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Igor Sataev

Zelenaya st., 38, Saratov, 410019, Russia
Institute of Radio Engineering and Electronics, Russian Academy of Sciences, Saratov Branch


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
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
Kuznetsov A. P., Migunova N. A., Sataev I. R., Sedova Y. V., Turukina L. V.
From Chaos to Quasi-Periodicity
2015, vol. 20, no. 2, pp.  189-204
Ensembles of several Rössler chaotic oscillators are considered. It is shown that a typical phenomenon for such systems is the emergence of different and sufficiently high dimensional invariant tori. The possibility of a quasi-periodic Hopf bifurcation and a cascade of such bifurcations based on tori of increasing dimension is demonstrated. The domains of resonance tori are revealed. Boundaries of these domains correspond to the saddle-node bifurcations. Inside the domains of resonance modes, torus-doubling bifurcations and destruction of tori are observed.
Keywords: chaos, quasi-periodic oscillation, invariant torus, Lyapunov exponent, bifurcation
Citation: Kuznetsov A. P., Migunova N. A., Sataev I. R., Sedova Y. V., Turukina L. V.,  From Chaos to Quasi-Periodicity, Regular and Chaotic Dynamics, 2015, vol. 20, no. 2, pp. 189-204
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
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
Borisov A. V., Jalnine A. Y., Kuznetsov S. P., Sataev I. R., Sedova Y. V.
Dynamical Phenomena Occurring due to Phase Volume Compression in Nonholonomic Model of the Rattleback
2012, vol. 17, no. 6, pp.  512-532
We study numerically the dynamics of the rattleback, a rigid body with a convex surface on a rough horizontal plane, in dependence on the parameters, applying methods used earlier for treatment of dissipative dynamical systems, and adapted here for the nonholonomic model. Charts of dynamical regimes on the parameter plane of the total mechanical energy and the angle between the geometric and dynamic principal axes of the rigid body are presented. Characteristic structures in the parameter space, previously observed only for dissipative systems, are revealed. A method for calculating the full spectrum of Lyapunov exponents is developed and implemented. Analysis of the Lyapunov exponents of the nonholonomic model reveals two classes of chaotic regimes. For the model reduced to a 3D map, the first one corresponds to a strange attractor with one positive and two negative Lyapunov exponents, and the second to the chaotic dynamics of quasi-conservative type, when positive and negative Lyapunov exponents are close in magnitude, and the remaining exponent is close to zero. The transition to chaos through a sequence of period-doubling bifurcations relating to the Feigenbaum universality class is illustrated. Several examples of strange attractors are considered in detail. In particular, phase portraits as well as the Lyapunov exponents, the Fourier spectra, and fractal dimensions are presented.
Keywords: rattleback, rigid body dynamics, nonholonomic mechanics, strange attractor, Lyapunov exponents, bifurcation, fractal dimension
Citation: Borisov A. V., Jalnine A. Y., Kuznetsov S. P., Sataev I. R., Sedova Y. V.,  Dynamical Phenomena Occurring due to Phase Volume Compression in Nonholonomic Model of the Rattleback, Regular and Chaotic Dynamics, 2012, vol. 17, no. 6, pp. 512-532
Kuznetsov A. P., Sataev I. R., Sedova Y. V.
Dynamics of Coupled Non-Identical Systems with Period-Doubling Cascade
2008, vol. 13, no. 1, pp.  9-18
We discuss the structure of bifurcation diagram in the plane of parameters controlling period-doublings for the system of coupled logistic maps. The analysis is carried out by computing the charts of dynamical regimes and charts of Lyapunov exponents giving showy and effective illustrations. The critical point of codimension two at the border of chaos is found. It is a terminal point for the Feigenbaum critical line. The bifurcation analysis in the vicinity of this point is presented.
Keywords: criticality, universality, transition to chaos, coupled maps, bifurcation, terminal point
Citation: Kuznetsov A. P., Sataev I. R., Sedova Y. V.,  Dynamics of Coupled Non-Identical Systems with Period-Doubling Cascade, Regular and Chaotic Dynamics, 2008, vol. 13, no. 1, pp. 9-18
Kuznetsov A. P., Kuznetsov S. P., Sataev I. R.
Codimension and typicity in a context of description of transition to chaos via period-doubling in dissipative dynamical systems
1997, vol. 2, nos. 3-4, pp.  90-105
While considering multiparameter families of nonlinear systems, types of behavior at the onset of chaos may appear which are distinct from Feigenbaum's universality. We present a review of such situations which can be met in families of one-dimensional maps and discuss a possibility of their realization and observation in nonlinear dissipative systems of more general form.
Citation: Kuznetsov A. P., Kuznetsov S. P., Sataev I. R.,  Codimension and typicity in a context of description of transition to chaos via period-doubling in dissipative dynamical systems, Regular and Chaotic Dynamics, 1997, vol. 2, nos. 3-4, pp. 90-105

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