Nonintegrability of parametrically forced nonlinear oscillators
type: Item
instance of: scholarly article
title: Nonintegrability of parametrically forced nonlinear oscillators
zbMATH Open document ID: 1402.37068
DOI: 10.1134/S156035471803005X
author: Shoya Motonaga
author: Kazuyuki Yagasaki
published in: Regular and Chaotic Dynamics
publication date: 2018-10-18T00:00:00Z
review text: The authors discuss the nonintegrability of parametrically forced nonlinear oscillators represented by second-order homogeneous differential equations of the form \[ \ddot{x}+f(x,\dot{x},\cos (\omega t),\sin (\omega t))x+g(x,\dot{x},\cos (\omega t),\sin (\omega t))\dot{x}=0, \] where \(\dot{x}=dx/dt\) and \(f\), \(g\) are rational functions satisfying the conditions \(f(0,0,y_1,y_2)=f_0+f_1y_1\), \(g(0,0,y_1,y_2)=g_0+g_1y_1+g_2y_2\). These contain the Duffing and van der Pol oscillators as special cases. Sufficient conditions for their rational nonintegrability are given in the sense of Bogoyavlenskij, using the Kovacic algorithm and an extension of the Morales-Ramis theory. The latter is due to \textit{M. Ayoul} and \textit{N. T. Zung} [C. R., Math., Acad. Sci. Paris 348, No. 23--24, 1323--1326 (2010; Zbl 1210.37076)] and is applied to the associated variational equations. The identity components of the corresponding differential Galois groups are shown to be noncommutative even if the Galois group is triangularizable (i.e., solvable by quadratures).
Mathematics Subject Classification ID: 37J30
Mathematics Subject Classification ID: 34C15
Mathematics Subject Classification ID: 70K40
zbMATH DE Number: 6958430
zbMATH Keywords: nonintegrability
zbMATH Keywords: nonlinear oscillator
zbMATH Keywords: Morales-Ramis theory
zbMATH Keywords: differential Galois theory
reviewed by: Vladimir P. Kostov
MaRDI profile type: Publication
cites work: Nonautonomous Hamiltonian Systems and Morales–Ramis Theory I. The Case $\ddot{x}=f(x,t)$
cites work: Non-integrability criterium for normal variational equations around an integrable subsystem and an example: the Wilberforce spring-pendulum
cites work: Non-integrability of some Hamiltonians with rational potentials
cites work: Differential Galois theory and non-integrability of planar polynomial vector fields
cites work: Chaotic motion under parametric excitation
cites work: Galoisian obstructions to non-Hamiltonian integrability
cites work: Extended integrability and bi-Hamiltonian systems
cites work: Picard-Vessiot Extensions of Real Differential Fields
cites work: Nonlinear oscillations, dynamical systems, and bifurcations of vector fields
cites work: Global Bifurcation Analysis of a Duffing–Van der Pol Oscillator with Parametric Excitation
cites work: Q3255437
cites work: An algorithm for solving second order linear homogeneous differential equations
cites work: Differential Galois theory and non-integrability of Hamiltonian systems
cites work: Galoisian obstructions to integrability of Hamiltonian systems. I.
cites work: Prediction of bifurcations in a parametrically excited Duffing oscillator
cites work: Hypergeometric first integrals of the Duffing and van der Pol oscillators
cites work: Q4790565
cites work: Introduction to Applied Nonlinear Dynamical Systems and Chaos
cites work: Nonintegrability of dynamical systems with homo- and heteroclinic orbits
full work available at URL: s156035471803005x
OpenAlex ID: W2806402867
Wikidata QID: Q129728454
description: scientific article; zbMATH DE number 6958430