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Regular and Chaotic Dynamics

, Volume 23, Issue 4, pp 480–502 | Cite as

Dynamics of a Smooth Profile in a Medium with Friction in the Presence of Parametric Excitation

  • Alexey V. Borisov
  • Ivan S. Mamaev
  • Eugeny V. Vetchanin
Article
  • 20 Downloads

Abstract

This paper addresses the problem of self-propulsion of a smooth profile in a medium with viscous dissipation and circulation by means of parametric excitation generated by oscillations of the moving internal mass. For the case of zero dissipation, using methods of KAM theory, it is shown that the kinetic energy of the system is a bounded function of time, and in the case of nonzero circulation the trajectories of the profile lie in a bounded region of the space. In the general case, using charts of dynamical regimes and charts of Lyapunov exponents, it is shown that the system can exhibit limit cycles (in particular, multistability), quasi-periodic regimes (attracting tori) and strange attractors. One-parameter bifurcation diagrams are constructed, and Neimark–Sacker bifurcations and period-doubling bifurcations are found. To analyze the efficiency of displacement of the profile depending on the circulation and parameters defining the motion of the internal mass, charts of values of displacement for a fixed number of periods are plotted. A hypothesis is formulated that, when nonzero circulation arises, the trajectories of the profile are compact. Using computer calculations, it is shown that in the case of anisotropic dissipation an unbounded growth of the kinetic energy of the system (Fermi-like acceleration) is possible.

Keywords

self-propulsion in a fluid motion with speed-up parametric excitation viscous dissipation, circulation period-doubling bifurcation Neimark–Sacker bifurcation Poincaré map chart of dynamical regimes chart of Lyapunov exponents strange attractor KAM curves anisotropic dissipation Fermi-like acceleration 

MSC2010 numbers

70H08 70Exx 76Bxx 76Dxx 

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Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  • Alexey V. Borisov
    • 1
  • Ivan S. Mamaev
    • 1
    • 2
  • Eugeny V. Vetchanin
    • 1
    • 2
  1. 1.Udmurt State UniversityIzhevskRussia
  2. 2.Izhevsk State Technical UniversityIzhevskRussia

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