@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{LevnajicPikovskij2014,
  author    = {Levnajic, Zoran and Pikovskij, Arkadij},
  title     = {Untangling complex dynamical systems via derivative-variable correlations},
  series = {Scientific reports},
  volume    = {4},
  journal   = {Scientific reports},
  publisher = {Nature Publ. Group},
  address   = {London},
  issn      = {2045-2322},
  doi       = {10.1038/srep05030},
  pages     = {6},
  year      = {2014},
  abstract  = {Inferring the internal interaction patterns of a complex dynamical system is a challenging problem. Traditional methods often rely on examining the correlations among the dynamical units. However, in systems such as transcription networks, one unit's variable is also correlated with the rate of change of another unit's variable. Inspired by this, we introduce the concept of derivative-variable correlation, and use it to design a new method of reconstructing complex systems (networks) from dynamical time series. Using a tunable observable as a parameter, the reconstruction of any system with known interaction functions is formulated via a simple matrix equation. We suggest a procedure aimed at optimizing the reconstruction from the time series of length comparable to the characteristic dynamical time scale. Our method also provides a reliable precision estimate. We illustrate the method's implementation via elementary dynamical models, and demonstrate its robustness to both model error and observation error.},
  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{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{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{Pikovskij2021,
  author    = {Pikovskij, Arkadij},
  title     = {Transition to synchrony in chiral active particles},
  series = {Journal of physics. Complexity},
  volume    = {2},
  journal   = {Journal of physics. Complexity},
  number    = {2},
  publisher = {IOP Publ. Ltd.},
  address   = {Bristol},
  issn      = {2632-072X},
  doi       = {10.1088/2632-072X/abdadb},
  pages     = {8},
  year      = {2021},
  abstract  = {I study deterministic dynamics of chiral active particles in two dimensions. Particles are considered as discs interacting with elastic repulsive forces. An ensemble of particles, started from random initial conditions, demonstrates chaotic collisions resulting in their normal diffusion. This chaos is transient, as rather abruptly a synchronous collisionless state establishes. The life time of chaos grows exponentially with the number of particles. External forcing (periodic or chaotic) is shown to facilitate the synchronization transition.},
  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{PeterPikovskij2018,
  author    = {Peter, Franziska and Pikovskij, Arkadij},
  title     = {Transition to collective oscillations in finite Kuramoto ensembles},
  series = {Physical review : E, Statistical, nonlinear and soft matter physics},
  volume    = {97},
  journal   = {Physical review : E, Statistical, nonlinear and soft matter physics},
  number    = {3},
  publisher = {American Physical Society},
  address   = {College Park},
  issn      = {2470-0045},
  doi       = {10.1103/PhysRevE.97.032310},
  pages     = {10},
  year      = {2018},
  abstract  = {We present an alternative approach to finite-size effects around the synchronization transition in the standard Kuramoto model. Our main focus lies on the conditions under which a collective oscillatory mode is well defined. For this purpose, the minimal value of the amplitude of the complex Kuramoto order parameter appears as a proper indicator. The dependence of this minimum on coupling strength varies due to sampling variations and correlates with the sample kurtosis of the natural frequency distribution. The skewness of the frequency sample determines the frequency of the resulting collective mode. The effects of kurtosis and skewness hold in the thermodynamic limit of infinite ensembles. We prove this by integrating a self-consistency equation for the complex Kuramoto order parameter for two families of distributions with controlled kurtosis and skewness, respectively.},
  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{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{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}
}
@misc{KomarovPikovskij2015,
  author    = {Komarov, Maxim and Pikovskij, Arkadij},
  title     = {The Kuramoto model of coupled oscillators with a bi-harmonic coupling function (vol 289, pg 18, 2014)},
  series = {Physica :D, Nonlinear phenomena},
  volume    = {313},
  journal   = {Physica :D, Nonlinear phenomena},
  publisher = {Elsevier},
  address   = {Amsterdam},
  issn      = {0167-2789},
  doi       = {10.1016/j.physd.2015.11.001},
  pages     = {117 -- 117},
  year      = {2015},
  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{WittKurthsPikovskij1998,
  author    = {Witt, Annette and Kurths, J{\"u}rgen and Pikovskij, Arkadij},
  title     = {Testing stationarity in time series},
  year      = {1998},
