TY - JOUR A1 - Mulansky, Mario A1 - Ahnert, Karsten A1 - Pikovskij, Arkadij A1 - Shepelyansky, Dima L. T1 - Dynamical thermalization of disordered nonlinear lattices N2 - We study numerically how the energy spreads over a finite disordered nonlinear one-dimensional lattice, where all linear modes are exponentially localized by disorder. We establish emergence of dynamical thermalization characterized as an ergodic chaotic dynamical state with a Gibbs distribution over the modes. Our results show that the fraction of thermalizing modes is finite and grows with the nonlinearity strength. Y1 - 2009 UR - http://pre.aps.org/ U6 - https://doi.org/10.1103/Physreve.80.056212 SN - 1539-3755 ER - TY - JOUR A1 - Abel, M. W. A1 - Shepelyansky, Dima L. T1 - Google matrix of business process management JF - The European physical journal : B, Condensed matter and complex systems N2 - Development of efficient business process models and determination of their characteristic properties are subject of intense interdisciplinary research. Here, we consider a business process model as a directed graph. Its nodes correspond to the units identified by the modeler and the link direction indicates the causal dependencies between units. It is of primary interest to obtain the stationary flow on such a directed graph, which corresponds to the steady-state of a firm during the business process. Following the ideas developed recently for the World Wide Web, we construct the Google matrix for our business process model and analyze its spectral properties. The importance of nodes is characterized by PageRank and recently proposed CheiRank and 2DRank, respectively. The results show that this two-dimensional ranking gives a significant information about the influence and communication properties of business model units. We argue that the Google matrix method, described here, provides a new efficient tool helping companies to make their decisions on how to evolve in the exceedingly dynamic global market. Y1 - 2011 U6 - https://doi.org/10.1140/epjb/e2010-10710-y SN - 1434-6028 VL - 84 IS - 4 SP - 493 EP - 500 PB - Springer CY - New York ER - 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 - Letellier, Christophe A1 - Abraham, Ralph A1 - Shepelyansky, Dima L. A1 - Rossler, Otto E. A1 - Holmes, Philip A1 - Lozi, Rene A1 - Glass, Leon A1 - Pikovsky, Arkady A1 - Olsen, Lars F. A1 - Tsuda, Ichiro A1 - Grebogi, Celso A1 - Parlitz, Ulrich A1 - Gilmore, Robert A1 - Pecora, Louis M. A1 - Carroll, Thomas L. T1 - Some elements for a history of the dynamical systems theory JF - Chaos : an interdisciplinary journal of nonlinear science N2 - Writing a history of a scientific theory is always difficult because it requires to focus on some key contributors and to "reconstruct" some supposed influences. In the 1970s, a new way of performing science under the name "chaos" emerged, combining the mathematics from the nonlinear dynamical systems theory and numerical simulations. To provide a direct testimony of how contributors can be influenced by other scientists or works, we here collected some writings about the early times of a few contributors to chaos theory. The purpose is to exhibit the diversity in the paths and to bring some elements-which were never published-illustrating the atmosphere of this period. Some peculiarities of chaos theory are also discussed. Y1 - 2021 U6 - https://doi.org/10.1063/5.0047851 SN - 1054-1500 SN - 1089-7682 VL - 31 IS - 5 PB - AIP Publishing CY - Melville 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 - Shepelyansky, Dima L. A1 - Pikovskij, Arkadij A1 - Schmidt, Jürgen A1 - Spahn, Frank T1 - Synchronization mechanism of sharp edges in rings of Saturn N2 - 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. Y1 - 2009 UR - http://www3.interscience.wiley.com/cgi-bin/issn?DESCRIPTOR=PRINTISSN&VALUE=0035-8711 U6 - https://doi.org/10.1111/j.1365-2966.2009.14719.x SN - 0035-8711 ER -