@article{MulanskyAhnertPikovskijetal.2009, author = {Mulansky, Mario and Ahnert, Karsten and Pikovskij, Arkadij and Shepelyansky, Dima L.}, title = {Dynamical thermalization of disordered nonlinear lattices}, issn = {1539-3755}, doi = {10.1103/Physreve.80.056212}, year = {2009}, abstract = {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.}, language = {en} } @article{AbelShepelyansky2011, author = {Abel, M. W. and Shepelyansky, Dima L.}, title = {Google matrix of business process management}, series = {The European physical journal : B, Condensed matter and complex systems}, volume = {84}, journal = {The European physical journal : B, Condensed matter and complex systems}, number = {4}, publisher = {Springer}, address = {New York}, issn = {1434-6028}, doi = {10.1140/epjb/e2010-10710-y}, pages = {493 -- 500}, year = {2011}, abstract = {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.}, 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{LetellierAbrahamShepelyanskyetal.2021, author = {Letellier, Christophe and Abraham, Ralph and Shepelyansky, Dima L. and Rossler, Otto E. and Holmes, Philip and Lozi, Rene and Glass, Leon and Pikovsky, Arkady and Olsen, Lars F. and Tsuda, Ichiro and Grebogi, Celso and Parlitz, Ulrich and Gilmore, Robert and Pecora, Louis M. and Carroll, Thomas L.}, title = {Some elements for a history of the dynamical systems theory}, series = {Chaos : an interdisciplinary journal of nonlinear science}, volume = {31}, journal = {Chaos : an interdisciplinary journal of nonlinear science}, number = {5}, publisher = {AIP Publishing}, address = {Melville}, issn = {1054-1500}, doi = {10.1063/5.0047851}, pages = {20}, year = {2021}, abstract = {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.}, 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} } @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} }