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Regular and Chaotic Dynamics

, Volume 22, Issue 8, pp 955–975 | Cite as

The Chaplygin Sleigh with Parametric Excitation: Chaotic Dynamics and Nonholonomic Acceleration

  • Ivan A. Bizyaev
  • Alexey V. Borisov
  • Ivan S. Mamaev
Article

Abstract

This paper is concerned with the Chaplygin sleigh with time-varying mass distribution (parametric excitation). The focus is on the case where excitation is induced by a material point that executes periodic oscillations in a direction transverse to the plane of the knife edge of the sleigh. In this case, the problem reduces to investigating a reduced system of two first-order equations with periodic coefficients, which is similar to various nonlinear parametric oscillators. Depending on the parameters in the reduced system, one can observe different types of motion, including those accompanied by strange attractors leading to a chaotic (diffusion) trajectory of the sleigh on the plane. The problem of unbounded acceleration (an analog of Fermi acceleration) of the sleigh is examined in detail. It is shown that such an acceleration arises due to the position of the moving point relative to the line of action of the nonholonomic constraint and the center of mass of the platform. Various special cases of existence of tensor invariants are found.

Keywords

nonholonomic mechanics Fermi acceleration Chaplygin sleigh parametric oscillator tensor invariants involution strange attractor Lyapunov exponents reversible systems chaotic dynamics 

MSC2010 numbers

37J60 34A34 

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Copyright information

© Pleiades Publishing, Ltd. 2017

Authors and Affiliations

  • Ivan A. Bizyaev
    • 1
  • Alexey V. Borisov
    • 2
  • Ivan S. Mamaev
    • 3
  1. 1.Moscow Institute of Physics and TechnologyDolgoprudnyiRussia
  2. 2.Udmurt State UniversityIzhevskRussia
  3. 3.Izhevsk State Technical UniversityIzhevskRussia

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