Alexey Kazakov
ul. Bolshaya Pecherskaya 25/12, Nizhny Novgorod, 603155, Russia
National Research University Higher School of Economics
Chief Researcher
National Research University Higher School of Economics
International Laboratory of Dynamical Systems and Applications
Born: 07.07.1987
2008: Bachelor's degree in Applied Mathematics and Informatics, Lobachevsky State University of Nizhny Novgorod
2010: Master's degree in Applied Mathematics and Informatics, Lobachevsky State University of Nizhny Novgorod
2011-2016: junior researcher, Udmurt State University, Institute of Computer Science
2014: Ph.D. (Candidate of Science) in physics and mathematics, National Research Nuclear University MEPhI, Moscow
since 2015: Senior Researcher, Leading Researcher, and then Chief Researcher, National Research University Higher School of Economics
2021: Doctor of Science in applied mathematics, National Research University Higher School of Economics, Moscow
Advisory board member in Chaos, Review Editor in Frontiers in Applied Mathematics and Statistics (in Dynamical Systems).
Publications:
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Borisov A. V., Kazakov A. O., Pivovarova E. N.
Abstract
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.
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Bizyaev I. A., Borisov A. V., Kazakov A. O.
Abstract
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 system parameters, we find cases of regular (in particular, integrable) behavior and detect various attracting sets (including strange attractors) that are typical of dissipative systems.We construct a chart of regimes with regions characterizing chaotic and regular regimes depending on the degree of conservativeness. We examine in detail the effect of reversal, which was observed previously in the motion of rattlebacks.
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Sataev I. R., Kazakov A. O.
Abstract
We study the dynamics in the Suslov problem which describes 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 system parameters, we find cases of regular (in particular, integrable) motions and, using a new method for constructing charts of Lyapunov exponents, detect different types of chaotic behavior such as conservative chaos, strange attractors and mixed dynamics, which are typical of reversible systems. In the paper we also examine the phenomenon of reversal, which was observed previously in the motion of Celtic stones.
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Borisov A. V., Kazakov A. O., Sataev I. R.
Abstract
We study both analytically and numerically the dynamics of an inhomogeneous ball on a rough horizontal plane under the infuence of gravity. A nonholonomic constraint of zero velocity at the point of contact of the ball with the plane is imposed. In the case of an arbitrary displacement of the center of mass of the ball, the system is nonintegrable without the property of phase volume conservation. We show that at certain parameter values the unbalanced ball exhibits the effect of reversal (the direction of the ball rotation reverses). Charts of dynamical regimes on the parameter plane are presented. The system under consideration exhibits diverse chaotic dynamics, in particular, the figure-eight chaotic attractor, which is a special type of pseudohyperbolic chaos.
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Vetchanin E. V., Kazakov A. O.
Abstract
This paper is concerned with the dynamics of two point vortices of the same intensity which are affected by an acoustic wave. Typical bifurcations of fixed points have been identified by constructing charts of dynamical regimes, and bifurcation diagrams have been plotted.
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Kazakov A. O.
Abstract
In this paper we study a problem of rolling of the dynamically asymmetric ball with displacement center of gravity on a plane without slipping and vertical rotating. It is shown that the dynamics of the ball is significantly affected by the type of reversibility. Depending on the type of the reversibility we found two different types of dynamical chaos: strange attractors and mixed chaotic dynamics. In this paper we describe a strange attractor development, and then its basic properties. A set of criteria by which in numerical experiments mixed dynamics may be distinguished from other types of dynamical chaos are given.
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Bizyaev I. A., Kazakov A. O.
Abstract
In this paper, we investigate the dynamics of systems describing the rolling without slipping and spinning (rubber rolling) of an ellipsoid on a plane and a sphere. We research these problems using Poincare maps, which investigation helps to discover a new integrable case.
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Bolsinov A. V., Kilin A. A., Kazakov A. O.
Abstract
The phenomenon of a topological monodromy in integrable Hamiltonian and nonholonomic systems is discussed. An efficient method for computing and visualizing the monodromy is developed. The comparative analysis of the topological monodromy is given for the rolling ellipsoid of revolution problem in two cases, namely, on a smooth and on a rough plane. The first of these systems is Hamiltonian, the second is nonholonomic. We show that, from the viewpoint of monodromy, there is no difference between the two systems, and thus disprove the conjecture by Cushman and Duistermaat stating that the topological monodromy gives a topological obstruction for Hamiltonization of the rolling ellipsoid of revolution on a rough plane.
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Gonchenko A. S., Gonchenko S. V., Kazakov A. O.
Abstract
We study chaotic dynamics of a nonholonomic model of celtic stone movement on the plane. Scenarious of appearance and development of chaos are investigated.
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