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    Igor Sataev

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


    Sataev I. R., Kazakov A. O.
    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.
    Keywords: nonholonomic model, Chaplygin top, Afraimovich – Shilnikov torus-breakdown, cascade of period-doubling bifurcations, scenario of period doublings of tori, figure-eight attractor
    Citation: Sataev I. R., Kazakov A. O.,  Scenarios of transition to chaos in the nonholonomic model of a Chaplygin top, Rus. J. Nonlin. Dyn., 2016, Vol. 12, No. 2, pp.  235-250
    Kuznetsov A. P., Migunova N. A., Sataev I. R., Sedova Y. V., Turukina L. V.
    Ensembles of several chaotic R¨ossler oscillators are considered. It is shown that a typical phenomenon for such systems is the emergence of invariant tori of different and sufficiently high dimension. The possibility of a quasi-periodic Hopf bifurcation and of the cascade of such bifurcations based on tori of increasing dimension is demonstrated. The domains of resonant tori are revealed whose boundaries correspond to a saddle-node bifurcation. Within areas of resonant modes the torus-doubling bifurcations and tori destruction are observed.
    Keywords: chaos, quasiperiodic oscillations, invariant tori, Lyapunov exponents, bifurcations
    Citation: Kuznetsov A. P., Migunova N. A., Sataev I. R., Sedova Y. V., Turukina L. V.,  Dynamics of coupled chaotic oscillators: from chaos to quasiperiodicity, Rus. J. Nonlin. Dyn., 2014, Vol. 10, No. 4, pp.  387-405
    Borisov A. V., Kazakov A. O., Sataev I. R.
    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.
    Keywords: Chaplygin’s top, rolling without slipping, reversibility, involution, integrability, reverse, chart of dynamical regimes, strange attractor
    Citation: Borisov A. V., Kazakov A. O., Sataev I. R.,  Regular and Chaotic Attractors in the Nonholonomic Model of Chapygin's ball, Rus. J. Nonlin. Dyn., 2014, Vol. 10, No. 3, pp.  361-380
    Isaeva O. B., Kuznetsov S. P., Sataev I. R., Pikovsky A.
    We describe one possible scenario of destruction or of a birth of the hyperbolic attractors considering the Smale—Williams solenoid as an example. The content of the transition observed under variation of the control parameter is the pairwise merge of the orbits belonging to the attractor and to the unstable invariant set on the border of the basin of attraction, in the course of the set of bifurcations of the saddle-node type. The transition is not a single event, but occupies a finite interval on the control parameter axis. In an extended space of the state variables and the control parameter this scenario can be regarded as a mutual transformation of the stable and unstable solenoids one to each other. Several model systems are discussed manifesting this scenario e.g. the specially designed iterative maps and the physically realizable system of coupled alternately activated non-autonomous van der Pol oscillators. Detailed studies of inherent features and of the related statistical and scaling properties of the scenario are provided.
    Keywords: strange attractor, chaos, bifurcation, self-sustained oscillator, hyperbolic chaos
    Citation: Isaeva O. B., Kuznetsov S. P., Sataev I. R., Pikovsky A.,  On a bifurcation scenario of a birth of attractor of Smale–Williams type, Rus. J. Nonlin. Dyn., 2013, Vol. 9, No. 2, pp.  267-294
    Kuznetsov A. P., Turukina L. V., Kuznetsov S. P., Sataev I. R.
    The conditions are discussed for which the ensemble of interacting oscillators may demonstrate Landau–Hopf scenario of successive birth of multi-frequency regimes. A model is proposed in the form of a network of five globally coupled oscillators, characterized by varying degree of excitement of individual oscillators. Illustrations are given for the birth of the tori of increasing dimension by successive quasi-periodic Hopf bifurcation.
    Keywords: synchronization, bifurcations, quasi-periodic dynamics, chaos
    Citation: Kuznetsov A. P., Turukina L. V., Kuznetsov S. P., Sataev I. R.,  Landau–Hopf scenario in the ensemble of interacting oscillators, Rus. J. Nonlin. Dyn., 2012, Vol. 8, No. 5, pp.  863-873
    Kuznetsov S. P., Jalnine A. Y., Sataev I. R., Sedova Y. V.
