Stability for systems of 1-D coupled nonlinear oscillators
type: Item
instance of: scholarly article
title: Stability for systems of 1-D coupled nonlinear oscillators
zbMATH Open document ID: 1469.34055
author: Cristian Vladimirescu
author: Gheorghe Moroşanu
published in: Nonlinear Analysis. Real World Applications
publication date: 2021-05-14T00:00:00Z
review text: In this paper, the authors firstly define a 1-D mechanical system, next they discuss the stability of the null solution of different systems, which correspond to the motion of two coupled nonlinear oscillators, both being under the action of some external forces of differential equations. Each of three cases will be given below is studied by two approaches based on classical arguments, by using differential inequalities and the method of Lyapunov's function. The following ODEs are associated with the given physical system, which describe the motion of oscillators: \[\begin{aligned}\ddot{x}+2f_{1}(t)\dot{x}+\beta(t)x-\gamma(t)y+g_{1}(t,x,y)&=0,\\ \ddot{y}+2f_{2}(t)\dot{y}+\delta(t)y-\gamma(t)x+g_{2}(t,x,y)&=0,\end{aligned}\] where \(\beta(t)=\frac{1}{m}(k_{1}(t)+k_{2}(t))\), \(\delta(t)=\frac{1}{m}k_{2}(t)\), \(f_{1}(t)=\frac{1}{m}\hat{f}_{1}(t)\),\(f_{2}(t)=\frac{1}{m}\hat{f}_{2}(t)\), \(\gamma(t)=\frac{1}{m}k_{2}(t)\),\(g_{1}(t,x,y)=-\frac{1}{m}\hat{g}_{1}(t,x,y)\),\(g_{2}(t,x,y)=-\frac{1}{m}\hat{g}_{2}(t,x,y)\). In Section 2, the main elements of stability theory that will be used throughout the paper are introduced. In Section 3, the stability of the null solution of two coupled damped nonlinear oscillators described by the system above is investigated under the given hypotheses. Later, they give two theorems on uniform stability and asymptotical stability of the null solution to the above equations. In addition, an example for the illustration is also given. In Section 4, the following ODEs, which are associated with a given physical system to describe the motion of oscillators, are given \[\begin{aligned}\ddot{x}+\beta(t)x-\gamma(t)y+g_{1}(t,x,y)&=0,\\ \ddot{y}+2f(t)\dot{y}+\delta(t)y-\gamma(t)x+g_{2}(t,x,y)&=0,\end{aligned}\] here \(\beta(t)=\frac{1}{m}(k_{1}(t)+k_{2}(t))\), \(\delta(t)=\frac{1}{m}k_{2}(t)\), \(\forall t\in\mathbb{R}_{+}\), \(\gamma(t)=\frac{1}{m}k_{2}(t)\),\(\forall t\in\mathbb{R}_{+}\) is the coupling factor of the oscillators. The function …
reviewed by: Cemil Tunç
Mathematics Subject Classification ID: 34C15
Mathematics Subject Classification ID: 34D20
Mathematics Subject Classification ID: 47N20
Mathematics Subject Classification ID: 37C60
Mathematics Subject Classification ID: 34A40
zbMATH DE Number: 7347888
zbMATH Keywords: coupled oscillators
zbMATH Keywords: uniform stability
zbMATH Keywords: asymptotic stability
describes a project that uses: Matlab
MaRDI profile type: Publication
full work available at URL: j.nonrwa.2020.103242
OpenAlex ID: W3109514913
cites work: Q4824307
cites work: Stability for a Nonlinear Second Order ODE
cites work: Stability for a damped nonlinear oscillator
cites work: Q5619320
cites work: Q3215593
cites work: A necessary and sufficient condition for the asymptotic stability of the damped oscillator
cites work: Integral conditions on the asymptotic stability for the damped linear oscillator with small damping
cites work: ASYMPTOTIC STABILITY OF x″+a(t)x′+x=0
cites work: Asymptotic stability of coupled oscillators with time-dependent damping
cites work: Stability for a system of two coupled nonlinear oscillators with partial lack of damping
cites work: Q3946471
cites work: Precise damping conditions for global asymptotic stability for nonlinear second order systems
cites work: Precise damping conditions for global asymptotic stability for nonlinear second order systems. II
cites work: Asymptotic Stability for Intermittently Controlled Nonlinear Oscillators
cites work: Asymptotic stability for ordinary differential systems with time-dependent restoring potentials
cites work: Q5484946
cites work: Q3843227
cites work: Q5608953
cites work: Q3871201
cites work: Q5569338
cites work: Ordinary differential equations in \(R^ n\). Problems and methods. Transl. from the Italian by A. LoBello
cites work: Q5848416
cites work: Q5330830
cites work: Q5520199
cites work: Q5822590
cites work: Q5665783
cites work: Q4692955
DOI: 10.1016/J.NONRWA.2020.103242
description: scientific article; zbMATH DE number 7347888