@article{AbelFlachPikovskij1998, author = {Abel, Markus and Flach, S. and Pikovskij, Arkadij}, title = {Localisation in a coupled standard map lattice}, year = {1998}, abstract = {We study spatially localized excitations in a lattice of coupled standard maps. Time-periodic solutions (breathers) exist in a range of coupling that is shown to shrink as the period grows to infinity, thus excluding the possibility of time-quasiperiodic breathers. The evolution of initially localized chaotic and quasiperiodic states in a lattice is studied numerically. Chaos is demonstrated to spread}, language = {en} } @article{AbelFlachPikovskij1998, author = {Abel, Markus and Flach, S. and Pikovskij, Arkadij}, title = {Localization in a coupled standard map lattice}, year = {1998}, language = {en} } @article{AbelPikovskij1997, author = {Abel, Markus and Pikovskij, Arkadij}, title = {Parametric excitation of breathers in a nonlinear lattice}, year = {1997}, abstract = {We investigate localized periodic solutions (breathers) in a lattice of parametrically driven, nonlinear dissipative oscillators. These breathers are demonstrated to be exponentially localized, with two characteristic localization lengths. The crossover between the two lengths is shown to be related to the transition in the phase of the lattice oscillations.}, language = {en} } @article{AhlersPikovskij2002, author = {Ahlers, Volker and Pikovskij, Arkadij}, title = {Critical Properties of the Synchronization Transition in Space-Time Chaos}, year = {2002}, abstract = {We study two coupled spatially extended dynamical systems which exhibit space-time chaos. The transition to the synchronized state is treated as a nonequilibrium phase transition, where the average synchronization error is the order parameter. The transition in one-dimensional systems is found to be generically in the universality class of the Kardar- Parisi-Zhang equation with a growth-limiting term ("bounded KPZ"). For systems with very strong nonlinearities in the local dynamics, however, the transition is found to be in the universality class of directed percolation.}, language = {en} } @article{AhlersZillmerPikovskij2001, author = {Ahlers, Volker and Zillmer, R{\"u}diger and Pikovskij, Arkadij}, title = {Lyapunov exponents in disordered chaotic systems : avoided crossing and level statistics}, year = {2001}, abstract = {The behavior of the Lyapunov exponents (LEs) of a disordered system consisting of mutually coupled chaotic maps with different parameters is studied. The LEs are demonstrated to exhibit avoided crossing and level repulsion, qualitatively similar to the behavior of energy levels in quantum chaos. Recent results for the coupling dependence of the LEs of two coupled chaotic systems are used to explain the phenomenon and to derive an approximate expression for the distribution functions of LE spacings. The depletion of the level spacing distribution is shown to be exponentially strong at small values. The results are interpreted in terms of the random matrix theory.}, language = {en} } @article{AhlersZillmerPikovskij2000, author = {Ahlers, Volker and Zillmer, R{\"u}diger and Pikovskij, Arkadij}, title = {Statistical theory for the coupling sensitivity of chaos}, isbn = {1-563-96915-7}, year = {2000}, language = {en} } @article{AhnertPikovskij2009, author = {Ahnert, Karsten and Pikovskij, Arkadij}, title = {Compactons and chaos in strongly nonlinear lattices}, issn = {1539-3755}, doi = {10.1103/Physreve.79.026209}, year = {2009}, abstract = {We study localized traveling waves and chaotic states in strongly nonlinear one-dimensional Hamiltonian lattices. We show that the solitary waves are superexponentially localized and present an accurate numerical method allowing one to find them for an arbitrary nonlinearity index. Compactons evolve from rather general initially localized perturbations and collide nearly elastically. Nevertheless, on a long time scale for finite lattices an extensive chaotic state is generally observed. Because of the system's scaling, these dynamical properties are valid for any energy.}, language = {en} } @article{AransonPikovskij2022, author = {Aranson, Igor S. and Pikovskij, Arkadij}, title = {Confinement and collective escape of active particles}, series = {Physical review letters}, volume = {128}, journal = {Physical review letters}, number = {10}, publisher = {American Physical Society}, address = {College Park, Md.}, issn = {0031-9007}, doi = {10.1103/PhysRevLett.128.108001}, pages = {6}, year = {2022}, abstract = {Active matter broadly covers the dynamics of self-propelled particles. While the onset of collective behavior in homogenous active systems is relatively well understood, the effect of inhomogeneities such as obstacles and traps lacks overall clarity. Here, we study how interacting, self-propelled particles become trapped and released from a trap. We have found that captured particles aggregate into an orbiting condensate with a crystalline structure. As more particles are added, the trapped condensates escape as a whole. Our results shed light on the effects of confinement and quenched disorder in active matter.}, language = {en} } @article{BaibolatovRosenblumZhanabaevetal.2009, author = {Baibolatov, Yernur and Rosenblum, Michael and Zhanabaev, Zeinulla Zh. and Kyzgarina, Meyramgul and Pikovskij, Arkadij}, title = {Periodically forced ensemble of nonlinearly coupled oscillators : from partial to full synchrony}, issn = {1539-3755}, doi = {10.1103/PhysRevE.80.046211}, year = {2009}, abstract = {We analyze the dynamics of a periodically forced oscillator ensemble with global nonlinear coupling. Without forcing, the system exhibits complicated collective dynamics, even for the simplest case of identical phase oscillators: due to nonlinearity, the synchronous state becomes unstable for certain values of the coupling parameter, and the system settles at the border between synchrony and asynchrony, what can be denoted as partial synchrony. We find that an external common forcing can result in two synchronous states: (i) a weak forcing entrains only the mean field, whereas the individual oscillators remain unlocked to the force and, correspondingly, to the mean field; (ii) a strong forcing fully synchronizes the system, making the phases of all oscillators identical. Analytical results are confirmed by numerics.