Global attraction to the origin in a parametrically driven nonlinear oscillator
type: Item
instance of: scholarly article
title: Global attraction to the origin in a parametrically driven nonlinear oscillator
zbMATH Open document ID: 1068.34053
DOI: 10.1016/S0096-3003(03)00317-5
published in: Applied Mathematics and Computation
publication date: 2004-08-06T00:00:00Z
review text: Sufficient conditions are given for the coefficients in the equation for a parametrically driven nonlinear oscillator so that the origin is a global attractor and that the trajectories converge monotonically to the origin. The equation considered is \(\dot x = y\), \(\dot y = -\gamma - f(t)x^{2m-1}\), \((x,y)\in \mathbb R^2\), \(t\geq 0\), where \(f(t)\) is bounded from above \(0\leq f\leq f^+\). It is proved that the origin is a global attractor if \(\gamma >-\min(f'(t)/2f(t))\) and letting \(u(t)= x(t)^{-2m}\), \(v(t) =\dot u(t)\), the trajectories \(x(t)\), \(y(t)\) for which \(u(t),v(t)\) enter the region \(u>0\), \(v>0\), \(u>4f^+/\gamma^2\), \(v/u < m(\gamma +\sqrt{\gamma^2-4f^+/u)}\) converge monotonically to zero. Further behavior of the trajectories is discussed based on numerical simulations.
Mathematics Subject Classification ID: 34D45
Mathematics Subject Classification ID: 34D05
zbMATH DE Number: 2081712
zbMATH Keywords: parametric forcing
zbMATH Keywords: asymptotics
zbMATH Keywords: numerical examples
zbMATH Keywords: convergence
zbMATH Keywords: nonlinear oscillator
zbMATH Keywords: global attractivity
author: Michele Bartuccelli
author: Jonathan Deane
author: Guido Gentile
author: Stephen A. Gourley
reviewed by: John B. Butler jun.
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 2081712