@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}
}
@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{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{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{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{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{StraubePikovskij2011,
  author    = {Straube, Arthur V. and Pikovskij, Arkadij},
  title     = {Pattern formation induced by time-dependent advection},
  series = {Mathematical modelling of natural phenomena},
  volume    = {6},
  journal   = {Mathematical modelling of natural phenomena},
  number    = {1},
  publisher = {EDP Sciences},
  address   = {Les Ulis},
  issn      = {0973-5348},
  doi       = {10.1051/mmnp/20116107},
  pages     = {138 -- 148},
  year      = {2011},
  abstract  = {We study pattern-forming instabilities in reaction-advection-diffusion systems. We develop an approach based on Lyapunov-Bloch exponents to figure out the impact of a spatially periodic mixing flow on the stability of a spatially homogeneous state. We deal with the flows periodic in space that may have arbitrary time dependence. We propose a discrete in time model, where reaction, advection, and diffusion act as successive operators, and show that a mixing advection can lead to a pattern-forming instability in a two-component system where only one of the species is advected. Physically, this can be explained as crossing a threshold of Turing instability due to effective increase of one of the diffusion constants.},
  language  = {en}
}
@misc{StraubePikovskij2011,
  author    = {Straube, Arthur V. and Pikovskij, Arkadij},
  title     = {Pattern formation induced by time-dependent advection},
  series = {Postprints der Universit{\"a}t Potsdam : Mathematisch Naturwissenschaftliche Reihe},
  journal   = {Postprints der Universit{\"a}t Potsdam : Mathematisch Naturwissenschaftliche Reihe},
  number    = {575},
  issn      = {1866-8372},
  doi       = {10.25932/publishup-41314},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-413140},
  pages     = {138-147},
  year      = {2011},
  abstract  = {We study pattern-forming instabilities in reaction-advection-diffusion systems. We develop an approach based on Lyapunov-Bloch exponents to figure out the impact of a spatially periodic mixing flow on the stability of a spatially homogeneous state. We deal with the flows periodic in space that may have arbitrary time dependence. We propose a discrete in time model, where reaction, advection, and diffusion act as successive operators, and show that a mixing advection can lead to a pattern-forming instability in a two-component system where only one of the species is advected. Physically, this can be explained as crossing a threshold of Turing instability due to effective increase of one of the diffusion constants.},
  language  = {en}
}
@article{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{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}
}