}, language = {en} } @article{BaibolatovRosenblumZhanabaevetal.2010, author = {Baibolatov, Yernur and Rosenblum, Michael and Zhanabaev, Zeinulla Zh. and Pikovskij, Arkadij}, title = {Complex dynamics of an oscillator ensemble with uniformly distributed natural frequencies and global nonlinear coupling}, issn = {1539-3755}, doi = {10.1103/Physreve.82.016212}, year = {2010}, abstract = {We consider large populations of phase oscillators with global nonlinear coupling. For identical oscillators such populations are known to demonstrate a transition from completely synchronized state to the state of self-organized quasiperiodicity. In this state phases of all units differ, yet the population is not completely incoherent but produces a nonzero mean field; the frequency of the latter differs from the frequency of individual units. Here we analyze the dynamics of such populations in case of uniformly distributed natural frequencies. We demonstrate numerically and describe theoretically (i) states of complete synchrony, (ii) regimes with coexistence of a synchronous cluster and a drifting subpopulation, and (iii) self-organized quasiperiodic states with nonzero mean field and all oscillators drifting with respect to it. We analyze transitions between different states with the increase of the coupling strength; in particular we show that the mean field arises via a discontinuous transition. For a further illustration we compare the results for the nonlinear model with those for the Kuramoto-Sakaguchi model.}, language = {en} } @article{BlahaPikovskijRosenblumetal.2011, author = {Blaha, Karen A. and Pikovskij, Arkadij and Rosenblum, Michael and Clark, Matthew T. and Rusin, Craig G. and Hudson, John L.}, title = {Reconstruction of two-dimensional phase dynamics from experiments on coupled oscillators}, series = {Physical review : E, Statistical, nonlinear and soft matter physics}, volume = {84}, journal = {Physical review : E, Statistical, nonlinear and soft matter physics}, number = {4}, publisher = {American Physical Society}, address = {College Park}, issn = {1539-3755}, doi = {10.1103/PhysRevE.84.046201}, pages = {7}, year = {2011}, abstract = {Phase models are a powerful method to quantify the coupled dynamics of nonlinear oscillators from measured data. We use two phase modeling methods to quantify the dynamics of pairs of coupled electrochemical oscillators, based on the phases of the two oscillators independently and the phase difference, respectively. We discuss the benefits of the two-dimensional approach relative to the one-dimensional approach using phase difference. We quantify the dependence of the coupling functions on the coupling magnitude and coupling time delay. We show differences in synchronization predictions of the two models using a toy model. We show that the two-dimensional approach reveals behavior not detected by the one-dimensional model in a driven experimental oscillator. This approach is broadly applicable to quantify interactions between nonlinear oscillators, especially where intrinsic oscillator sensitivity and coupling evolve with time.}, language = {en} } @article{BolotovBolotovSmirnovetal.2019, author = {Bolotov, Dmitry and Bolotov, Maxim I. and Smirnov, Lev A. and Osipov, Grigory V. and Pikovskij, Arkadij}, title = {Twisted States in a System of Nonlinearly Coupled Phase Oscillators}, series = {Regular and chaotic dynamics : international scientific journal}, volume = {24}, journal = {Regular and chaotic dynamics : international scientific journal}, number = {6}, publisher = {Pleiades publishing inc}, address = {Moscow}, issn = {1560-3547}, doi = {10.1134/S1560354719060091}, pages = {717 -- 724}, year = {2019}, abstract = {We study the dynamics of the ring of identical phase oscillators with nonlinear nonlocal coupling. Using the Ott - Antonsen approach, the problem is formulated as a system of partial derivative equations for the local complex order parameter. In this framework, we investigate the existence and stability of twisted states. Both fully coherent and partially coherent stable twisted states were found (the latter ones for the first time for identical oscillators). We show that twisted states can be stable starting from a certain critical value of the medium length, or on a length segment. The analytical results are confirmed with direct numerical simulations in finite ensembles.}, language = {en} } @article{BolotovOsipovPikovskij2016, author = {Bolotov, M. I. and Osipov, Grigory V. and Pikovskij, Arkadij}, title = {Marginal chimera state at cross-frequency locking of pulse-coupled neural networks}, series = {Physical review : E, Statistical, nonlinear and soft matter physics}, volume = {93}, journal = {Physical review : E, Statistical, nonlinear and soft matter physics}, publisher = {American Physical Society}, address = {College Park}, issn = {2470-0045}, doi = {10.1103/PhysRevE.93.032202}, pages = {6}, year = {2016}, abstract = {We consider two coupled populations of leaky integrate-and-fire neurons. Depending on the coupling strength, mean fields generated by these populations can have incommensurate frequencies or become frequency locked. In the observed 2:1 locking state of the mean fields, individual neurons in one population are asynchronous with the mean fields, while in another population they have the same frequency as the mean field. These synchronous neurons form a chimera state, where part of them build a fully synchronized cluster, while other remain scattered. We explain this chimera as a marginal one, caused by a self-organized neutral dynamics of the effective circle map.}, language = {en} } @article{BolotovSmirnovBubnovaetal.2021, author = {Bolotov, Maxim I. and Smirnov, Lev A. and Bubnova, E. S. and Osipov, Grigory V. and Pikovskij, Arkadij}, title = {Spatiotemporal regimes in the Kuramoto-Battogtokh system of nonidentical oscillators}, series = {Journal of experimental and theoretical physics}, volume = {132}, journal = {Journal of experimental and theoretical physics}, number = {1}, publisher = {Springer}, address = {Heidelberg [u.a.]}, issn = {1063-7761}, doi = {10.1134/S1063776121010106}, pages = {127 -- 147}, year = {2021}, abstract = {We consider the spatiotemporal states of an ensemble of nonlocally coupled nonidentical phase oscillators, which correspond to different regimes of the long-term evolution of such a system. We have obtained homogeneous, twisted, and nonhomogeneous stationary solutions to the Ott-Antonsen equations corresponding to key variants of the realized collective rotational motion of elements of the medium in question with nonzero mesoscopic characteristics determining the degree of coherence of the dynamics of neighboring particles. We have described the procedures of the search for the class of nonhomogeneous solutions as stationary points of the auxiliary point map and of determining the stability based on analysis of the eigenvalue spectrum of the composite operator. Static and breather cluster regimes have been demonstrated and described, as well as the regimes with an irregular behavior of averaged complex fields including, in particular, the local order parameter.