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## Diffusion for chaotic sections of 3-periodic surfaces

Cornell University
,
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. Using thermodynamical formalism, we construct an invariant measure for systems of isometries of a special class called the Rauzy gasket, and investigate main properties of the Lyapunov spectrum of the corresponding suspension flow.

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

Karatetskaia E., Шыхмамедов А. И., Kazakov A., Chaos 2021 Vol. 31 Article 011102

A Shilnikov homoclinic attractor of a three-dimensional diffeomorphism contains a saddle-focus fixed point with a two-dimensional unstable invariant manifold and homoclinic orbits to this saddle-focus. The orientation-reversing property of the diffeomorphism implies a symmetry between two branches of the one-dimensional stable manifold. This symmetry leads to a significant difference between Shilnikov attractors in the orientation-reversing ...

Added: September 8, 2021

Stankevich N., Kuznetsov A., Popova E. et al., Nonlinear Dynamics 2019 Vol. 97 P. 2355-2370

Using an example of a radiophysical generator model, scenarios for the formation of various chaotic attractors are described, including chaos and hyperchaos. It is shown that as a result of a secondary Neimark–Sacker bifurcation, a hyperchaos with two positive Lyapunov exponents can occur in the system. A comparative analysis of chaotic attractors born as a ...

Added: December 2, 2019

Stankevich N., Volkov E., Nonlinear Dynamics 2018 Vol. 94 No. 4 P. 2455-2467

The emergence of multistability in a simplethree-dimensionalautonomousoscillatorisinvestigatedusingnumericalsimulations,calculationsofLyapunov exponents and bifurcation analysis over a broad area of two-dimensional plane of control parameters. Using Neimark–Sacker bifurcation of 1:1 limit cycle asthestartingregime,manyparameterislandswiththe coexisting attractors were detected in the phase diagram,includingthecoexistenceoftorus,resonantlimit cycles and chaos; and transitions between the regimes were considered in detail. The overlapping between resonant limit cycles ...

Added: December 2, 2019

Stankevich N., Shchegoleva N. A., Sataev I. R. et al., Journal of Computational and Nonlinear Dynamics 2020 Vol. 15 No. 11 P. 111001

Using an example a system of two coupled generators of quasiperiodic oscillations, we study the occurrence of chaotic dynamics with one positive, two zero and several negative Lyapunov exponents. It is shown that such dynamic arises as a result of a sequence of bifurcations of two-frequency torus doubling and involve saddle tori occurring at their ...

Added: September 4, 2020

Avila A., Hubert P., Skripchenko A., Bulletin de la Société Mathématique de France 2016 Vol. 144 No. 3 P. 539-568

In this paper, we prove that the Hausdor dimension of the Rauzy gasket is less than 2. By this result, we answer a question addressed by Pierre Arnoux. Also, this question is a very particular case of the conjecture stated by S.P. Novikov and A. Maltsev in 2003. ...

Added: November 9, 2016

Kruglov V., Krylosova D., Sataev I. R. et al., Chaos 2021 Vol. 31 No. 7 Article 073118

Transition to chaos via the destruction of a two-dimensional torus is studied numerically using an example of the Hénon map and the Toda oscillator under quasiperiodic forcing and also experimentally using an example of a quasi-periodically excited RL–diode circuit. A feature of chaotic dynamics in these systems is the fact that the chaotic attractor in ...

Added: July 15, 2021

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

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

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. ...

Added: March 20, 2015

Stankevich N., Kuznetsov A. P., Seleznev E. P., Chaos, Solitons and Fractals 2021 Vol. 147 Article 110998

Appearance of chaotic dynamics as a result of multi-frequency tori destruction is carried out on the example of a model of a multimode generator. Quasiperiodic bifurcations occurring with multi-frequency tori are discussed in the context of the Landau-Hopf scenario. Structure of the parameter space is studied, areas with various chaotic dynamics, including chaos and hyperchaos, ...

