TY - JOUR A1 - Zhirov, O. V. A1 - Pikovskij, Arkadij A1 - Shepelyansky, Dima L. T1 - Quantum vacuum of strongly nonlinear lattices JF - Physical review : E, Statistical, nonlinear and soft matter physics N2 - 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. Y1 - 2011 U6 - https://doi.org/10.1103/PhysRevE.83.016202 SN - 1539-3755 VL - 83 IS - 1 PB - American Physical Society CY - College Park ER - TY - JOUR A1 - Turukina, L. V. A1 - Pikovskij, Arkadij T1 - Hyperbolic chaos in a system of resonantly coupled weakly nonlinear oscillators JF - Modern physics letters : A, Particles and fields, gravitation, cosmology, nuclear physics N2 - 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. KW - Coupled oscillators KW - Hyperbolic chaos Y1 - 2011 U6 - https://doi.org/10.1016/j.physleta.2011.02.017 SN - 0375-9601 VL - 375 IS - 11 SP - 1407 EP - 1411 PB - Elsevier CY - Amsterdam ER - TY - JOUR A1 - Straube, Arthur V. A1 - Pikovskij, Arkadij T1 - Pattern formation induced by time-dependent advection JF - Mathematical modelling of natural phenomena N2 - 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. KW - pattern formation KW - reaction-advection-diffusion equation Y1 - 2011 U6 - https://doi.org/10.1051/mmnp/20116107 SN - 0973-5348 VL - 6 IS - 1 SP - 138 EP - 148 PB - EDP Sciences CY - Les Ulis ER - TY - GEN A1 - Straube, Arthur V. A1 - Pikovskij, Arkadij T1 - Pattern formation induced by time-dependent advection T2 - Postprints der Universität Potsdam : Mathematisch Naturwissenschaftliche Reihe N2 - 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. T3 - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - 575 KW - pattern formation KW - reaction-advection-diffusion equation Y1 - 2019 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-413140 SN - 1866-8372 IS - 575 SP - 138-147 ER - TY - JOUR A1 - Pikovskij, Arkadij A1 - Rosenblum, Michael T1 - Dynamics of heterogeneous oscillator ensembles in terms of collective variables JF - Physica :D, Nonlinear phenomena N2 - 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. KW - Coupled oscillators KW - Oscillator ensembles KW - Kuramoto model KW - Nonlinear coupling KW - Watanabe-Strogatz theory KW - Ott-Antonsen theory Y1 - 2011 U6 - https://doi.org/10.1016/j.physd.2011.01.002 SN - 0167-2789 VL - 240 IS - 9-10 SP - 872 EP - 881 PB - Elsevier CY - Amsterdam ER - TY - JOUR A1 - Pikovskij, Arkadij A1 - Fishman, Shmuel T1 - Scaling properties of weak chaos in nonlinear disordered lattices JF - Physical review : E, Statistical, nonlinear and soft matter physics N2 - 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. Y1 - 2011 U6 - https://doi.org/10.1103/PhysRevE.83.025201 SN - 1539-3755 SN - 1550-2376 VL - 83 IS - 2 PB - American Physical Society CY - College Park ER - TY - JOUR A1 - Mulansky, Mario A1 - Ahnert, Karsten A1 - Pikovskij, Arkadij A1 - Shepelyansky, Dima L. T1 - Strong and weak chaos in weakly nonintegrable many-body hamiltonian systems JF - Journal of statistical physics N2 - 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. KW - Lyapunov exponent KW - Arnold diffusion KW - Chaos spreading Y1 - 2011 U6 - https://doi.org/10.1007/s10955-011-0335-3 SN - 0022-4715 VL - 145 IS - 5 SP - 1256 EP - 1274 PB - Springer CY - New York ER - TY - JOUR A1 - Mulansky, Mario A1 - Ahnert, Karsten A1 - Pikovskij, Arkadij T1 - Scaling of energy spreading in strongly nonlinear disordered lattices JF - Physical review : E, Statistical, nonlinear and soft matter physics N2 - 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. Y1 - 2011 U6 - https://doi.org/10.1103/PhysRevE.83.026205 SN - 1539-3755 VL - 83 IS - 2 PB - American Physical Society CY - College Park ER - TY - JOUR A1 - Lueck, S. A1 - Pikovskij, Arkadij T1 - Dynamics of multi-frequency oscillator ensembles with resonant coupling JF - Modern physics letters : A, Particles and fields, gravitation, cosmology, nuclear physics N2 - 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. KW - Oscillator populations KW - Kuramoto model KW - Resonant interaction Y1 - 2011 U6 - https://doi.org/10.1016/j.physleta.2011.06.016 SN - 0375-9601 VL - 375 IS - 28-29 SP - 2714 EP - 2719 PB - Elsevier CY - Amsterdam ER - TY - JOUR A1 - Levnajic, Zoran A1 - Pikovskij, Arkadij T1 - Network reconstruction from random phase resetting JF - Physical review letters N2 - 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. Y1 - 2011 U6 - https://doi.org/10.1103/PhysRevLett.107.034101 SN - 0031-9007 VL - 107 IS - 3 PB - American Physical Society CY - College Park ER -