}, language = {en} } @article{BolotovSmirnovOsipovetal.2018, author = {Bolotov, Maxim I. and Smirnov, Lev A. and Osipov, Grigory V. and Pikovskij, Arkadij}, title = {Simple and complex chimera states in a nonlinearly coupled oscillatory medium}, series = {Chaos : an interdisciplinary journal of nonlinear science}, volume = {28}, journal = {Chaos : an interdisciplinary journal of nonlinear science}, number = {4}, publisher = {American Institute of Physics}, address = {Melville}, issn = {1054-1500}, doi = {10.1063/1.5011678}, pages = {9}, year = {2018}, abstract = {We consider chimera states in a one-dimensional medium of nonlinear nonlocally coupled phase oscillators. In terms of a local coarse-grained complex order parameter, the problem of finding stationary rotating nonhomogeneous solutions reduces to a third-order ordinary differential equation. This allows finding chimera-type and other inhomogeneous states as periodic orbits of this equation. Stability calculations reveal that only some of these states are stable. We demonstrate that an oscillatory instability leads to a breathing chimera, for which the synchronous domain splits into subdomains with different mean frequencies. Further development of instability leads to turbulent chimeras. Published by AIP Publishing.}, language = {en} } @article{BolotovSmirnovOsipovetal.2017, author = {Bolotov, Maxim I. and Smirnov, Lev A. and Osipov, Grigory V. and Pikovskij, Arkadij}, title = {Breathing chimera in a system of phase oscillators}, series = {JETP Letters}, volume = {106}, journal = {JETP Letters}, publisher = {Pleiades Publ.}, address = {New York}, issn = {0021-3640}, doi = {10.1134/S0021364017180059}, pages = {393 -- 399}, year = {2017}, abstract = {Chimera states consisting of synchronous and asynchronous domains in a medium of nonlinearly coupled phase oscillators have been considered. Stationary inhomogeneous solutions of the Ott-Antonsen equation for a complex order parameter that correspond to fundamental chimeras have been constructed. The direct numerical simulation has shown that these structures under certain conditions are transformed to oscillatory (breathing) chimera regimes because of the development of instability.}, language = {en} } @article{BordyugovPikovskijRosenblum2010, author = {Bordyugov, Grigory and Pikovskij, Arkadij and Rosenblum, Michael}, title = {Self-emerging and turbulent chimeras in oscillator chains}, issn = {1539-3755}, doi = {10.1103/Physreve.82.035205}, year = {2010}, abstract = {We report on a self-emerging chimera state in a homogeneous chain of nonlocally and nonlinearly coupled oscillators. This chimera, i.e., a state with coexisting regions of complete and partial synchrony, emerges via a supercritical bifurcation from a homogeneous state. We develop a theory of chimera based on the Ott-Antonsen equations for the local complex order parameter. Applying a numerical linear stability analysis, we also describe the instability of the chimera and transition to phase turbulence with persistent patches of synchrony.}, language = {en} } @article{BraunPikovskijMatiasetal.2012, author = {Braun, W. and Pikovskij, Arkadij and Matias, M. A. and Colet, P.}, title = {Global dynamics of oscillator populations under common noise}, series = {epl : a letters journal exploring the frontiers of physics}, volume = {99}, journal = {epl : a letters journal exploring the frontiers of physics}, number = {2}, publisher = {EDP Sciences}, address = {Mulhouse}, issn = {0295-5075}, doi = {10.1209/0295-5075/99/20006}, pages = {6}, year = {2012}, abstract = {Common noise acting on a population of identical oscillators can synchronize them. We develop a description of this process which is not limited to the states close to synchrony, but provides a global picture of the evolution of the ensembles. The theory is based on the Watanabe-Strogatz transformation, allowing us to obtain closed stochastic equations for the global variables. We show that at the initial stage, the order parameter grows linearly in time, while at the later stages the convergence to synchrony is exponentially fast. Furthermore, we extend the theory to nonidentical ensembles with the Lorentzian distribution of natural frequencies and determine the stationary values of the order parameter in dependence on driving noise and mismatch.}, language = {en} } @article{BurylkoPikovskij2011, author = {Burylko, Oleksandr and Pikovskij, Arkadij}, title = {Desynchronization transitions in nonlinearly coupled phase oscillators}, series = {Physica :D, Nonlinear phenomena}, volume = {240}, journal = {Physica :D, Nonlinear phenomena}, number = {17}, publisher = {Elsevier}, address = {Amsterdam}, issn = {0167-2789}, doi = {10.1016/j.physd.2011.05.016}, pages = {1352 -- 1361}, year = {2011}, abstract = {We consider the nonlinear extension of the Kuramoto model of globally coupled phase oscillators where the phase shift in the coupling function depends on the order parameter. A bifurcation analysis of the transition from fully synchronous state to partial synchrony is performed. We demonstrate that for small ensembles it is typically mediated by stable cluster states, that disappear with creation of heteroclinic cycles, while for a larger number of oscillators a direct transition from full synchrony to a periodic or a quasiperiodic regime occurs.}, language = {en} } @article{CestnikPikovskij2022, author = {Cestnik, Rok and Pikovskij, Arkadij}, title = {Exact finite-dimensional reduction for a population of noisy oscillators and its link to Ott-Antonsen and Watanabe-Strogatz theories}, series = {Chaos : an interdisciplinary journal of nonlinear science}, volume = {32}, journal = {Chaos : an interdisciplinary journal of nonlinear science}, number = {11}, publisher = {AIP}, address = {Melville}, issn = {1054-1500}, doi = {10.1063/5.0106171}, pages = {15}, year = {2022}, abstract = {Populations of globally coupled phase oscillators are described in the thermodynamic limit by kinetic equations for the distribution densities or, equivalently, by infinite hierarchies of equations for the order parameters. Ott and Antonsen [Chaos 18, 037113 (2008)] have found an invariant finite-dimensional subspace on which the dynamics is described by one complex variable per population. For oscillators with Cauchy distributed frequencies or for those driven by Cauchy white noise, this subspace is weakly stable and, thus, describes the asymptotic dynamics. Here, we report on an exact finite-dimensional reduction of the dynamics outside of the Ott-Antonsen subspace. We show that the evolution from generic initial states can be reduced to that of three complex variables, plus a constant function. For identical noise-free oscillators, this reduction corresponds to the Watanabe-Strogatz system of equations [Watanabe and Strogatz, Phys. Rev. Lett. 70, 2391 (1993)]. We discuss how the reduced system can be used to explore the transient dynamics of perturbed ensembles. Published under an exclusive license by AIP Publishing.