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2013
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    Valeri Smirnov

    ul. Kosygina 4, Moscow, 119991 Russia
    Semenov Institute of Chemical Physics RAS

    Publications:

    Smirnov V. V., Kovaleva M. A., Manevitch L. I.
    Nonlinear Dynamics of Torsion Lattices
    2018, Vol. 14, no. 2, pp.  179-193
    Abstract
    We present an analysis of torsion oscillations in quasi-one-dimensional lattices with periodic potentials of the nearest neighbor interaction. A one-dimensional chain of point dipoles (spins) under an external field and without the latter is the simplest realization of such a system. We obtained dispersion relations for the nonlinear normal modes for a wide range of oscillation amplitudes and wave numbers. The features of the short wavelength part of the spectrum at large-amplitude oscillations are discussed. The problem of localized excitations near the edges of the spectrum is studied by the asymptotic method. We show that the localized oscillations (breathers) appear near the long wavelength edge, while the short wavelength edge of the spectrum contains only dark solitons. The continuum limit of the dynamic equations leads to a generalization of the nonlinear Schrödinger equation and can be considered as a complex representation of the sine-Gordon equation.
    Keywords: essentially nonlinear systems, coupled pendulums, nonlinear normal modes, limiting phase trajectories
    Citation: Smirnov V. V., Kovaleva M. A., Manevitch L. I.,  Nonlinear Dynamics of Torsion Lattices, Rus. J. Nonlin. Dyn., 2018, Vol. 14, no. 2, pp.  179-193
    DOI:10.20537/nd180203
    Kovaleva M., Smirnov V. V., Manevitch L. I.
    Abstract
    An analysis is presented of the nonlinear dynamics of harmonically coupled pendulums without restrictions to oscillation amplitudes. This is a basic model in many areas of mechanics and physics (paraffin crystals, DNA molecules etc.). Stationary solutions of equations of motion corresponding to nonlinear normal modes (NNMs) are obtained. The inversion of the NNM frequencies with increasing oscillation amplitude is found. An essentially nonstationary process of the resonant energy exchange is described in terms of limiting phase trajectories (LPTs), for which an effective analytic representation is obtained in slow time-scale. Explicit expressions of threshold values of dimensionless parameters are found which correspond to the instability of NNMs and to the transition (in parametric space) from the full energy exchange between the pendulums to the localization of energy. The analytic results obtained are verified by analysis of the Poincar´e sections describing evolution of the initial system.
    Keywords: essentially nonlinear systems, coupled pendulums, nonlinear normal modes, limiting phase trajectories
    Citation: Kovaleva M., Smirnov V. V., Manevitch L. I.,  Stationary and nonstationary dynamics of the system of two harmonically coupled pendulums, Rus. J. Nonlin. Dyn., 2017, Vol. 13, No. 1, pp.  105-115
    DOI:10.20537/nd1701007

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