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    Alexey Jalnine

    Zelenaya 38, Saratov, 410019, Russia
    Saratov Branch of Kotelnikov’s Institute of Radio-Engineering and Electronics of RAS


    Jalnine A. Y., Kuznetsov S. P.
    We investigate strange nonchaotic self-oscillations in a dissipative system consisting of three mechanical rotators driven by a constant torque applied to one of them. The external driving is nonoscillatory; the incommensurable frequency ratio in vibrational-rotational dynamics arises due to an irrational ratio of diameters of the rotating elements involved. It is shown that, when losing stable equilibrium, the system can demonstrate two- or three-frequency quasi-periodic, chaotic and strange nonchaotic self-oscillations. The conclusions of the work are confirmed by numerical calculations of Lyapunov exponents, fractal dimensions, spectral analysis, and by special methods of detection of a strange nonchaotic attractor (SNA): phase sensitivity and analysis using rational approximation for the frequency ratio. In particular, SNA possesses a zero value of the largest Lyapunov exponent (and negative values of the other exponents), a capacitive dimension close to “2” and a singular continuous power spectrum. In general, the results of this work shed a new light on the occurrence of strange nonchaotic dynamics.
    Keywords: autonomous dynamical system, mechanical rotators, quasi-periodic oscillations, strange nonchaotic attractor, chaos
    Citation: Jalnine A. Y., Kuznetsov S. P.,  Autonomous strange non-chaotic oscillations in a system of mechanical rotators, Rus. J. Nonlin. Dyn., 2017, Vol. 13, No. 2, pp.  257-275
    Jalnine A. Y.
    In the present paper we consider a family of coupled self-oscillatory systems presented by pairs of coupled van der Pol generators and FitzHugh–Nagumo neural models, with the parameters being periodically modulated in anti-phase, so that the subsystems undergo alternate excitation with a successive transmission of the phase of oscillations from one subsystem to another. It is shown that, due to the choice of the parameter modulation and coupling methods, one can observe a whole spectrum of robust chaotic dynamical regimes, taking the form ranging from quasiharmonic ones (with a chaotically floating phase) to the well-defined neural oscillations, which represent a sequence of amplitude bursts, in which the phase dynamics of oscillatory spikes is described by a chaotic mapping of Bernoulli type. It is also shown that 4D maps arising in a stroboscopic Poincaré section of the model flow systems universally possess a hyperbolic strange attractor of the Smale–Williams type. The results are confirmed by analysis of phase portraits and time series, by numerical calculation of Lyapunov exponents and their parameter dependencies, as well as by direct computation of the distributions of angles between stable and unstable tangent subspaces of chaotic trajectories.
    Keywords: chaos, hyperbolicity, Smale–Williams attractor, neurons, FitzHugh–Nagumo model
    Citation: Jalnine A. Y.,  From quasiharmonic oscillations to neural spikes and bursts: a variety of hyperbolic chaotic regimes based on Smale – Williams attractor, Rus. J. Nonlin. Dyn., 2016, Vol. 12, No. 1, pp.  53-73
    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

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