}, language = {en} } @article{ChatePikovskijRudzick1999, author = {Chat{\´e}, Hugues and Pikovskij, Arkadij and Rudzick, Oliver}, title = {Forcing oscillatory media : phase kinks vs. synchronization}, year = {1999}, language = {en} } @article{ChigarevKazakovPikovskij2021, author = {Chigarev, Vladimir and Kazakov, Alexey and Pikovskij, Arkadij}, title = {Mutual singularities of overlapping attractor and repeller}, series = {Chaos : an interdisciplinary journal of nonlinear science}, volume = {31}, journal = {Chaos : an interdisciplinary journal of nonlinear science}, number = {8}, publisher = {American Institute of Physics}, address = {Melville}, issn = {1054-1500}, doi = {10.1063/5.0056891}, pages = {10}, year = {2021}, abstract = {We apply the concepts of relative dimensions and mutual singularities to characterize the fractal properties of overlapping attractor and repeller in chaotic dynamical systems. We consider one analytically solvable example (a generalized baker's map); two other examples, the Anosov-Mobius and the Chirikov-Mobius maps, which possess fractal attractor and repeller on a two-dimensional torus, are explored numerically. We demonstrate that although for these maps the stable and unstable directions are not orthogonal to each other, the relative Renyi and Kullback-Leibler dimensions as well as the mutual singularity spectra for the attractor and repeller can be well approximated under orthogonality assumption of two fractals.}, language = {en} } @article{CimponeriuRosenblumPikovskij2004, author = {Cimponeriu, Laura and Rosenblum, Michael and Pikovskij, Arkadij}, title = {Estimation of delay in coupling from time series}, issn = {1063-651X}, year = {2004}, abstract = {We demonstrate that a tune delay in weak coupling between two self-sustained oscillators can be estimated from the observed time series data. We present two methods which are. based on the analysis of interrelations between the phases of the signals. We show analytically and numerically that irregularity of the phase dynamics (due to the intrinsic noise or chaos) is essential for determination,of the delay. We compare and contrast both methods to the standard cross-correlation analysis}, language = {en} } @article{DolmatovaGoldobinPikovskij2017, author = {Dolmatova, Anastasiya V. and Goldobin, Denis S. and Pikovskij, Arkadij}, title = {Synchronization of coupled active rotators by common noise}, series = {Physical review : E, Statistical, nonlinear and soft matter physics}, volume = {96}, journal = {Physical review : E, Statistical, nonlinear and soft matter physics}, publisher = {American Physical Society}, address = {College Park}, issn = {2470-0045}, doi = {10.1103/PhysRevE.96.062204}, pages = {E10648 -- E10657}, year = {2017}, abstract = {We study the effect of common noise on coupled active rotators. While such a noise always facilitates synchrony, coupling may be attractive (synchronizing) or repulsive (desynchronizing). We develop an analytical approach based on a transformation to approximate angle-action variables and averaging over fast rotations. For identical rotators, we describe a transition from full to partial synchrony at a critical value of repulsive coupling. For nonidentical rotators, the most nontrivial effect occurs at moderate repulsive coupling, where a juxtaposition of phase locking with frequency repulsion (anti-entrainment) is observed. We show that the frequency repulsion obeys a nontrivial power law.}, language = {en} } @article{EhrichPikovskijRosenblum2013, author = {Ehrich, Sebastian and Pikovskij, Arkadij and Rosenblum, Michael}, title = {From complete to modulated synchrony in networks of identical Hindmarsh-Rose neurons}, series = {European physical journal special topics}, volume = {222}, journal = {European physical journal special topics}, number = {10}, publisher = {Springer}, address = {Heidelberg}, issn = {1951-6355}, doi = {10.1140/epjst/e2013-02025-8}, pages = {2407 -- 2416}, year = {2013}, abstract = {In most cases tendency to synchrony in networks of oscillatory units increases with the coupling strength. Using the popular Hindmarsh-Rose neuronal model, we demonstrate that even for identical neurons and simple coupling the dynamics can be more complicated. Our numerical analysis for globally coupled systems and oscillator lattices reveals a new scenario of synchrony breaking with the increase of coupling, resulting in a quasiperiodic, modulated synchronous state.}, language = {en} } @article{EngbertMergenthalerSinnetal.2011, author = {Engbert, Ralf and Mergenthaler, Konstantin and Sinn, Petra and Pikovskij, Arkadij}, title = {An integrated model of fixational eye movements and microsaccades}, series = {Proceedings of the National Academy of Sciences of the United States of America}, volume = {108}, journal = {Proceedings of the National Academy of Sciences of the United States of America}, number = {39}, publisher = {National Acad. of Sciences}, address = {Washington}, issn = {0027-8424}, doi = {10.1073/pnas.1102730108}, pages = {E765 -- E770}, year = {2011}, abstract = {When we fixate a stationary target, our eyes generate miniature (or fixational) eye movements involuntarily. These fixational eye movements are classified as slow components (physiological drift, tremor) and microsaccades, which represent rapid, small-amplitude movements. Here we propose an integrated mathematical model for the generation of slow fixational eye movements and microsaccades. The model is based on the concept of self-avoiding random walks in a potential, a process driven by a self-generated activation field. The self-avoiding walk generates persistent movements on a short timescale, whereas, on a longer timescale, the potential produces antipersistent motions that keep the eye close to an intended fixation position. We introduce microsaccades as fast movements triggered by critical activation values. As a consequence, both slow movements and microsaccades follow the same law of motion; i.e., movements are driven by the self-generated activation field. Thus, the model contributes a unified explanation of why it has been a long-standing problem to separate slow movements and microsaccades with respect to their motion-generating principles. We conclude that the concept of a self-avoiding random walk captures fundamental properties of fixational eye movements and provides a coherent theoretical framework for two physiologically distinct movement types.