Dynamics of nonlinear oscillators under simultaneous internal and external resonances
type: Item
instance of: scholarly article
title: Dynamics of nonlinear oscillators under simultaneous internal and external resonances
zbMATH Open document ID: 0924.70019
DOI: 10.1023/A:1008264104238
author: S. Mitsi
author: S. Natsiavas
author: I. Tsiafis
published in: Nonlinear Dynamics
publication date: 1999-11-10T00:00:00Z
review text: The main goal of this paper is to study the interaction between \(1:3\) internal resonance and two external resonances for a class of weakly nonlinear oscillators with two important degrees of freedom. The examined oscillators possess symmetric restoring force, involving linear and cubic displacement terms. First of all, the authors apply the method of multiple time scales which results in four slow-flow equations for amplitudes and phases of approximate solutions. Several types of constant solutions of these equations are identified and determined for both single-mode and two-mode forcing conditions. The first section includes also a procedure revealing the stability properties of all types of located constant solutions. In a particular case (constant solutions for single-mode forcing) the solutions are found in closed form. The second section examines numerically a mechanical oscillator. This verifies the coexistence of both periodic motions with large period and quasiperiodic and chaotic motions which exhibit torus doubling, boundary crisis and transient chaos.
Mathematics Subject Classification ID: 70K30
Mathematics Subject Classification ID: 34C15
zbMATH DE Number: 1185504
zbMATH Keywords: \(1:3\) internal resonance
zbMATH Keywords: weakly nonlinear oscillators
zbMATH Keywords: symmetric restoring force
zbMATH Keywords: method of multiple time scales
zbMATH Keywords: constant solutions
zbMATH Keywords: stability
zbMATH Keywords: torus doubling
zbMATH Keywords: boundary crisis
zbMATH Keywords: transient chaos
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1185504