@article{StraubePikovskij2011, author = {Straube, Arthur V. and Pikovskij, Arkadij}, title = {Pattern formation induced by time-dependent advection}, series = {Mathematical modelling of natural phenomena}, volume = {6}, journal = {Mathematical modelling of natural phenomena}, number = {1}, publisher = {EDP Sciences}, address = {Les Ulis}, issn = {0973-5348}, doi = {10.1051/mmnp/20116107}, pages = {138 -- 148}, year = {2011}, abstract = {We study pattern-forming instabilities in reaction-advection-diffusion systems. We develop an approach based on Lyapunov-Bloch exponents to figure out the impact of a spatially periodic mixing flow on the stability of a spatially homogeneous state. We deal with the flows periodic in space that may have arbitrary time dependence. We propose a discrete in time model, where reaction, advection, and diffusion act as successive operators, and show that a mixing advection can lead to a pattern-forming instability in a two-component system where only one of the species is advected. Physically, this can be explained as crossing a threshold of Turing instability due to effective increase of one of the diffusion constants.}, language = {en} } @article{ZhirovPikovskijShepelyansky2011, author = {Zhirov, O. V. and Pikovskij, Arkadij and Shepelyansky, Dima L.}, title = {Quantum vacuum of strongly nonlinear lattices}, series = {Physical review : E, Statistical, nonlinear and soft matter physics}, volume = {83}, journal = {Physical review : E, Statistical, nonlinear and soft matter physics}, number = {1}, publisher = {American Physical Society}, address = {College Park}, issn = {1539-3755}, doi = {10.1103/PhysRevE.83.016202}, pages = {7}, year = {2011}, abstract = {We study the properties of classical and quantum strongly nonlinear chains by means of extensive numerical simulations. Due to strong nonlinearity, the classical dynamics of such chains remains chaotic at arbitrarily low energies. We show that the collective excitations of classical chains are described by sound waves whose decay rate scales algebraically with the wave number with a generic exponent value. The properties of the quantum chains are studied by the quantum Monte Carlo method and it is found that the low-energy excitations are well described by effective phonon modes with the sound velocity dependent on an effective Planck constant. Our results show that at low energies the quantum effects lead to a suppression of chaos and drive the system to a quasi-integrable regime of effective phonon modes.}, language = {en} } @article{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} } @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{TurukinaPikovskij2011, author = {Turukina, L. V. and Pikovskij, Arkadij}, title = {Hyperbolic chaos in a system of resonantly coupled weakly nonlinear oscillators}, series = {Modern physics letters : A, Particles and fields, gravitation, cosmology, nuclear physics}, volume = {375}, journal = {Modern physics letters : A, Particles and fields, gravitation, cosmology, nuclear physics}, number = {11}, publisher = {Elsevier}, address = {Amsterdam}, issn = {0375-9601}, doi = {10.1016/j.physleta.2011.02.017}, pages = {1407 -- 1411}, year = {2011}, abstract = {We show that a hyperbolic chaos can be observed in resonantly coupled oscillators near a Hopf bifurcation, described by normal-form-type equations for complex amplitudes. The simplest example consists of four oscillators, comprising two alternatively activated, due to an external periodic modulation, pairs. In terms of the stroboscopic Poincare map, the phase differences change according to an expanding Bernoulli map that depends on the coupling type. Several examples of hyperbolic chaos for different types of coupling are illustrated numerically.