}, language = {en} } @article{FreitasMacauPikovskij2015, author = {Freitas, Celso and Macau, Elbert and Pikovskij, Arkadij}, title = {Partial synchronization in networks of non-linearly coupled oscillators: The Deserter Hubs Model}, series = {Chaos : an interdisciplinary journal of nonlinear science}, volume = {25}, journal = {Chaos : an interdisciplinary journal of nonlinear science}, number = {4}, publisher = {American Institute of Physics}, address = {Melville}, issn = {1054-1500}, doi = {10.1063/1.4919246}, pages = {8}, year = {2015}, abstract = {We study the Deserter Hubs Model: a Kuramoto-like model of coupled identical phase oscillators on a network, where attractive and repulsive couplings are balanced dynamically due to nonlinearity of interactions. Under weak force, an oscillator tends to follow the phase of its neighbors, but if an oscillator is compelled to follow its peers by a sufficient large number of cohesive neighbors, then it actually starts to act in the opposite manner, i.e., in anti-phase with the majority. Analytic results yield that if the repulsion parameter is small enough in comparison with the degree of the maximum hub, then the full synchronization state is locally stable. Numerical experiments are performed to explore the model beyond this threshold, where the overall cohesion is lost. We report in detail partially synchronous dynamical regimes, like stationary phase-locking, multistability, periodic and chaotic states. Via statistical analysis of different network organizations like tree, scale-free, and random ones, we found a measure allowing one to predict relative abundance of partially synchronous stationary states in comparison to time-dependent ones. (C) 2015 AIP Publishing LLC.}, language = {en} } @article{GengelPikovskij2019, author = {Gengel, Erik and Pikovskij, Arkadij}, title = {Phase demodulation with iterative Hilbert transform embeddings}, series = {Signal processing}, volume = {165}, journal = {Signal processing}, publisher = {Elsevier}, address = {Amsterdam}, issn = {0165-1684}, doi = {10.1016/j.sigpro.2019.07.005}, pages = {115 -- 127}, year = {2019}, abstract = {We propose an efficient method for demodulation of phase modulated signals via iterated Hilbert transform embeddings. We show that while a usual approach based on one application of the Hilbert transform provides only an approximation to a proper phase, with iterations the accuracy is essentially improved, up to precision limited mainly by discretization effects. We demonstrate that the method is applicable to arbitrarily complex waveforms, and to modulations fast compared to the basic frequency. Furthermore, we develop a perturbative theory applicable to a simple cosine waveform, showing convergence of the technique.}, language = {en} } @article{GengelPikovskij2022, author = {Gengel, Erik and Pikovskij, Arkadij}, title = {Phase reconstruction from oscillatory data with iterated Hilbert transform embeddings}, series = {Physica : D, Nonlinear phenomena}, volume = {429}, journal = {Physica : D, Nonlinear phenomena}, publisher = {Elsevier}, address = {Amsterdam}, issn = {0167-2789}, doi = {10.1016/j.physd.2021.133070}, pages = {9}, year = {2022}, abstract = {In the data analysis of oscillatory systems, methods based on phase reconstruction are widely used to characterize phase-locking properties and inferring the phase dynamics. The main component in these studies is an extraction of the phase from a time series of an oscillating scalar observable. We discuss a practical procedure of phase reconstruction by virtue of a recently proposed method termed iterated Hilbert transform embeddings. We exemplify the potential benefits and limitations of the approach by applying it to a generic observable of a forced Stuart-Landau oscillator. Although in many cases, unavoidable amplitude modulation of the observed signal does not allow for perfect phase reconstruction, in cases of strong stability of oscillations and a high frequency of the forcing, iterated Hilbert transform embeddings significantly improve the quality of the reconstructed phase. We also demonstrate that for significant amplitude modulation, iterated embeddings do not provide any improvement.}, language = {en} } @article{GinelliAhlersLivietal.2003, author = {Ginelli, F. and Ahlers, Volker and Livi, R. and Mukamel, D. and Pikovskij, Arkadij and Politi, Antonio and Torcini, A.}, title = {From multiplicative noise to directed percolation in wetting transitions}, issn = {1063-651X}, year = {2003}, abstract = {A simple one-dimensional microscopic model of the depinning transition of an interface from an attractive hard wall is introduced and investigated. Upon varying a control parameter, the critical behavior observed along the transition line changes from a directed-percolation type to a multiplicative-noise type. Numerical simulations allow for a quantitative study of the multicritical point separating the two regions. Mean-field arguments and the mapping on yet a simpler model provide some further insight on the overall scenario}, language = {en} } @article{GlendinningFeudelPikovskijetal.2000, author = {Glendinning, P. A. and Feudel, Ulrike and Pikovskij, Arkadij and Stark, J.}, title = {The structure of mode-locking regions in quasi-periodically forced circle maps}, year = {2000}, language = {en} } @article{GoldobinPikovskij2006, author = {Goldobin, Denis S. and Pikovskij, Arkadij}, title = {Antireliability of noise-driven neurons}, issn = {1539-3755}, doi = {10.1103/Physreve.73.061906}, year = {2006}, abstract = {We demonstrate, within the framework of the FitzHugh-Nagumo model, that a firing neuron can respond to a noisy driving in a nonreliable manner: the same Gaussian white noise acting on identical neurons evokes different patterns of spikes. The effect is characterized via calculations of the Lyapunov exponent and the event synchronization correlations. We construct a theory that explains the antireliability as a combined effect of a high sensitivity to noise of some stages of the dynamics and nonisochronicity of oscillations. Geometrically, the antireliability is described by a random noninvertible one-dimensional map}, language = {en} } @article{GoldobinPikovskij2006, author = {Goldobin, Denis S. and Pikovskij, Arkadij}, title = {Effects of delayed feedback on Kuramoto transition}, issn = {0375-9687}, doi = {10.1143/PTPS.161.43}, year = {2006}, abstract = {We develop a weakly nonlinear theory of the Kuramoto transition in an ensemble of globally coupled oscillators in presence of additional time-delayed coupling terms. We show that a linear delayed feedback not only controls the transition point, but effectively changes the nonlinear terms near the transition. A purely nonlinear delayed coupling does not effect the transition point, but can reduce or enhance the amplitude of collective oscillations}, language = {en} } @article{GoldobinPikovskij2005, author = {Goldobin, Denis S. and Pikovskij, Arkadij}, title = {Synchronization and desynchronization of self-sustained oscillators by common noise}, year = {2005}, abstract = {We consider the effect of external noise on the dynamics of limit cycle oscillators. The Lyapunov exponent becomes negative under influence of small white noise, what means synchronization of two or more