    We perform a numerical study of the motion of the rattleback, a rigid body with a convex surface on a rough horizontal plane in dependence on the parameters, applying the methods used previously for the treatment of dissipative dynamical systems, and adapted for the nonholonomic model. Charts of dynamical regimes are presented on the parameter plane of the total mechanical energy and the angle between the geometric and dynamic principal axes of the rigid body. Presence of characteristic structures in the parameter space, previously observed only for dissipative systems, is demonstrated. A method of calculating for the full spectrum of Lyapunov exponents is developed and implemented. It is shown that analysis of the Lyapunov exponents of chaotic regimes of the nonholonomic model reveals two classes, one of which is typical for relatively high energies, and the second for the relatively small energies. For the model reduced to a three-dimensional 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 the quasiconservative type, with close in magnitude positive and negative Lyapunov exponents, and the rest one about zero. The transition to chaos through a sequence of period-doubling bifurcations is illustrated, and the observed scaling corresponds to that intrinsic to the dissipative systems. A study of strange attractors is provided, in particularly, phase portraits are presented as well as the Lyapunov exponents, the Fourier spectra, the results of calculating the fractal dimensions.
    Keywords: rattleback, rigid body dynamics, nonholonomic mechanics, strange attractor, Lyapunov exponents, bifurcation, fractal dimension
    Citation: Kuznetsov S. P., Jalnine A. Y., Sataev I. R., Sedova Y. V.,  Phenomena of nonlinear dynamics of dissipative systems in nonholonomic mechanics of the rattleback, Rus. J. Nonlin. Dyn., 2012, Vol. 8, No. 4, pp.  735-762
    Kuznetsov A. P., Sataev I. R., Turukina L. V.
    The problem of external driving by the harmonic signal of two coupled self-oscillators is investigated. Comparison with the synchronization picture for phase oscillators is given. We discuss the configuration of periodic, two- and three-frequency regimes in the parameter space of external signal. The illustrations of three-frequency tori and resonance two-frequency tori are given. A number of significant differences from the bifurcation mechanisms for the destruction of synchronization are found compared with the case of phase oscillators.
    Keywords: synchronization, bifurcations, quasi-periodic dynamics, chaos
    Citation: Kuznetsov A. P., Sataev I. R., Turukina L. V.,  Forced synchronization of two coupled van der Pol self-oscillators, Rus. J. Nonlin. Dyn., 2011, Vol. 7, No. 3, pp.  411-425
    Kuznetsov A. P., Sataev I. R., Turukina L. V.
    The problem of the dynamics of phase oscillators is discussed with an increasing their numbers. We discuss the organization of the parameters plane responsible for the frequency detunings of the oscillators and amplitude of the dissipative coupled. The region of complete synchronization, quasi-periodic oscillations of different dimension and chaos are are observed. We discuss the changing of the synchronization picture with an increasing of the number of oscillators in the chain. We use the method of charts of Lyapunov exponents and modification of the method of charts of dynamical regimes visualized two-frequency resonant tori of different types.
    Keywords: synchronization, phase oscillators, quasi-periodical dynamics, chaos
    Citation: Kuznetsov A. P., Sataev I. R., Turukina L. V.,  Synchronization and multi-frequency oscillations in the chain of phase oscillators, Rus. J. Nonlin. Dyn., 2010, Vol. 6, No. 4, pp.  693-717
    Kuznetsov A. P., Kuznetsov S. P., Sataev I. R., Sedova Y. V.
    In paper we suggest an example of system which dynamics is answered to conception of a «critical quasi-attractor». Besides the brief review of earlier obtained results the new results are presented, namely the illustrations of scaling for basins of attraction of elements of critical quasi-attractor, the renormalization group approach in the presence of additive uncorrelated noise, the calculation of universal constant responsible for the scaling regularities of the noise effect, the illustrations of transitions initialized by noise that are realized between coexisted attractors.
    Keywords: quasi-attractor, renormalization group method, type of criticality, bifurcation, scaling, noise
    Citation: Kuznetsov A. P., Kuznetsov S. P., Sataev I. R., Sedova Y. V.,  Critical point of accumulation of fold-flip bifurcation points and critical quasi-attractor (the review and new results), Rus. J. Nonlin. Dyn., 2008, Vol. 4, No. 2, pp.  113-132

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