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## Динамика перемежающихся функций

Доклады Российской академии наук. Математика, информатика, процессы управления (ранее - Доклады Академии Наук. Математика). 2015. Т. 461. № 2. С. 1-5.

In press

We study the functional properties of the concept of interlacing introduced by I.M. Gelfand and show that in the context of collective random walks, this property leads to synchronization.

Dmitrichev A., Zakharov D., Nekorkin V., Radiophysics and Quantum Electronics 2017 Vol. 60 No. 6 P. 506-512

We study stability of a synchronous regime in hub clusters of the power networks, which are simulated by ensembles of phase oscillators. An approach allowing one to estimate the regions in the parameter space, which correspond to the global asymptotic stability of this regime, is presented. The method is illustrated by an example of a ...

Added: October 4, 2018

Bizyaev I. A., Борисов А. В., Мамаев И. С., Труды Математического института им. В.А. Стеклова РАН 2016 Т. 294 С. 268-292

This paper is concerned with the nonholonomic Suslov problem and its generalization proposed by Chaplygin. The issue of the existence of an invariant measure with singular density (having singularities at some points of phase space) is discussed. ...

Added: April 4, 2017

Blank M., Доклады Академии наук 2013 Т. 448 № 6 С. 629-632

We give conditions for unique ergodicity for a discrete time collective
random walk on a continuous circle. Individual particles in this collective
motion perform independent (and different) random walks conditioned
by the assumption that the particles cannot overrun each other.
Deterministic version of this system is studied as well. ...

Added: November 25, 2014

Skripchenko A., Hubert P., Avila A., / Cornell University. Series math "arxiv.org". 2014. No. 1412.7913.

We study chaotic plane sections of some particular family of triply periodic surfaces. The question about possible behavior of such sections was posed by S. P. Novikov. We prove some estimations on diffusion rate of these sections using the connection between Novikov's problem and systems of isometries - some natural generalization of interval exchange transformations. ...

Added: January 27, 2015

Manita A., Lobachevskii Journal of Mathematics 2017 Vol. 38 No. 5 P. 948-953

We introduce a class of stochastic networks in which synchronization between nodes is modelled by a message passing mechanism with heterogeneous Markovian routing. We present a series of results about probability distribution related to steady states of such models. ...

Added: June 20, 2017

Blank M., Problems of Information Transmission 2014 Vol. 50 No. 4 P. 350-363

We study functional consequences of the interlacing property consisting in
that a new configuration of “particles” occurs in gaps between elements of
a previous configuration. This property was introduced by I.M. Gelfand in
terms of spectra of sequences of matrices of increasing dimensions and turned
out to be highly needed in many areas of modern mathematics. We examine
conditions under ...

Added: March 20, 2015

Bolotov M., Smirnov L., Osipov G. et al., Physical Review E - Statistical, Nonlinear, and Soft Matter Physics 2020 Vol. 102 No. 4 Article 042218

We study how a chimera state in a one-dimensional medium of nonlocally coupled oscillators responds to a homogeneous in space periodic in time external force. On a macroscopic level, where a chimera can be considered as an oscillating object, forcing leads to entrainment of the chimera’s basic frequency inside an Arnold tongue. On a mesoscopic ...

Added: October 31, 2020

Volk D., Kleptsyn V., Gorodetski A. et al., Moscow Mathematical Journal 2014 Vol. 14 No. 2 P. 291-308

We consider a minimal action of a finitely generated semigroup by homeomorphisms of the circle, and show that the collection of translation numbers of individual elements completely determines the set of generators (up to a common continuous change of coordinates). One of the main tools used in the proof is the synchronization properties of random ...

Added: December 30, 2015

Dogonasheva O., Радушев Д. О., Гуткин Б. С. et al., Physical Review E - Statistical, Nonlinear, and Soft Matter Physics 2023

Methods that distinguish dynamical regimes in networks of active elements make it possible to design the dynamics of models of realistic networks. A particularly salient example is partial synchronization, which may play a pivotal role in elucidating the dynamics of biological neural networks. Such emergent partial synchronization in structurally homogeneous networks is commonly denoted as ...

Added: December 12, 2023

Zakharov D., Krupa M., Гуткин Б. С., Communications in Nonlinear Science and Numerical Simulation 2020 Vol. 82 P. 105086

Gamma rhythm (20-100 Hz) plays a key role in numerous cognitive tasks: working memory, sensory processing and in routing of information across neural circuits. In comparison with lower frequency oscillations in the brain, gamma-rhythm associated firing of the individual neurons is sparse and the activity is locally distributed in the cortex. Such “weak” gamma rhythm ...

Added: October 23, 2019

Blank M., Nonlinearity 2012 Vol. 25 No. 12 P. 3389-3408

We study ergodic properties of a family of traffic maps acting in
the space of bi-infinite sequences of real numbers. The corresponding
dynamics mimics the motion of vehicles in a simple traffic flow, which
explains the name. Using connections to topological Markov chains we obtain
nontrivial invariant measures, prove their stochastic stability, and
calculate the topological entropy. Technically these results ...

Added: November 26, 2014

Blank M., Russian Mathematical Surveys 2016 Vol. 71 No. 3 P. 588-590

We present sufficient (and in some cases necessary) conditions under
which the time average along a trajectory of a measurable dynamical
system coincides with the space average for almost all initial points
with respect to a given reference measure (rather than to an ergodic one,
which may not exist in general). ...