identical systems subject to common noise. We analytically study the effect of small nonidentities in the oscillators and in the noise, and derive statistical characteristics of deviations from the perfect synchrony. Large white noise can lead to desynchronization of oscillators, provided they are nonisochronous. This is demonstrated for the Van der Pol-Duffing system}, language = {en} } @article{GoldobinPikovskij2005, author = {Goldobin, Denis S. and Pikovskij, Arkadij}, title = {Synchronization of self-sustained oscillators by common white noise}, year = {2005}, abstract = {We study the stability of self-sustained oscillations under the influence of external noise. For small-noise amplitude a phase approximation for the Langevin dynamics is valid. A stationary distribution of the phase is used for an analytic calculation of the maximal Lyapunov exponent. We demonstrate that for small noise the exponent is negative, which corresponds to synchronization of oscillators. (c) 2004 Elsevier B.V. All rights reserved}, language = {en} } @article{GoldobinPimenovaRosenblumetal.2017, author = {Goldobin, Denis S. and Pimenova, Anastasiya V. and Rosenblum, Michael and Pikovskij, Arkadij}, title = {Competing influence of common noise and desynchronizing coupling on synchronization in the Kuramoto-Sakaguchi ensemble}, 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/e2017-70039-y}, pages = {1921 -- 1937}, year = {2017}, abstract = {We describe analytically synchronization and desynchronization effects in an ensemble of phase oscillators driven by common noise and by global coupling. Adopting the Ott-Antonsen ansatz, we reduce the dynamics to closed stochastic equations for the order parameters, and study these equations for the cases of populations of identical and nonidentical oscillators. For nonidentical oscillators we demonstrate a counterintuitive effect of divergence of individual frequencies for moderate repulsive coupling, while the order parameter remains large.}, language = {en} } @article{GoldobinTyulkinaKlimenkoetal.2018, author = {Goldobin, Denis S. and Tyulkina, Irina V. and Klimenko, Lyudmila S. and Pikovskij, Arkadij}, title = {Collective mode reductions for populations of coupled noisy oscillators}, series = {Chaos : an interdisciplinary journal of nonlinear science}, volume = {28}, journal = {Chaos : an interdisciplinary journal of nonlinear science}, number = {10}, publisher = {American Institute of Physics}, address = {Melville}, issn = {1054-1500}, doi = {10.1063/1.5053576}, pages = {6}, year = {2018}, abstract = {We analyze the accuracy of different low-dimensional reductions of the collective dynamics in large populations of coupled phase oscillators with intrinsic noise. Three approximations are considered: (i) the Ott-Antonsen ansatz, (ii) the Gaussian ansatz, and (iii) a two-cumulant truncation of the circular cumulant representation of the original system's dynamics. For the latter, we suggest a closure, which makes the truncation, for small noise, a rigorous first-order correction to the Ott-Antonsen ansatz, and simultaneously is a generalization of the Gaussian ansatz. The Kuramoto model with intrinsic noise and the population of identical noisy active rotators in excitable states with the Kuramoto-type coupling are considered as examples to test the validity of these approximations. For all considered cases, the Gaussian ansatz is found to be more accurate than the Ott-Antonsen one for high-synchrony states only. The two-cumulant approximation is always superior to both other approximations. Synchrony of large ensembles of coupled elements can be characterised by the order parameters—the mean fields. Quite often, the evolution of these collective variables is surprisingly simple, which makes a description with only a few order parameters feasible. Thus, one tries to construct accurate closed low-dimensional mathematical models for the dynamics of the first few order parameters. These models represent useful tools for gaining insight into the underlaying mechanisms of some more sophisticated collective phenomena: for example, one describes coupled populations by virtue of coupled equations for the relevant order parameters. A regular approach to the construction of closed low-dimensional systems is also beneficial for dealing with phenomena, which are beyond the applicability scope of these models; for instance, with such an approach, one can determine constraints on clustering in populations. There are two prominent types of situations, where the low-dimensional models can be constructed: (i) for a certain class of ideal paradigmatic systems of coupled phase oscillators, the Ott-Antonsen ansatz yields an exact equation for the main order parameter and (ii) the Gaussian approximation for the probability density of the phases, also yielding a low-dimensional closure, is frequently quite accurate. In this paper, we compare applications of these two model reductions for situations, where neither of them is perfectly accurate. Furthermore, we construct a new reduction approach which practically works as a first-order correction to the best of the two basic approximations.}, language = {en} } @article{GoldschmidtPikovskijPoliti2019, author = {Goldschmidt, Richard Janis and Pikovskij, Arkadij and Politi, Antonio}, title = {Blinking chimeras in globally coupled rotators}, series = {Chaos : an interdisciplinary journal of nonlinear science}, volume = {29}, journal = {Chaos : an interdisciplinary journal of nonlinear science}, number = {7}, publisher = {American Institute of Physics}, address = {Melville}, issn = {1054-1500}, doi = {10.1063/1.5105367}, pages = {7}, year = {2019}, abstract = {In globally coupled ensembles of identical oscillators so-called chimera states can be observed. The chimera state is a symmetry-broken regime, where a subset of oscillators forms a cluster, a synchronized population, while the rest of the system remains a collection of nonsynchronized, scattered units. We describe here a blinking chimera regime in an ensemble of seven globally coupled rotators (Kuramoto oscillators with inertia). It is characterized by a death-birth process, where a long-term stable cluster of four oscillators suddenly dissolves and is very quickly reborn with a new reshuffled configuration. We identify three different kinds of rare blinking events and give a quantitative characterization by applying stability analysis to the long-lived chaotic state and to the short-lived regular regimes that arise when the cluster dissolves.