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Регулярная и хаотическая динамика в «резиновой» модели волчка Чаплыгина
Нелинейная динамика. 2017. Т. 13. № 2. С. 277-297.
This paper is concerned with the rolling motion of a dynamically asymmetric unbalanced ball (Chaplygin top) in a gravitational field on a plane under the assumption that there is no slipping and spinning at the point of contact. We give a description of strange attractors existing in the system and discuss in detail the scenario of how one of them arises via a sequence of perioddoubling bifurcations. In addition, we analyze the dynamics of the system in absolute space and show that in the presence of strange attractors in the system the behavior of the point of contact considerably depends on the characteristics of the attractor and can be both chaotic and nearly quasi-periodic.
Research target:
Mathematics
Language:
Russian
Keywords: странный аттракторstrange attractornonholonomic constraintнеголономная связьChaplygin Topволчок Чаплыгинаrubber model
Publication based on the results of:
Kazakov A., Борисов А. В., Пивоварова Е. Н., Regular and Chaotic Dynamics 2016 Vol. 21 No. 7-8 P. 885-901
This paper is concerned with the rolling motion of a dynamically asymmetric unbalanced ball (Chaplygin top) in a gravitational field on a plane under the assumption that there is no slipping and spinning at the point of contact. We give a description of strange attractors existing in the system and discuss in detail the scenario ...
Added: January 30, 2017
Kazakov A., Известия высших учебных заведений. Радиофизика 2018 Т. 61 № 8-9 С. 729-738
In this paper, a new scenario of the appearance of mixed dynamics in two-dimensional reversible diffeomorphisms is proposed. The key point of the scenario is a sharp increase of the sizes of both strange attractor and strange repeller which appears due to heteroclinic bifurcations of the invariant manifolds of saddle fixed points belonging to these ...
Added: October 26, 2018
Бизяев И. А., Борисов А. В., Kazakov A., Нелинейная динамика 2016 Т. 12 № 2 С. 263-287
In this paper, we present some results on chaotic dynamics in the Suslov problem which describe the motion of a heavy rigid body with a fixed point, subject to a nonholonomic constraint, which is expressed by the condition that the projection of angular velocity onto the body-fixed axis is equal to zero. Depending on the ...
Added: October 29, 2016
Kazakov A., Гонченко А. С., Гонченко С. В. et al., Известия высших учебных заведений. Радиофизика 2018 Т. 61 № 10 С. 867-882
We study dynamical properties of a Celtic stone moving along the plane. Both one- and two-parameter families of the corresponding nonholonomic models are considered, in which bifurcations are studied that lead to changing types of stable motions of the stone as well as to the onset of chaotic dynamics. It is shown that multistability phenomena ...
Added: October 26, 2018
Kazakov A., Борисов А. В., Кузнецов С. П., Успехи физических наук 2014 Т. 184 № 5 С. 493-500
Based on the results of numerical simulations we discuss and illustrate dynamical phenomena characteristic for the rattleback, a solid body of convex surface moving on a rough horizontal plane, which are associated with the lack of conservation for the phase volume in the nonholonomic mechanical system. Due to local compression of the phase volume, behaviors ...
Added: October 22, 2015
Kazakov A., Баханова Ю. В., Коротков А. Г., Журнал Средневолжского математического общества 2017 Т. 19 № 2 С. 13-24
Investigations of spiral chaos in generalized Lotka-Volterra systems and Rosenzweig-MacArthur systems that describe the interaction of three species are made in this work. It is shown that in systems under study the spiral chaos appears in agreement with Shilnikov's scenario, that is when changing a parameter in system a stable limit cycle and a saddle-focus ...
Added: October 13, 2017
Kazakov A., Korotkov A., Levanova T. et al., IFAC-PapersOnLine 2018
We study the peculiarities of chaotic dynamics in the phenomenological model of the ensemble of two FitzHugh-Nagumo elements with weak excitatory couplings. This model was recently proposed as a suitable model for describing the behaviour of two coupled neurons. A rich diversity of different types of neuron-like behaviour, including regular in-phase, anti-phase, sequential spiking activities ...
Added: October 26, 2018
Kuryzhov E., Karatetskaia E., Mints D., Russian Journal of Nonlinear Dynamics 2021 Vol. 17 No. 2 P. 165-174
We consider the system of two coupled one-dimensional parabola maps. It is well known that the parabola map is the simplest map that can exhibit chaotic dynamics, chaos in this map appears through an infinite cascade of period-doubling bifurcations. For two coupled parabola maps we focus on studying attractors of two types: those which resemble ...
Added: September 8, 2021
Bizyaev I. A., Borisov A. V., Kazakov A., Regular and Chaotic Dynamics 2015 Vol. 20 No. 5 P. 605-626
In this paper, we present some results on chaotic dynamics in the Suslov problem which describe the motion of a heavy rigid body with a fixed point, subject to a nonholonomic constraint, which is expressed by the condition that the projection of angular velocity onto the body-fixed axis is equal to zero. Depending on the ...
