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Stability of autoresonance models

Abstract

We consider systems of two nonlinear nonautonomous differential equations on the real line which arise when averaging rapid nonlinear vibrations. We study the Lyapunov stability of solutions with infinitely increasing amplitude. Such solutions are related to the description of the initial stage of autoresonance or resonance trapping in oscillating nonlinear systems with a small perturbation. We obtain conditions on the coefficients of the equations under which the increasing solutions are stable or unstable. The problem is reduced to the analysis of an equilibrium. The stability of the equilibrium is studied by the Lyapunov second method. The construction of Lyapunov functions is based the presence of dissipative terms with coefficients moderately decaying in time.

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Correspondence to L. A. Kalyakin.

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Original Russian Text © L.A. Kalyakin, O.A. Sultanov, 2013, published in Differentsial’nye Uravneniya, 2013, Vol. 49, No. 3, pp. 279–293.

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Kalyakin, L.A., Sultanov, O.A. Stability of autoresonance models. Diff Equat 49, 267–281 (2013). https://doi.org/10.1134/S0012266113030014

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Keywords

  • Quadratic Form
  • Asymptotic Expansion
  • Hamiltonian System
  • Lyapunov Function
  • Total Derivative