}, language = {en} } @article{GongPikovskij2019, author = {Gong, Chen Chris and Pikovskij, Arkadij}, title = {Low-dimensional dynamics for higher-order harmonic, globally coupled phase-oscillator ensembles}, series = {Physical review : E, Statistical, nonlinear and soft matter physics}, volume = {100}, journal = {Physical review : E, Statistical, nonlinear and soft matter physics}, number = {6}, publisher = {American Physical Society}, address = {College Park}, issn = {2470-0045}, doi = {10.1103/PhysRevE.100.062210}, pages = {10}, year = {2019}, abstract = {The Kuramoto model, despite its popularity as a mean-field theory for many synchronization phenomenon of oscillatory systems, is limited to a first-order harmonic coupling of phases. For higher-order coupling, there only exists a low-dimensional theory in the thermodynamic limit. In this paper, we extend the formulation used by Watanabe and Strogatz to obtain a low-dimensional description of a system of arbitrary size of identical oscillators coupled all-to-all via their higher-order modes. To demonstrate an application of the formulation, we use a second harmonic globally coupled model, with a mean-field equal to the square of the Kuramoto mean-field. This model is known to exhibit asymmetrical clustering in previous numerical studies. We try to explain the phenomenon of asymmetrical clustering using the analytical theory developed here, as well as discuss certain phenomena not observed at the level of first-order harmonic coupling.}, language = {en} } @article{GongZhengToenjesetal.2019, author = {Gong, Chen Chris and Zheng, Chunming and Toenjes, Ralf and Pikovskij, Arkadij}, title = {Repulsively coupled Kuramoto-Sakaguchi phase oscillators ensemble subject to common noise}, series = {Chaos : an interdisciplinary journal of nonlinear science}, volume = {29}, journal = {Chaos : an interdisciplinary journal of nonlinear science}, number = {3}, publisher = {American Institute of Physics}, address = {Melville}, issn = {1054-1500}, doi = {10.1063/1.5084144}, pages = {11}, year = {2019}, abstract = {We consider the Kuramoto-Sakaguchi model of identical coupled phase oscillators with a common noisy forcing. While common noise always tends to synchronize the oscillators, a strong repulsive coupling prevents the fully synchronous state and leads to a nontrivial distribution of oscillator phases. In previous numerical simulations, the formation of stable multicluster states has been observed in this regime. However, we argue here that because identical phase oscillators in the Kuramoto-Sakaguchi model form a partially integrable system according to the Watanabe-Strogatz theory, the formation of clusters is impossible. Integrating with various time steps reveals that clustering is a numerical artifact, explained by the existence of higher order Fourier terms in the errors of the employed numerical integration schemes. By monitoring the induced change in certain integrals of motion, we quantify these errors. We support these observations by showing, on the basis of the analysis of the corresponding Fokker-Planck equation, that two-cluster states are non-attractive. On the other hand, in ensembles of general limit cycle oscillators, such as Van der Pol oscillators, due to an anharmonic phase response function as well as additional amplitude dynamics, multiclusters can occur naturally. Published under license by AIP Publishing.}, language = {en} } @article{GrinesOsipovPikovskij2018, author = {Grines, Evgeny and Osipov, Grigory V. and Pikovskij, Arkadij}, title = {Describing dynamics of driven multistable oscillators with phase transfer curves}, 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 = {10.1063/1.5037290}, pages = {6}, year = {2018}, abstract = {Phase response curve is an important tool in the studies of stable self-sustained oscillations; it describes a phase shift under action of an external perturbation. We consider multistable oscillators with several stable limit cycles. Under a perturbation, transitions from one oscillating mode to another one may occur. We define phase transfer curves to describe the phase shifts at such transitions. This allows for a construction of one-dimensional maps that characterize periodically kicked multistable oscillators. We show that these maps are good approximations of the full dynamics for large periods of forcing. Published by AIP Publishing.}, language = {en} } @article{KatzorkePikovskij2000, author = {Katzorke, Ines and Pikovskij, Arkadij}, title = {Chaos and complexity in a simple model of production dynamics}, issn = {1026-0226}, year = {2000}, language = {en} } @article{KomarovGuptaPikovskij2014, author = {Komarov, Maxim and Gupta, Shamik and Pikovskij, Arkadij}, title = {Synchronization transitions in globally coupled rotors in the presence of noise and inertia: Exact results}, series = {epl : a letters journal exploring the frontiers of physics}, volume = {106}, journal = {epl : a letters journal exploring the frontiers of physics}, number = {4}, publisher = {EDP Sciences}, address = {Mulhouse}, issn = {0295-5075}, doi = {10.1209/0295-5075/106/40003}, pages = {6}, year = {2014}, abstract = {We study a generic model of globally coupled rotors that includes the effects of noise, phase shift in the coupling, and distributions of moments of inertia and natural frequencies of oscillation. As particular cases, the setup includes previously studied Sakaguchi-Kuramoto, Hamiltonian and Brownian mean-field, and Tanaka-Lichtenberg-Oishi and Acebron-Bonilla-Spigler models. We derive an exact solution of the self-consistent equations for the order parameter in the stationary state, valid for arbitrary parameters in the dynamics, and demonstrate nontrivial phase transitions to synchrony that include reentrant synchronous regimes. Copyright (C) EPLA, 2014}, language = {en} } @article{KomarovPikovskij2013, author = {Komarov, Maxim and Pikovskij, Arkadij}, title = {Multiplicity of singular synchronous States in the kuramoto model of coupled oscillators}, series = {Physical review letters}, volume = {111}, journal = {Physical review letters}, number = {20}, publisher = {American Physical Society}, address = {College Park}, issn = {0031-9007}, doi = {10.1103/PhysRevLett.111.204101}, pages = {5}, year = {2013}, abstract = {We study the Kuramoto model of globally coupled oscillators with a biharmonic coupling function. We develop an analytic self-consistency approach to find stationary synchronous states in the thermodynamic limit and demonstrate that there is a huge multiplicity of such states, which differ microscopically in the distributions of locked phases. These synchronous regimes already exist prior to the linear instability transition of the fully asynchronous state. In the presence of white Gaussian noise, the multiplicity is lifted, but the dependence of the order parameters on coupling constants remains nontrivial.