Added: November 13, 2016

Bizyaev I. A., Borisov A., Mamaev I., Symmetry, Integrability and Geometry: Methods and Applications (SIGMA) 2016 Vol. 12 P. 1-19

In this paper, using the Hojman construction, we give examples of various Poisson brackets which differ from those which are usually analyzed in Hamiltonian mechanics. They possess a nonmaximal rank, and in the general case an invariant measure and Casimir functions can be globally absent for them. ...

Added: April 5, 2017

Blank M., Nonlinearity 2014 Vol. 27 No. 5 P. 953-971

We discuss conditions for unique ergodicity of a collective random walk on a continuous circle. Individual particles in this collective motion perform independent (and different in general) random walks conditioned by the assumption that the particles cannot overrun each other. Additionally to sufficient conditions for the unique ergodicity we discover a new and unexpected way ...

Added: November 21, 2014

Zaev D., / Cornell University. Series math "arxiv.org". 2015.

We consider L^p-Wasserstein distances on a subset of probability measures. If the subset of interest appears to be a simplex, these distances are determined by their values on extreme points of the simplex. We show that this fact is a corollary of the following decomposition result: an optimal transport plan can be represented as a mixture ...

Added: May 25, 2015

Bogachev V., Veretennikov A., Shaposhnikov S., Доклады Российской академии наук. Математика, информатика, процессы управления (ранее - Доклады Академии Наук. Математика) 2015 Т. 460 № 5 С. 507-511

Методами уравнений в частных производных установлены достаточные условия дифференцируемости инвариантных мер диффузионных процессов по параметру ...

Added: October 11, 2015

Пиковский А., Physical Review E - Statistical, Nonlinear, and Soft Matter Physics 2019 Vol. 100 No. 062210 P. 1-10

The Kuramoto model, despite its popularity as a mean-field theory for many synchronization phenomenon of oscillatory systems, is limited to a first-order harmonic coupling of phases. For higher-order coupling, there only exists a low-dimensional theory in the thermodynamic limit. In this paper, we extend the formulation used by Watanabe and Strogatz to obtain a low-dimensional ...

Added: October 31, 2020

O. Dogonasheva, Kasatkin D., Boris Gutkin et al., Chaos 2022 Vol. 32 No. 10 Article 101101

Over the past decades, one of the most exciting and fast developed area of modern synchronization theory is the study of chimera states. Such chimera states states are characterized by the coexistence of multiple synchronous and asynchronous domains, despite that the network topology does not at all predict such structures. Moreover, these states are of interest ...

Added: September 16, 2022

Zakharov D., Гуткин Б. С., Krupa M. et al., Physical Review E - Statistical, Nonlinear, and Soft Matter Physics 2018 Vol. 97 No. 6 P. 062211-1-062211-5

We analyzed a generic relaxation oscillator under moderately strong forcing at a frequency much greater that the natural intrinsic frequency of the oscillator. Additionally, the forcing is of the same sign and, thus, has a nonzero average, matching neuroscience applications. We found that, first, the transition to high-frequency synchronous oscillations occurs mostly through periodic solutions ...

Added: October 4, 2018

Munyaev V., Smirnov L., Kostin V. et al., New Journal of Physics 2020 Vol. 22 Article 023036

We study populations of globally coupled noisy rotators(oscillators with inertia) allowing a nonequilibrium transition from a desynchronized state to a synchronous one (with the nonvanishing order parameter). The newly developed analytical approaches resulted in solutions describing the synchronous state with constant order parameter for weakly inertial rotators, including the case of zero inertia, when the ...

Added: October 31, 2020

Кузнецов А. П., Stankevich N., Щеголева Н. А., Известия высших учебных заведений. Прикладная нелинейная динамика 2021 Т. 29 № 1 С. 136-159

The purpose of this study is to describe the complete picture of synchronization of two coupled generators of quasi-periodic oscillations, to classify various types of synchronization, to study features of occurrence and destruction of multi-frequency quasi-periodic oscillations. Methods. The object of the research is systems of ordinary differential equations of various dimensions. The work uses the fourth-order Runge–Kutta ...

Added: February 2, 2021

Shaposhnikov S., Bogachev V., Агафонцев Б. В., Доклады Академии наук 2011 Т. 438 № 3 С. 295-299

Получены точные условия положительности плотности инвариантной меры диффузионного процесса. ...

Added: October 14, 2014

Kuznetsov A., Kuznetsov S., Shchegoleva N. et al., Physica D: Nonlinear Phenomena 2019 Vol. 398 P. 1-12

A problem of synchronization of quasiperiodic oscillations is discussed in application to an example of coupled systems with autonomous quasiperiodic dynamics. Charts of Lyapunov exponents are presented that reveal characteristic domains on the parameter plane such as oscillator death, complete synchronization, phase synchronization of quasiperiodic oscillations, broadband synchronization, broadband quasiperiodicity. Features of each kind of ...

Added: December 2, 2019

Bizyaev I. A., Borisov A., Mamaev I., Regular and Chaotic Dynamics 2016 Vol. 21 No. 1 P. 136-146

This paper is concerned with the motion of the Chaplygin sleigh on the surface of a circular cylinder. In the case of inertial motion, the problem reduces to the study of the dynamical system on a (two-dimensional) torus and to the classification of singular points. Particular cases in which the system admits an invariant measure ...

Added: April 5, 2017