Alexey Kazakov
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
20112016: 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:
Gonchenko M. S., Kazakov A. O., Samylina E. A., Shykhmamedov A.
On 1:3 Resonance Under Reversible Perturbations of Conservative Cubic Hénon Maps
2022, vol. 27, no. 2, pp. 198216
Abstract
We consider reversible nonconservative perturbations of the conservative cubic Hénon maps $H_3^{\pm}: \bar x = y, \bar y = x + M_1 + M_2 y \pm y^3$ and study their influence on the 1:3 resonance, i.e., bifurcations of fixed points with eigenvalues $e^{\pm i 2\pi/3}$. It
follows from [1] that this resonance
is degenerate for $M_1=0$, $M_2=1$ when the corresponding
fixed point is elliptic. We show that bifurcations of this
point
under reversible perturbations give rise to four 3periodic orbits, two of them are symmetric
and conservative (saddles in the case of map $H_3^+$ and elliptic orbits in the case of map $H_3^$),
the other two orbits are nonsymmetric and they compose symmetric couples of dissipative orbits
(attracting and repelling orbits in the case of map $H_3^+$ and saddles with the Jacobians less
than 1 and greater than 1 in the case of map $H_3^$). We show that these local symmetrybreaking
bifurcations can lead to mixed dynamics due to accompanying global reversible bifurcations of
symmetric nontransversal homo and heteroclinic cycles. We also generalize the results
of [1] to the case of the $p:q$ resonances with odd $q$ and show that
all of them are also degenerate for the
maps $H_3^{\pm}$ with $M_1=0$.

Borisov A. V., Kazakov A. O., Pivovarova E. N.
Regular and Chaotic Dynamics in the Rubber Model of a Chaplygin Top
2016, vol. 21, no. 78, pp. 885901
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 quasiperiodic.

Borisov A. V., Kazakov A. O., Sataev I. R.
Spiral Chaos in the Nonholonomic Model of a Chaplygin Top
2016, vol. 21, no. 78, pp. 939954
Abstract
This paper presents a numerical study of the chaotic dynamics of a dynamically asymmetric unbalanced ball (Chaplygin top) rolling on a plane. It is well known that the dynamics of such a system reduces to the investigation of a threedimensional map, which in the general case has no smooth invariant measure. It is shown that homoclinic strange attractors of discrete spiral type (discrete Shilnikov type attractors) arise in this model for certain parameters. From the viewpoint of physical motions, the trace of the contact point of a Chaplygin top on a plane is studied for the case where the phase trajectory sweeps out a discrete spiral attractor. Using the analysis of the trajectory of this trace, a conclusion is drawn
about the influence of “strangeness” of the attractor on the motion pattern of the top.

Korotkov A. G., Kazakov A. O., Osipov G. V.
Sequential Dynamics in the Motif of Excitatory Coupled Elements
2015, vol. 20, no. 6, pp. 701715
Abstract
In this article a new model of motif (small ensemble) of neuronlike elements is proposed. It is built with the use of the generalized Lotka–Volterra model with excitatory couplings. The main motivation for this work comes from the problems of neuroscience where excitatory couplings are proved to be the predominant type of interaction between neurons of the brain. In this paper it is shown that there are two modes depending on the type of coupling between the elements: the mode with a stable heteroclinic cycle and the mode with a stable limit cycle. Our second goal is to examine the chaotic dynamics of the generalized threedimensional Lotka–Volterra model.

Bizyaev I. A., Borisov A. V., Kazakov A. O.
Dynamics of the Suslov Problem in a Gravitational Field: Reversal and Strange Attractors
2015, vol. 20, no. 5, pp. 605626
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 bodyfixed 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.

Borisov A. V., Kazakov A. O., Sataev I. R.
The Reversal and Chaotic Attractor in the Nonholonomic Model of Chaplygin’s Top
2014, vol. 19, no. 6, pp. 718733
Abstract
In this paper we consider the motion of a dynamically asymmetric unbalanced ball on a plane in a gravitational field. The point of contact of the ball with the plane is subject to a nonholonomic constraint which forbids slipping. The motion of the ball is governed by the nonholonomic reversible system of 6 differential equations. In the case of arbitrary displacement of the center of mass of the ball the system under consideration is a nonintegrable system without an invariant measure. Using qualitative and quantitative analysis we show that the unbalanced ball exhibits reversal (the phenomenon of reversal of the direction of rotation) for some parameter values. Moreover, by constructing charts of Lyaponov exponents we find a few types of strange attractors in the system, including the socalled figureeight attractor which belongs to the genuine strange attractors of pseudohyperbolic type.

Kazakov A. O.
Strange Attractors and Mixed Dynamics in the Problem of an Unbalanced Rubber Ball Rolling on a Plane
2013, vol. 18, no. 5, pp. 508520
Abstract
We consider the dynamics of an unbalanced rubber ball rolling on a rough plane. The term rubber means that the vertical spinning of the ball is impossible. The roughness of the plane means that the ball moves without slipping. The motions of the ball are described by a nonholonomic system reversible with respect to several involutions whose number depends on the type of displacement of the center of mass. This system admits a set of first integrals, which helps to reduce its dimension. Thus, the use of an appropriate twodimensional Poincaré map is enough to describe the dynamics of our system. We demonstrate for this system the existence of complex chaotic dynamics such as strange attractors and mixed dynamics. The type of chaotic behavior depends on the type of reversibility. In this paper we describe the development of a strange attractor and then its basic properties. After that we show the existence of another interesting type of chaos — the socalled mixed dynamics. In numerical experiments, a set of criteria by which the mixed dynamics may be distinguished from other types of dynamical chaos in twodimensional maps is given.

Gonchenko A. S., Gonchenko S. V., Kazakov A. O.
Richness of Chaotic Dynamics in Nonholonomic Models of a Celtic Stone
2013, vol. 18, no. 5, pp. 521538
Abstract
We study the regular and chaotic dynamics of two nonholonomic models of a Celtic stone. We show that in the first model (the socalled BMmodel of a Celtic stone) the chaotic dynamics arises sharply, during a subcritical period doubling bifurcation of a stable limit cycle, and undergoes certain stages of development under the change of a parameter including the appearance of spiral (Shilnikovlike) strange attractors and mixed dynamics. For the second model, we prove (numerically) the existence of Lorenzlike attractors (we call them discrete Lorenz attractors) and trace both scenarios of development and breakdown of these attractors.