}, language = {en} } @article{KomarovPikovskij2013, author = {Komarov, Maxim and Pikovskij, Arkadij}, title = {Dynamics of multifrequency oscillator communities}, series = {Physical review letters}, volume = {110}, journal = {Physical review letters}, number = {13}, publisher = {American Physical Society}, address = {College Park}, issn = {0031-9007}, doi = {10.1103/PhysRevLett.110.134101}, pages = {5}, year = {2013}, abstract = {We consider a generalization of the Kuramoto model of coupled oscillators to the situation where communities of oscillators having essentially different natural frequencies interact. General equations describing possible resonances between the communities' frequencies are derived. The simplest situation of three resonantly interacting groups is analyzed in detail. We find conditions for the mutual coupling to promote or suppress synchrony in individual populations and present examples where the interaction between communities leads to their synchrony or to a partially asynchronous state or to a chaotic dynamics of order parameters. DOI: 10.1103/PhysRevLett.110.134101}, language = {en} } @article{KomarovPikovskij2011, author = {Komarov, Maxim and Pikovskij, Arkadij}, title = {Effects of nonresonant interaction in ensembles of phase oscillators}, series = {Physical review : E, Statistical, nonlinear and soft matter physics}, volume = {84}, 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.84.016210}, pages = {12}, year = {2011}, abstract = {We consider general properties of groups of interacting oscillators, for which the natural frequencies are not in resonance. Such groups interact via nonoscillating collective variables like the amplitudes of the order parameters defined for each group. We treat the phase dynamics of the groups using the Ott-Antonsen ansatz and reduce it to a system of coupled equations for the order parameters. We describe different regimes of cosynchrony in the groups. For a large number of groups, heteroclinic cycles, corresponding to a sequential synchronous activity of groups and chaotic states where the order parameters oscillate irregularly, are possible.}, language = {en} } @article{KomarovPikovskij2014, author = {Komarov, Maxim and Pikovskij, Arkadij}, title = {The Kuramoto model of coupled oscillators with a bi-harmonic coupling function}, series = {Physica : D, Nonlinear phenomena}, volume = {289}, journal = {Physica : D, Nonlinear phenomena}, publisher = {Elsevier}, address = {Amsterdam}, issn = {0167-2789}, doi = {10.1016/j.physd.2014.09.002}, pages = {18 -- 31}, year = {2014}, abstract = {We study synchronization in a Kuramoto model of globally coupled phase oscillators with a bi-harmonic coupling function, in the thermodynamic limit of large populations. We develop a method for an analytic solution of self-consistent equations describing uniformly rotating complex order parameters, both for single-branch (one possible state of locked oscillators) and multi-branch (two possible values of locked phases) entrainment. We show that synchronous states coexist with the neutrally linearly stable asynchronous regime. The latter has a finite life time for finite ensembles, this time grows with the ensemble size as a power law. (C) 2014 Elsevier B.V. All rights reserved.}, language = {en} } @article{KomarovPikovskij2015, author = {Komarov, Maxim and Pikovskij, Arkadij}, title = {Finite-size-induced transitions to synchrony in oscillator ensembles with nonlinear global coupling}, series = {Physical review : E, Statistical, nonlinear and soft matter physics}, volume = {92}, 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.92.020901}, pages = {5}, year = {2015}, abstract = {We report on finite-sized-induced transitions to synchrony in a population of phase oscillators coupled via a nonlinear mean field, which microscopically is equivalent to a hypernetwork organization of interactions. Using a self-consistent approach and direct numerical simulations, we argue that a transition to synchrony occurs only for finite-size ensembles and disappears in the thermodynamic limit. For all considered setups, which include purely deterministic oscillators with or without heterogeneity in natural oscillatory frequencies, and an ensemble of noise-driven identical oscillators, we establish scaling relations describing the order parameter as a function of the coupling constant and the system size.}, language = {en} } @article{KomarovPikovskij2015, author = {Komarov, Maxim and Pikovskij, Arkadij}, title = {Intercommunity resonances in multifrequency ensembles of coupled oscillators}, 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.92.012906}, pages = {11}, year = {2015}, abstract = {We generalize the Kuramoto model of globally coupled oscillators to multifrequency communities. A situation when mean frequencies of two subpopulations are close to the resonance 2 : 1 is considered in detail. We construct uniformly rotating solutions describing synchronization inside communities and between them. Remarkably, cross coupling across the frequencies can promote synchrony even when ensembles are separately asynchronous. We also show that the transition to synchrony due to the cross coupling is accompanied by a huge multiplicity of distinct synchronous solutions, which is directly related to a multibranch entrainment. On the other hand, for synchronous populations, the cross-frequency coupling can destroy phase locking and lead to chaos of mean fields.}, language = {en} } @article{KralemannFruehwirthPikovskijetal.2013, author = {Kralemann, Bjoern and Fruehwirth, Matthias and Pikovskij, Arkadij and Rosenblum, Michael and Kenner, Thomas and Schaefer, Jochen and Moser, Maximilian}, title = {In vivo cardiac phase response curve elucidates human respiratory heart rate variability}, series = {Nature Communications}, volume = {4}, journal = {Nature Communications}, publisher = {Nature Publ. Group}, address = {London}, issn = {2041-1723}, doi = {10.1038/ncomms3418}, pages = {9}, year = {2013}, abstract = {Recovering interaction of endogenous rhythms from observations is challenging, especially if a mathematical model explaining the behaviour of the system is unknown. The decisive information for successful reconstruction of the dynamics is the sensitivity of an oscillator to external influences, which is quantified by its phase response curve. Here we present a technique that allows the extraction of the phase response curve from a non-invasive observation of a system consisting of two interacting oscillators-in this case heartbeat and respiration-in its natural environment and under free-running conditions. We use this method to obtain the phase-coupling functions describing cardiorespiratory interactions and the phase response curve of 17 healthy humans. We show for the first time the phase at which the cardiac beat is susceptible to respiratory drive and extract the respiratory-related component of heart rate variability. This non-invasive method for the determination of phase response curves of coupled oscillators may find application in many scientific disciplines.}, language = {en} }