TY - JOUR A1 - Cestnik, Rok A1 - Pikovsky, Arkady T1 - Hierarchy of exact low-dimensional reductions for populations of coupled oscillators JF - Physical review letters N2 - We consider an ensemble of phase oscillators in the thermodynamic limit, where it is described by a kinetic equation for the phase distribution density. We propose an Ansatz for the circular moments of the distribution (Kuramoto-Daido order parameters) that allows for an exact truncation at an arbitrary number of modes. In the simplest case of one mode, the Ansatz coincides with that of Ott and Antonsen [Chaos 18, 037113 (2008)]. Dynamics on the extended manifolds facilitate higher-dimensional behavior such as chaos, which we demonstrate with a simulation of a Josephson junction array. The findings are generalized for oscillators with a Cauchy-Lorentzian distribution of natural frequencies. Y1 - 2022 U6 - https://doi.org/10.1103/PhysRevLett.128.054101 SN - 0031-9007 SN - 1079-7114 VL - 128 IS - 5 PB - American Physical Society CY - College Park ER - TY - JOUR A1 - Pikovsky, Arkady T1 - Scaling of energy spreading in a disordered Ding-Dong lattice JF - Journal of statistical mechanics: theory and experiment N2 - We study numerical propagation of energy in a one-dimensional Ding-Dong lattice composed of linear oscillators with elastic collisions. Wave propagation is suppressed by breaking translational symmetry, and we consider three ways to do this: position disorder, mass disorder, and a dimer lattice with alternating distances between the units. In all cases the spreading of an initially localized wavepacket is irregular, due to the appearance of chaos, and subdiffusive. For a range of energies and of weak and moderate levels of disorder, we focus on the macroscopic statistical characterization of spreading. Guided by a nonlinear diffusion equation, we establish that the mean waiting times of spreading obey a scaling law in dependence of energy. Moreover, we show that the spreading exponents very weakly depend on the level of disorder. KW - connections between chaos and statistical physics KW - nonlinear dynamics KW - transport properties Y1 - 2020 U6 - https://doi.org/10.1088/1742-5468/ab7e30 SN - 1742-5468 VL - 2020 IS - 5 PB - IOP Publishing Ltd. CY - Bristol ER - TY - JOUR A1 - Tönjes, Ralf A1 - Pikovsky, Arkady T1 - Low-dimensional description for ensembles of identical phase oscillators subject to Cauchy noise JF - Physical review : E, Statistical, nonlinear and soft matter physics N2 - We study ensembles of globally coupled or forced identical phase oscillators subject to independent white Cauchy noise. We demonstrate that if the oscillators are forced in several harmonics, stationary synchronous regimes can be exactly described with a finite number of complex order parameters. The corresponding distribution of phases is a product of wrapped Cauchy distributions. For sinusoidal forcing, the Ott-Antonsen low-dimensional reduction is recovered. Y1 - 2020 U6 - https://doi.org/10.1103/PhysRevE.102.052315 SN - 2470-0045 SN - 2470-0053 VL - 102 IS - 5 PB - American Physical Society CY - College Park ER - TY - JOUR A1 - Gong, Chen Chris A1 - Tönjes, Ralf A1 - Pikovsky, Arkady T1 - Coupled Möbius maps as a tool to model Kuramoto phase synchronization JF - Physical review : E, Statistical, nonlinear and soft matter physics N2 - We propose Mobius maps as a tool to model synchronization phenomena in coupled phase oscillators. Not only does the map provide fast computation of phase synchronization, it also reflects the underlying group structure of the sinusoidally coupled continuous phase dynamics. We study map versions of various known continuous-time collective dynamics, such as the synchronization transition in the Kuramoto-Sakaguchi model of nonidentical oscillators, chimeras in two coupled populations of identical phase oscillators, and Kuramoto-Battogtokh chimeras on a ring, and demonstrate similarities and differences between the iterated map models and their known continuous-time counterparts. Y1 - 2020 U6 - https://doi.org/10.1103/PhysRevE.102.022206 SN - 2470-0045 SN - 2470-0053 SN - 1063-651X SN - 2470-0061 SN - 1550-2376 VL - 102 IS - 2 PB - American Physical Society CY - College Park ER - TY - GEN A1 - Smirnov, Lev A. A1 - Bolotov, Maxim A1 - Bolotov, Dmitri A1 - Osipov, Grigory V. A1 - Pikovsky, Arkady T1 - Finite-density-induced motility and turbulence of chimera solitons T2 - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe N2 - We consider a one-dimensional oscillatory medium with a coupling through a diffusive linear field. In the limit of fast diffusion this setup reduces to the classical Kuramoto–Battogtokh model. We demonstrate that for a finite diffusion stable chimera solitons, namely localized synchronous domain in an infinite asynchronous environment, are possible. The solitons are stable also for finite density of oscillators, but in this case they sway with a nearly constant speed. This finite-density-induced motility disappears in the continuum limit, as the velocity of the solitons is inverse proportional to the density. A long-wave instability of the homogeneous asynchronous state causes soliton turbulence, which appears as a sequence of soliton mergings and creations. As the instability of the asynchronous state becomes stronger, this turbulence develops into a spatio-temporal intermittency. T3 - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - 1291 KW - chimera KW - soliton KW - finite-size effects Y1 - 2023 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-574281 SN - 1866-8372 IS - 1291 ER - TY - JOUR A1 - Chigarev, Vladimir A1 - Kazakov, Alexey A1 - Pikovsky, Arkady T1 - Kantorovich-Rubinstein-Wasserstein distance between overlapping attractor and repeller JF - Chaos : an interdisciplinary journal of nonlinear science N2 - We consider several examples of dynamical systems demonstrating overlapping attractor and repeller. These systems are constructed via introducing controllable dissipation to prototypic models with chaotic dynamics (Anosov cat map, Chirikov standard map, and incompressible three-dimensional flow of the ABC-type on a three-torus) and ergodic non-chaotic behavior (skew-shift map). We employ the Kantorovich-Rubinstein-Wasserstein distance to characterize the difference between the attractor and the repeller, in dependence on the dissipation level. Y1 - 2020 U6 - https://doi.org/10.1063/5.0007230 SN - 1054-1500 SN - 1089-7682 VL - 30 IS - 7 PB - American Institute of Physics CY - Melville ER - TY - JOUR A1 - Bolotov, Dmitry A1 - Bolotov, Maxim I. A1 - Smirnov, Lev A. A1 - Osipov, Grigory V. A1 - Pikovsky, Arkady T1 - Synchronization regimes in an ensemble of phase oscillators coupled through a diffusion field JF - Radiophysics and quantum electronics N2 - We consider an ensemble of identical phase oscillators coupled through a common diffusion field. Using the Ott-Antonsen reduction, we develop dynamical equations for the complex local order parameter and the mean field. The regions of the existence and stability are determined for the totally synchronous, partially synchronous, and asynchronous spatially homogeneous states. A procedure of searching for inhomogeneous states as periodic trajectories of an auxiliary system of the ordinary differential equations is demonstrated. A scenario of emergence of chimera structures from homogeneous synchronous solutions is described. Y1 - 2022 U6 - https://doi.org/10.1007/s11141-022-10173-4 SN - 0033-8443 SN - 1573-9120 VL - 64 IS - 10 SP - 709 EP - 725 PB - Springer CY - New York ER - TY - JOUR A1 - Smirnov, Lev A. A1 - Bolotov, Maxim A1 - Bolotov, Dmitri A1 - Osipov, Grigory V. A1 - Pikovsky, Arkady T1 - Finite-density-induced motility and turbulence of chimera solitons JF - New Journal of Physics N2 - We consider a one-dimensional oscillatory medium with a coupling through a diffusive linear field. In the limit of fast diffusion this setup reduces to the classical Kuramoto–Battogtokh model. We demonstrate that for a finite diffusion stable chimera solitons, namely localized synchronous domain in an infinite asynchronous environment, are possible. The solitons are stable also for finite density of oscillators, but in this case they sway with a nearly constant speed. This finite-density-induced motility disappears in the continuum limit, as the velocity of the solitons is inverse proportional to the density. A long-wave instability of the homogeneous asynchronous state causes soliton turbulence, which appears as a sequence of soliton mergings and creations. As the instability of the asynchronous state becomes stronger, this turbulence develops into a spatio-temporal intermittency. KW - chimera KW - soliton KW - finite-size effects Y1 - 2022 U6 - https://doi.org/10.1088/1367-2630/ac63d9 SN - 1367-2630 VL - 24 PB - IOP CY - London 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 -