compactons - solitary waves with a compact support - and kovatons - compact formations of glued together kink-antikink pairs that propagate with a unique speed, but may assume an arbitrary width. We demonstrate that lattice solitary waves, though not exactly compact, have tails which decay at a superexponential rate. They are robust and collide nearly elastically and together with wave sources are the building blocks of the dynamics that emerges from typical initial conditions. In finite lattices, after a long time, the dynamics becomes chaotic. Numerical studies of the complex Ginzburg-Landau lattice show that the non-dispersive coupling causes a damping and deceleration, or growth and acceleration, of compactons. A simple perturbation method is applied to study these effects. (c) 2006 Elsevier B.V. All rights reserved}, language = {en} } @article{PikovskijPopovychMaistrenko2001, author = {Pikovskij, Arkadij and Popovych, Orest and Maistrenko, Yu}, title = {Resolving Clusters in Chaotic Ensembles of Globally Coupled Identical Oscillators}, year = {2001}, abstract = {Clustering in ensembles of globally coupled identical chaotic oscillators is reconsidered using a twofold approach. Stability of clusters towards "emanation" of the elements is described with the evaporation Lyapunov exponents. It appears that direct numerical simulations of ensembles often lead to spurious clusters that have positive evaporation exponents, due to a numerical trap. We propose a numerical method that surmounts the spurious clustering. We also demonstrate that clustering can be very sensitive to the number of elements in the ensemble.}, language = {en} } @book{PikovskijPoliti2016, author = {Pikovskij, Arkadij and Politi, Antonio}, title = {Lyapunov Exponents}, publisher = {Cambridge University Press}, address = {Cambridge}, isbn = {978-1-107-03042-8}, publisher = {Universit{\"a}t Potsdam}, pages = {XII, 285}, year = {2016}, abstract = {Lyapunov exponents lie at the heart of chaos theory, and are widely used in studies of complex dynamics. Utilising a pragmatic, physical approach, this self-contained book provides a comprehensive description of the concept. Beginning with the basic properties and numerical methods, it then guides readers through to the most recent advances in applications to complex systems. Practical algorithms are thoroughly reviewed and their performance is discussed, while a broad set of examples illustrate the wide range of potential applications. The description of various numerical and analytical techniques for the computation of Lyapunov exponents offers an extensive array of tools for the characterization of phenomena such as synchronization, weak and global chaos in low and high-dimensional set-ups, and localization. This text equips readers with all the investigative expertise needed to fully explore the dynamical properties of complex systems, making it ideal for both graduate students and experienced researchers.}, language = {en} } @article{PikovskijPoliti2001, author = {Pikovskij, Arkadij and Politi, Antonio}, title = {Dynamic localization of Lyapunov vectors in Hamiltonian lattices}, year = {2001}, language = {en} } @article{PikovskijPoliti1998, author = {Pikovskij, Arkadij and Politi, Antonio}, title = {Dynamic localization of Lyapunov vectors in space-time chaos}, year = {1998}, abstract = {We study the dynamics of Lyapunov vectors in various models of one-dimensional distributed systems with spacetime chaos. We demonstrate that the vector corresponding to the maximum exponent is always localized and the localization region wanders irregularly. This localization is explained by interpreting the logarithm of the Lyapunov vector as a roughening interface. We show that for many systems, the `interface' belongs to the Kardar-Parisi- Zhang universality class. Accordingly, we discuss the scaling behaviour of finite-size effects and self-averaging properties of the Lyapunov exponents.}, language = {en} } @article{PikovskijNeumann2001, author = {Pikovskij, Arkadij and Neumann, Eireen}, title = {Comment on "Simple approach to the creation of a strange nonchaotic attractor in any chaotic system"}, year = {2001}, abstract = {We address the problem of existence of strange nonchaotic attractors (SNAs) in quasiperiodically forced dynamical systems. Recently, Shuai and Wong [Phys. Rev. E 59, 5338 (1999)] suggested a universal method for constructing a SNA in an arbitrary system possessing chaos. We demonstrate here that, in general, this method fails. For arbitrary systems, it gives a SNA only in a vicinity of transition to chaos. We discuss also a special example, where the method by Shuai and Wong indeed produces a SNA.}, language = {en} } @article{PikovskijKurths1997, author = {Pikovskij, Arkadij and Kurths, J{\"u}rgen}, title = {Coherence Resonance in a Noise-Driven Excitable System}, year = {1997}, abstract = {We study the dynamics of the excitable Fitz Hugh-Nagumo system under external noisy driving. Noise activates the system producing a sequence of pulses. The coherence of these noise-induced oscillations is shown to be maximal for a certain noise amplitude. This new effect of coherence resonance is explained by different noise dependencies of the activation and the excursion times. A simple one-dimensional model based on the Langevin dynamics is proposed for the quantitative description of this phenomenon.}, language = {en} } @article{PikovskijKurths1997, author = {Pikovskij, Arkadij and Kurths, J{\"u}rgen}, title = {Coherence resonance in a noise-driven excitable system}, year = {1997}, language = {en} } @article{PikovskijGuptaTelesetal.2014, author = {Pikovskij, Arkadij and Gupta, Shamik and Teles, Tarcisio N. and Benetti, Fernanda P. C. and Pakter, Renato and Levin, Yan and Ruffo, Stefano}, title = {Ensemble inequivalence in a mean-field XY model with ferromagnetic and nematic couplings}, series = {Physical review : E, Statistical, nonlinear and soft matter physics}, volume = {90}, journal = {Physical review : E, Statistical, nonlinear and soft matter physics}, number = {6}, publisher = {American Physical Society}, address = {College Park}, issn = {1539-3755}, doi = {10.1103/PhysRevE.90.062141}, pages = {5}, year = {2014}, abstract = {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.}, language = {en} } @article{PikovskijFishman2011, author = {Pikovskij, Arkadij and Fishman, Shmuel}, title = {Scaling properties of weak chaos in nonlinear disordered lattices}, series = {Physical review : E, Statistical, nonlinear and soft matter physics}, volume = {83}, journal = {Physical review : E, Statistical, nonlinear and soft matter physics}, number = {2}, publisher = {American Physical Society}, address = {College Park}, issn = {1539-3755}, doi = {10.1103/PhysRevE.83.025201}, pages = {4}, year = {2011}, abstract = {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.}, language = {en} } @unpublished{PikovskijFeudel1994, author = {Pikovskij, Arkadij and Feudel, Ulrike}, title = {Characterizing strange nonchaotic attractors}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-13405}, year = {1994}, abstract = {Strange nonchaotic attractors typically appear in quasiperiodically driven nonlinear systems. Two methods of their characterization are proposed. The first one is based on the bifurcation analysis of the systems, resulting from periodic approximations of the quasiperiodic forcing. Secondly, we propose th characterize their strangeness by calculating a phase sensitivity exponent, that measures the sensitivity with respect to changes of the phase of the external force. It is shown, that phase sensitivity appears if there is a non-zero probability for positive local Lyapunov exponents to occur.}, language = {en} }