Added: October 22, 2015
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
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
Kazakov A., Гонченко С. В., Гонченко А. С. et al., Известия высших учебных заведений. Прикладная нелинейная динамика 2017 Т. 25 № 2 С. 4-36
We consider important problems of modern theory of dynamical chaos and its applications. At present, it is customary to assume that in the finite-dimensional smooth dynamical systems three fundamentally different forms of chaos can be observed. This is the dissipative chaos, whose mathematical image is a strange attractor; the conservative chaos, for which the whole ...
Added: October 13, 2017
Revenko A., Kuznetsov S., Fundamenta Informaticae 2012 Vol. 4 No. 115 P. 377-394
An approach for studying relations between properties of functions on sets is proposed. The approach is based on Attribute Exploration. 16 properties of functions are considered, among them monotonicity, idempotency, path independence, exchange properties, convexity, etc. Example functions are partially computer generated on the powersets of sets with 2, 3 and 4 elements. Attribute Exploration ...
Added: December 31, 2012
Belomestny D., Iosipoi L., Mathematics and Computers in Simulation 2021 No. 181 P. 351-363
Markov Chain Monte Carlo methods become increasingly popular in applied mathematics as a tool for numerical integration with respect to complex and high-dimensional distributions. However, application of MCMC methods to heavy-tailed distributions and distributions with analytically intractable densities turns out to be rather problematic. In this paper, we propose a novel approach towards the use ...
Added: October 31, 2020
Popkov Y., Popkov A., Dubnov Y. A., Автоматика и телемеханика 2020 № 7 С. 148-172
A randomized forecasting method based on the generation of ensembles of entropy-optimal forecasting trajectories is developed. The latter are generated by randomized dynamic regression models containing random parameters, measurement noises, and a random input. The probability density functions of random parameters and measurement noises are estimated using real data within the randomized machine learning procedure. ...
Added: October 31, 2020
Malyshev D., Вестник Нижегородского университета им. Н.И. Лобачевского 2008 № 6 С. 141-146
Рассматривается понятие граничного класса, которое является полезным инструментом для анализа вычислительной сложности задач на графах. Исследуются два конкретных класса графов, и приводятся задачи, для которых эти классы являются граничными. ...
Added: August 31, 2012
Beklemishev L. D., Оноприенко А. А., Математический сборник 2015 Т. 206 № 9 С. 3-20
We formulate some term rewriting systems in which the number of computation steps is finite for each output, but this number cannot be bounded by a provably total computable function in Peano arithmetic PA. Thus, the termination of such systems is unprovable in PA. These systems are derived from an independent combinatorial result known as the Worm ...
Added: March 13, 2016
D. V. Gribanov, D.S. Malyshev, P. M. Pardalos et al., Journal of Combinatorial Optimization 2018 Vol. 35 No. 4 P. 1128-1146
In this paper, we present fixed-parameter tractable algorithms for special cases of the shortest lattice vector, integer linear programming, and simplex width computation problems, when matrices included in the problems’ formulations are near square. The parameter is the maximum absolute value of the rank minors in the corresponding matrices. Additionally, we present fixed-parameter tractable algorithms ...
Added: February 19, 2018
Kryuchkov M., Rusakov S. V., Вестник Ижевского государственного технического университета 2015 № 2(66) С. 110-112
This paper describes the results of testing the neuronal technical trend indicator according to the exchange rate of Brent oil in 2014. Testing of the model was carried out on three time series, which characterized by their features. ...
Added: August 31, 2015
Babash A. V., М. : ИНФРА-М, РИОР, 2013
Пособие предназначено для студентов высших учебных заведений, обучающихся по специальности «Прикладная информатика (в экономике)». Оно также содержит методический материал для ряда инновационных курсов лекций по профилю «Информационная безопасность» и может быть использовано и для блока дисциплин этого профиля. Ряд представленных результатов полезен специалистам и аспирантам, специализирующихся в указанной области. ...
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Feigin B. L., Finkelberg M. V., Rybnikov L. G. et al., Selecta Mathematica, New Series 2011 Vol. 17 No. 3 P. 573-607
Laumon moduli spaces are certain smooth closures of the moduli spaces of maps from the projective line to the flag variety of GLn. We construct the action of the Yangian of sln in the cohomology of Laumon spaces by certain natural correspondences. We construct the action of the affine Yangian (two-parametric deformation of the universal ...
Added: October 9, 2012
Malyshev D., Дискретный анализ и исследование операций 2012 Т. 19 № 4 С. 66-72
Рассматривается конструктивный подход к формированию новых случаев эффективной разрешимости задачи о независимом множестве в семействе наследственных частей множества графов Free({P5,C5}). Именно, доказывается, что если эта задача полиномиально разрешима в классе Free({P5,C5,G}), то для любого графа H, который может быть индуктивно получен из G применением к текущему графу сложения с K1 или умножения на K1, эта ...
Added: August 31, 2012
Vyalyi M., Дискретная математика 1991 Т. 3 № 3 С. 35-45
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Sirotkin D., Malyshev D., Дискретная математика 2017 Т. 29 № 3 С. 114-125
Задача о независимом множестве для заданного обыкновенного графа состоит в вычислении размера наибольшего множества его попарно несмежных вершин. Предлагается новый способ редукции графов. С его помощью получено новое доказательство NP-полноты задачи о независимом множестве в классе планарных графов и доказана NP-полнота данной задачи в классе плоских графов, имеющих только треугольные внутренние грани, с максимальной степенью ...
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