  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{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{MontaseriYazdanpanahPikovskijetal.2013,
  author    = {Montaseri, Ghazal and Yazdanpanah, Mohammad Javad and Pikovskij, Arkadij and Rosenblum, Michael},
  title     = {Synchrony suppression in ensembles of coupled oscillators via adaptive vanishing feedback},
  series = {Chaos : an interdisciplinary journal of nonlinear science},
  volume    = {23},
  journal   = {Chaos : an interdisciplinary journal of nonlinear science},
  number    = {3},
  publisher = {American Institute of Physics},
  address   = {Melville},
  issn      = {1054-1500},
  doi       = {10.1063/1.4817393},
  pages     = {12},
  year      = {2013},
  abstract  = {Synchronization and emergence of a collective mode is a general phenomenon, frequently observed in ensembles of coupled self-sustained oscillators of various natures. In several circumstances, in particular in cases of neurological pathologies, this state of the active medium is undesirable. Destruction of this state by a specially designed stimulation is a challenge of high clinical relevance. Typically, the precise effect of an external action on the ensemble is unknown, since the microscopic description of the oscillators and their interactions are not available. We show that, desynchronization in case of a large degree of uncertainty about important features of the system is nevertheless possible; it can be achieved by virtue of a feedback loop with an additional adaptation of parameters. The adaptation also ensures desynchronization of ensembles with non-stationary, time-varying parameters. We perform the stability analysis of the feedback-controlled system and demonstrate efficient destruction of synchrony for several models, including those of spiking and bursting neurons.},
  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}
}
@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{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{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{Pikovskij2021,
  author    = {Pikovskij, Arkadij},
  title     = {Synchronization of oscillators with hyperbolic chaotic phases},
  series = {Izvestija vysšich učebnych zavedenij : naučno-techničeskij žurnal = Izvestiya VUZ. Prikladnaja nelinejnaja dinamika = Applied nonlinear dynamics},
  volume    = {29},
  journal   = {Izvestija vysšich učebnych zavedenij : naučno-techničeskij žurnal = Izvestiya VUZ. Prikladnaja nelinejnaja dinamika = Applied nonlinear dynamics},
  number    = {1},
  publisher = {Saratov State University},
  address   = {Saratov},
  issn      = {0869-6632},
  doi       = {10.18500/0869-6632-2021-29-1-78-87},
  pages     = {78 -- 87},
  year      = {2021},
  abstract  = {Topic and aim. Synchronization in populations of coupled oscillators can be characterized with order parameters that describe collective order in ensembles. A dependence of the order parameter on the coupling constants is well-known for coupled periodic oscillators. The goal of the study is to extend this analysis to ensembles of oscillators with chaotic phases, moreover with phases possessing hyperbolic chaos. Models and methods. Two models are studied in the paper. One is an abstract discrete-time map, composed with a hyperbolic Bernoulli transformation and with Kuramoto dynamics. Another model is a system of coupled continuous-time chaotic oscillators, where each individual oscillator has a hyperbolic attractor of Smale-Williams type. Results. The discrete-time model is studied with the Ott-Antonsen ansatz, which is shown to be invariant under the application of the Bernoulli map. The analysis of the resulting map for the order parameter shows, that the asynchronouis state is always stable, but the synchronous one becomes stable above a certain coupling strength. Numerical analysis of the continuous-time model reveals a complex sequence of transitions from an asynchronous state to a completely synchronous hyperbolic chaos, with intermediate stages that include regimes with periodic in time mean field, as well as with weakly and strongly irregular mean field variations. Discussion. Results demonstrate that synchronization of systems with hyperbolic chaos of phases is possible, although a rather strong coupling is required. The approach can be applied to other systems of interacting units with hyperbolic chaotic dynamics.},
  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{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{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{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{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{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{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}
}
@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{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{MulanskyAhnertPikovskijetal.2011,
  author    = {Mulansky, Mario and Ahnert, Karsten and Pikovskij, Arkadij and Shepelyansky, Dima L.},
