On the periodic solutions of a class of Duffing differential equations
type: Item
instance of: scholarly article
title: On the periodic solutions of a class of Duffing differential equations
zbMATH Open document ID: 1272.34052
author: Jaume Llibre
author: Luci Any Roberto
published in: Discrete and Continuous Dynamical Systems
publication date: 2013-03-19T00:00:00Z
review text: Consider the differential equation \[ x''+\varepsilon Cx'+ \varepsilon^2A(t)x+ b(t) x^3= \varepsilon^3\Lambda h(t),\tag{\(*\)} \] where \(C\), \(\varepsilon\) and \(\Lambda\) are real parameters, \(C\) is positive, \(\varepsilon\) is small and positive, the functions \(A\), \(b\) and \(h\) are continuous and \(T\)-periodic satisfying \(A_0b_0> 0\) with \[ A_0= {1\over T} \int^T_0 A(t)\,dt,\quad b_0= {1\over T} \int^T_0 b(t)\,dt. \] The authors study the existence of periodic solutions of \((*)\) and their stability for sufficiently small \(\varepsilon\) in dependence on the parameter \(\Lambda\) and the averaged functions \(A\), \(b\) and \(h\). The proof is based on the averaging theory applied to the bifurcation of periodic solutions.
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 34C23
Mathematics Subject Classification ID: 34C29
Mathematics Subject Classification ID: 34D20
Mathematics Subject Classification ID: 37G15
zbMATH DE Number: 6146687
zbMATH Keywords: periodic solution
zbMATH Keywords: averaging method
zbMATH Keywords: Duffing differential equation
zbMATH Keywords: bifurcation
zbMATH Keywords: stability
reviewed by: Klaus R. Schneider
MaRDI profile type: Publication
full work available at URL: dcds.2013.33.277
OpenAlex ID: W2010140642
DOI: 10.3934/DCDS.2013.33.277
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description: scientific article; zbMATH DE number 6146687