High order Melnikov method: pendulums
type: Item
instance of: scholarly article
title: High order Melnikov method: pendulums
zbMATH Open document ID: 1502.34053
author: Ali Oksasoglu
author: Qiudong Wang
published in: Journal of Differential Equations
publication date: 2022-01-20T00:00:00Z
review text: This paper is concerned with the computation of the second order Melnikov function for periodically perturbed pendulum equations given by \[ \frac{dx}{dt}=y,\qquad \frac{dy}{dt}=-\sin x +\varepsilon\cos^2(x/2)\cdot P(t), \] where \(P(t)\) is a periodic function in \(t\) and \(\varepsilon\) is a small parameter. Let \(D(t_0 , \varepsilon)\) be the splitting distance of the stable and unstable manifold of a saddle fixed point of the unperturbed system. Then \(D(t_0 , \varepsilon)\) can be expanded into a formal power series in \(\varepsilon\) as \[ D(t_0 , \varepsilon) = D_0 (t_0 ) + \varepsilon D_1 (t_0 ) + \cdots + \varepsilon^n D_n (t_0 ) + \cdots. \] The authors present an explicit integral formula for \(D_1(t_0)\). This work extends the theory derived in [\textit{F. Chen} and \textit{Q. Wang}, Adv. Nonlinear Stud. 17, No. 4, 793--818 (2017; Zbl 1378.37121)] to cover the pendulum equations.
reviewed by: Yun Tian
Mathematics Subject Classification ID: 34C45
Mathematics Subject Classification ID: 34C15
Mathematics Subject Classification ID: 34E10
Mathematics Subject Classification ID: 37C60
zbMATH DE Number: 7460669
zbMATH Keywords: higher order Melnikov functions
zbMATH Keywords: pendulum equations
MaRDI profile type: Publication
full work available at URL: j.jde.2021.12.015
OpenAlex ID: W4206632475
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DOI: 10.1016/J.JDE.2021.12.015
description: scientific article; zbMATH DE number 7460669