Added: May 12, 2021

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

Avila A., Hubert P., Skripchenko A., Inventiones Mathematicae 2016 Vol. 206 No. 1 P. 109-146

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 the diffusion rate of these sections using the connection between Novikov’s problem and systems of isometries—some natural generalization of interval exchange transformations. Using ...

Added: November 9, 2016

Stankevich N., Kazakov A., Gonchenko S., Chaos 2020 Vol. 30 Article 123129

The generalized four-dimensional Rössler system is studied. Main bifurcation scenarios leading to a hyperchaos are described phenomenologically and their implementation in the model is demonstrated. In particular, we show that the formation of hyperchaotic invariant sets is related mainly to cascades (finite or infinite) of nondegenerate bifurcations of two types: period-doubling bifurcations of saddle cycles with a ...

Added: January 18, 2021

Kazakov A., Козлов А. Д., Журнал Средневолжского математического общества 2018 Т. 20 № 2 С. 187-198

In the paper a new method of constructing of three-dimensional flow systems with different chaotic attractors is presented. Using this method, an example of three-dimensional system possessing an asymmetric Lorenz attractor is obtained. Unlike the classical Lorenz attractor, the observed attractor does not have symmetry. However, the discovered asymmetric attractor, as well as classical one, ...

Added: October 26, 2018

Skripchenko A., Hubert P., Avila A., / Cornell University. Series math "arxiv.org". 2013. No. 1311.5361.

In this paper, we answer a question addressed by Pierre Arnoux. We prove that the Hausdorff dimension of the Rauzy gasket is less than 2. Also, this question is a very particular case of the conjecture stated by S.P. Novikov and A. Maltsev in 2003. ...

Added: March 3, 2014

Fougeron C., Skripchenko A., Monatshefte fur Mathematik 2021 Vol. 194 No. 4 P. 767-787

We introduce a new strategy to prove simplicity of the spectrum of Lyapunov exponents that can be applied to a wide class of Markovian multidimensional continued fraction algorithms. As an application we use it for Selmer algorithm in dimension 2 and for the Triangle sequence algorithm and show that these algorithms are not optimal.
There is ...

Added: February 10, 2021

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

A. Kilina, Panteleeva P., Stankevich N., Communications in Nonlinear Science and Numerical Simulation 2024 Vol. 135 Article 108041

A non-autonomous model of the Anishchenko–Astakhov generator in the regime of periodic and chaotic self-oscillations is considered. A periodic sequence of short pulses is considered as an external force. It is shown that the synchronization picture is close in structure to the classical synchronization picture observed in a two-dimensional system, but the pulse action leads ...

Added: May 3, 2024

Stankevich N., Volkov E., Chaos 2021 Vol. 31 No. 10 Article 103112

We investigate the dynamics of three identical three-dimensional ring synthetic genetic oscillators (repressilators) located in different cells and indirectly globally coupled by quorum sensing whereby it is meant that a mechanism in which special signal molecules are produced that, after the fast diffusion mixing and partial dilution in the environment, activate the expression of a ...

Added: October 12, 2021

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

Kuptsov P., Kuptsova A. V., Stankevich N., Russian Journal of Nonlinear Dynamics 2021 Vol. 17 No. 1 P. 5-21

We suggest a universal map capable of recovering the behavior of a wide range of dynamical systems given by ODEs. The map is built as an artificial neural network whose weights encode a modeled system. We assume that ODEs are known and prepare training datasets using the equations directly without computing numerical time series. Parameter ...

Added: April 3, 2021

Kuznetsov A. P., Sedova Y. V., Stankevich N., Chaos, Solitons and Fractals 2024 Vol. 186 Article 115237

We study numerically the dynamics of low–dimensional ensembles of discrete neuron models - Chialvo maps. We are focused on choosing the autonomous map parameters corresponding to the invariant curve. We consider two cases of coupling organization: (i) via a nonlinear function of models; (ii) linear coupling, which is an analog of electrical neuron interaction. For ...

Added: July 10, 2024

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