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Multi-clustered solutions for a singularly perturbed forced pendulum equation

Part of: Asymptotic theory General theory in ordinary differential equations

Published online by Cambridge University Press:  26 January 2019

Salomé Martínez  and
Dora Salazar
Salomé Martínez
Affiliation:
Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático, UMI 2807 CNRS, Universidad de Chile, Beauchef 851, Edificio Norte–Piso 5, Santiago, Chile (samartin@dim.uchile.cl)
Dora Salazar
Affiliation:
Escuela de Matemáticas, Universidad Nacional de Colombia Sede Medellín, Apartado Aéreo 3840, Medellín, Colombia (dcsalazarl@unal.edu.co)
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Abstract

In this paper, we are concerned with unbounded solutions of the singularly perturbed forced pendulum equation in the presence of friction, namely

$$\varepsilon ^2u_\varepsilon ^{{\prime}{\prime}} + \sin u_\varepsilon = \varepsilon ^2\alpha (t)u_\varepsilon + \varepsilon ^2\beta (t)u_\varepsilon ^{\prime} \quad {\rm in}\;(-L,L).{\rm }$$
Using a limiting energy function, we describe the behaviour of the solutions as the parameter ε approaches zero. We also prove the existence of a family of solutions having a prescribed asymptotic profile and exhibiting a highly rotatory behaviour alternated with a highly oscillatory behaviour in some open subsets of the domain. The proof relies on a combination of the Nehari finite dimensional reduction with the topological degree theory.

Keywords

forced pendulum equationunbounded solutions

MSC classification

Primary: 34A12: Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions
Secondary: 34E15: Singular perturbations, general theory 34E10: Perturbations, asymptotics
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 1 , February 2020 , pp. 387 - 417
Copyright
Copyright © Royal Society of Edinburgh 2019

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