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2013
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# 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:

 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.  198-216 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 3-periodic 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 symmetry-breaking 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 $M_1=0$. Keywords: cubic Hénon map, reversible system, 1:3 resonance, homoclinic tangencies, mixed dynamics Citation: Gonchenko M. S., Kazakov A. O., Samylina E. A., Shykhmamedov A.,  On 1:3 Resonance Under Reversible Perturbations of Conservative Cubic Hénon Maps, Regular and Chaotic Dynamics, 2022, vol. 27, no. 2, pp. 198-216 DOI:10.1134/S1560354722020058
 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. 7-8, pp.  885-901 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 period-doubling 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. Keywords: Chaplygin top, nonholonomic constraint, rubber model, strange attractor, bifurcation, trajectory of the point of contact Citation: Borisov A. V., Kazakov A. O., Pivovarova E. N.,  Regular and Chaotic Dynamics in the Rubber Model of a Chaplygin Top, Regular and Chaotic Dynamics, 2016, vol. 21, no. 7-8, pp. 885-901 DOI:10.1134/S156035471607011X
 Borisov A. V., Kazakov A. O., Sataev I. R. Spiral Chaos in the Nonholonomic Model of a Chaplygin Top 2016, vol. 21, no. 7-8, pp.  939-954 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 three-dimensional 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. Keywords: nonholonomic constraint, spiral chaos, discrete spiral attractor Citation: Borisov A. V., Kazakov A. O., Sataev I. R.,  Spiral Chaos in the Nonholonomic Model of a Chaplygin Top, Regular and Chaotic Dynamics, 2016, vol. 21, no. 7-8, pp. 939-954 DOI:10.1134/S1560354716070157
 Korotkov A. G., Kazakov A. O., Osipov G. V. Sequential Dynamics in the Motif of Excitatory Coupled Elements 2015, vol. 20, no. 6, pp.  701-715 Abstract In this article a new model of motif (small ensemble) of neuron-like 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 three-dimensional Lotka–Volterra model. Keywords: Neuronal motifs, Lotka–Volterra model, heteroclinic cycle, period-doubling bifurcation, Feigenbaum scenario, strange attractor, Lyapunov exponents Citation: Korotkov A. G., Kazakov A. O., Osipov G. V.,  Sequential Dynamics in the Motif of Excitatory Coupled Elements, Regular and Chaotic Dynamics, 2015, vol. 20, no. 6, pp. 701-715 DOI:10.1134/S1560354715060064
 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.  605-626 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. Keywords: Suslov problem, nonholonomic constraint, reversal, strange attractor Citation: Bizyaev I. A., Borisov A. V., Kazakov A. O.,  Dynamics of the Suslov Problem in a Gravitational Field: Reversal and Strange Attractors, Regular and Chaotic Dynamics, 2015, vol. 20, no. 5, pp. 605-626 DOI:10.1134/S1560354715050056
 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.  718-733 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 so-called figure-eight attractor which belongs to the genuine strange attractors of pseudohyperbolic type. Keywords: rolling without slipping, reversibility, involution, integrability, reversal, chart of Lyapunov exponents, strange attractor Citation: Borisov A. V., Kazakov A. O., Sataev I. R.,  The Reversal and Chaotic Attractor in the Nonholonomic Model of Chaplygin’s Top, Regular and Chaotic Dynamics, 2014, vol. 19, no. 6, pp. 718-733 DOI:10.1134/S1560354714060094
 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.  508-520 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 two-dimensional 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 so-called mixed dynamics. In numerical experiments, a set of criteria by which the mixed dynamics may be distinguished from other types of dynamical chaos in two-dimensional maps is given. Keywords: mixed dynamics, strange attractor, unbalanced ball, rubber rolling, reversibility, twodimensional Poincaré map, bifurcation, focus, saddle, invariant manifolds, homoclinic tangency, Lyapunov’s exponents Citation: Kazakov A. O.,  Strange Attractors and Mixed Dynamics in the Problem of an Unbalanced Rubber Ball Rolling on a Plane, Regular and Chaotic Dynamics, 2013, vol. 18, no. 5, pp. 508-520 DOI:10.1134/S1560354713050043
 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.  521-538 Abstract We study the regular and chaotic dynamics of two nonholonomic models of a Celtic stone. We show that in the first model (the so-called BM-model 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 (Shilnikov-like) strange attractors and mixed dynamics. For the second model, we prove (numerically) the existence of Lorenz-like attractors (we call them discrete Lorenz attractors) and trace both scenarios of development and break-down of these attractors. Keywords: celtic stone, nonholonomic model, strange attractor, discrete Lorenz attractor, Shilnikov-like spiral attractor, mixed dynamics Citation: Gonchenko A. S., Gonchenko S. V., Kazakov A. O.,  Richness of Chaotic Dynamics in Nonholonomic Models of a Celtic Stone, Regular and Chaotic Dynamics, 2013, vol. 18, no. 5, pp. 521-538 DOI:10.1134/S1560354713050055