}, language = {en} } @article{PikovskijRosenblum2011, author = {Pikovskij, Arkadij and Rosenblum, Michael}, title = {Dynamics of heterogeneous oscillator ensembles in terms of collective variables}, series = {Physica :D, Nonlinear phenomena}, volume = {240}, journal = {Physica :D, Nonlinear phenomena}, number = {9-10}, publisher = {Elsevier}, address = {Amsterdam}, issn = {0167-2789}, doi = {10.1016/j.physd.2011.01.002}, pages = {872 -- 881}, year = {2011}, abstract = {We consider general heterogeneous ensembles of phase oscillators, sine coupled to arbitrary external fields. Starting with the infinitely large ensembles, we extend the Watanabe-Strogatz theory, valid for identical oscillators, to cover the case of an arbitrary parameter distribution. The obtained equations yield the description of the ensemble dynamics in terms of collective variables and constants of motion. As a particular case of the general setup we consider hierarchically organized ensembles, consisting of a finite number of subpopulations, whereas the number of elements in a subpopulation can be both finite or infinite. Next, we link the Watanabe-Strogatz and Ott-Antonsen theories and demonstrate that the latter one corresponds to a particular choice of constants of motion. The approach is applied to the standard Kuramoto-Sakaguchi model, to its extension for the case of nonlinear coupling, and to the description of two interacting subpopulations, exhibiting a chimera state. With these examples we illustrate that, although the asymptotic dynamics can be found within the framework of the Ott-Antonsen theory, the transients depend on the constants of motion. The most dramatic effect is the dependence of the basins of attraction of different synchronous regimes on the initial configuration of phases.}, language = {en} } @article{KralemannPikovskijRosenblum2011, author = {Kralemann, Bj{\"o}rn and Pikovskij, Arkadij and Rosenblum, Michael}, title = {Reconstructing phase dynamics of oscillator networks}, series = {Chaos : an interdisciplinary journal of nonlinear science}, volume = {21}, journal = {Chaos : an interdisciplinary journal of nonlinear science}, number = {2}, publisher = {American Institute of Physics}, address = {Melville}, issn = {1054-1500}, doi = {10.1063/1.3597647}, pages = {10}, year = {2011}, abstract = {We generalize our recent approach to the reconstruction of phase dynamics of coupled oscillators from data [B. Kralemann et al., Phys. Rev. E 77, 066205 (2008)] to cover the case of small networks of coupled periodic units. Starting from a multivariate time series, we first reconstruct genuine phases and then obtain the coupling functions in terms of these phases. Partial norms of these coupling functions quantify directed coupling between oscillators. We illustrate the method by different network motifs for three coupled oscillators and for random networks of five and nine units. We also discuss nonlinear effects in coupling.}, language = {en} } @article{LueckPikovskij2011, author = {Lueck, S. and Pikovskij, Arkadij}, title = {Dynamics of multi-frequency oscillator ensembles with resonant coupling}, series = {Modern physics letters : A, Particles and fields, gravitation, cosmology, nuclear physics}, volume = {375}, journal = {Modern physics letters : A, Particles and fields, gravitation, cosmology, nuclear physics}, number = {28-29}, publisher = {Elsevier}, address = {Amsterdam}, issn = {0375-9601}, doi = {10.1016/j.physleta.2011.06.016}, pages = {2714 -- 2719}, year = {2011}, abstract = {We study dynamics of populations of resonantly coupled oscillators having different frequencies. Starting from the coupled van der Pol equations we derive the Kuramoto-type phase model for the situation, where the natural frequencies of two interacting subpopulations are in relation 2 : 1. Depending on the parameter of coupling, ensembles can demonstrate fully synchronous clusters, partial synchrony (only one subpopulation synchronizes), or asynchrony in both subpopulations. Theoretical description of the dynamics based on the Watanabe-Strogatz approach is developed.}, language = {en} } @article{LevnajicPikovskij2011, author = {Levnajic, Zoran and Pikovskij, Arkadij}, title = {Network reconstruction from random phase resetting}, series = {Physical review letters}, volume = {107}, journal = {Physical review letters}, number = {3}, publisher = {American Physical Society}, address = {College Park}, issn = {0031-9007}, doi = {10.1103/PhysRevLett.107.034101}, pages = {4}, year = {2011}, abstract = {We propose a novel method of reconstructing the topology and interaction functions for a general oscillator network. An ensemble of initial phases and the corresponding instantaneous frequencies is constructed by repeating random phase resets of the system dynamics. The desired details of network structure are then revealed by appropriately averaging over the ensemble. The method is applicable for a wide class of networks with arbitrary emergent dynamics, including full synchrony.}, 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} }