  title     = {Strong and weak chaos in weakly nonintegrable many-body hamiltonian systems},
  series = {Journal of statistical physics},
  volume    = {145},
  journal   = {Journal of statistical physics},
  number    = {5},
  publisher = {Springer},
  address   = {New York},
  issn      = {0022-4715},
  doi       = {10.1007/s10955-011-0335-3},
  pages     = {1256 -- 1274},
  year      = {2011},
  abstract  = {We study properties of chaos in generic one-dimensional nonlinear Hamiltonian lattices comprised of weakly coupled nonlinear oscillators by numerical simulations of continuous-time systems and symplectic maps. For small coupling, the measure of chaos is found to be proportional to the coupling strength and lattice length, with the typical maximal Lyapunov exponent being proportional to the square root of coupling. This strong chaos appears as a result of triplet resonances between nearby modes. In addition to strong chaos we observe a weakly chaotic component having much smaller Lyapunov exponent, the measure of which drops approximately as a square of the coupling strength down to smallest couplings we were able to reach. We argue that this weak chaos is linked to the regime of fast Arnold diffusion discussed by Chirikov and Vecheslavov. In disordered lattices of large size we find a subdiffusive spreading of initially localized wave packets over larger and larger number of modes. The relations between the exponent of this spreading and the exponent in the dependence of the fast Arnold diffusion on coupling strength are analyzed. We also trace parallels between the slow spreading of chaos and deterministic rheology.},
  language  = {en}
}
@book{FreudeKuznetsovPikovskij2006,
  author    = {Freude, Ulrike and Kuznetsov, Sergey P. and Pikovskij, Arkadij},
  title     = {Strange nonchaotic attractors : dynamics between order and chaos in Quasiperiodically Forced Systems},
  publisher = {World Scientific},
  address   = {Singapore},
  isbn      = {981-256633-3},
  pages     = {350 S.},
  year      = {2006},
  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{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{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{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{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}
}
@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{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{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}
}
@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{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{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{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{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}
}
@article{MulanskyPikovskij2012,
  author    = {Mulansky, Mario and Pikovskij, Arkadij},
  title     = {Scaling properties of energy spreading in nonlinear Hamiltonian two-dimensional lattices},
  series = {Physical review : E, Statistical, nonlinear and soft matter physics},
  volume    = {86},
  journal   = {Physical review : E, Statistical, nonlinear and soft matter physics},
  number    = {5},
  publisher = {American Physical Society},
  address   = {College Park},
  issn      = {1539-3755},
  doi       = {10.1103/PhysRevE.86.056214},
  pages     = {7},
  year      = {2012},
  abstract  = {In nonlinear disordered Hamiltonian lattices, where there are no propagating phonons, the spreading of energy is of subdiffusive nature. Recently, the universality class of the subdiffusive spreading according to the nonlinear diffusion equation (NDE) has been suggested and checked for one-dimensional lattices. Here, we apply this approach to two-dimensional strongly nonlinear lattices and find a nice agreement of the scaling predicted from the NDE with the spreading results from extensive numerical studies. Moreover, we show that the scaling works also for regular lattices with strongly nonlinear coupling, for which the scaling exponent is estimated analytically. This shows that the process of chaotic diffusion in such lattices does not require disorder.},
  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{MulanskyAhnertPikovskij2011,
  author    = {Mulansky, Mario and Ahnert, Karsten and Pikovskij, Arkadij},
  title     = {Scaling of energy spreading in strongly 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.83.026205},
  pages     = {4},
  year      = {2011},
  abstract  = {To characterize a destruction of Anderson localization by nonlinearity, we study the spreading behavior of initially localized states in disordered, strongly nonlinear lattices. Due to chaotic nonlinear interaction of localized linear or nonlinear modes, energy spreads nearly subdiffusively. Based on a phenomenological description by virtue of a nonlinear diffusion equation, we establish a one-parameter scaling relation between the velocity of spreading and the density, which is confirmed numerically. From this scaling it follows that for very low densities the spreading slows down compared to the pure power law.},
  language  = {en}
}