Continuation of homoclinic orbits in the suspension bridge equation: A computer-assisted proof

Jan Bouwe van den Berg; Maxime Breden; Jean Philippe Lessard; Maxime Murray

doi:10.1016/j.jde.2017.11.011

Continuation of homoclinic orbits in the suspension bridge equation: A computer-assisted proof

Jan Bouwe van den Berg, Maxime Breden, Jean Philippe Lessard^*, Maxime Murray

^*Corresponding author for this work

Mathematics

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Abstract

In this paper, we prove existence of symmetric homoclinic orbits for the suspension bridge equation u^⁗+βu^″+e^u−1=0 for all parameter values β∈[0.5,1.9]. For each β a parameterization of the stable manifold is computed and the symmetric homoclinic orbits are obtained by solving a projected boundary value problem using Chebyshev series. The proof is computer-assisted and combines the uniform contraction theorem and the radii polynomial approach, which provides an efficient means of determining a set, centered at a numerical approximation of a solution, on which a Newton-like operator is a contraction.

Original language	English
Pages (from-to)	3086-3130
Number of pages	45
Journal	Journal of Differential Equations
Volume	264
Issue number	5
Early online date	15 Nov 2017
DOIs	https://doi.org/10.1016/j.jde.2017.11.011
Publication status	Published - 1 Mar 2018

Funding

Jan Bouwe van den Berg was supported by the grant NWO Vici-grant 639.033.109 . Jean-Philippe Lessard was supported by Gouvernement du Canada/ Natural Sciences and Engineering Research Council of Canada (NSERC), CG100747 .

Funders	Funder number
NWO Vici-grant 639.033.109	Vici-grant 639.033.109
Natural Sciences and Engineering Research Council of Canada	CG100747
Gouvernement du Burundi

Keywords

Contraction mapping
Rigorous numerics
Stable manifolds
Suspension bridge equation
Symmetric homoclinic orbits
Traveling waves

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10.1016/j.jde.2017.11.011

Continuation of homoclinic orbits in the suspension bridge equation_A computerassisted proofFinal published version, 803 KBLicence: Article 25fa Dutch Copyright Act

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title = "Continuation of homoclinic orbits in the suspension bridge equation: A computer-assisted proof",

abstract = "In this paper, we prove existence of symmetric homoclinic orbits for the suspension bridge equation u⁗+βu″+eu−1=0 for all parameter values β∈[0.5,1.9]. For each β a parameterization of the stable manifold is computed and the symmetric homoclinic orbits are obtained by solving a projected boundary value problem using Chebyshev series. The proof is computer-assisted and combines the uniform contraction theorem and the radii polynomial approach, which provides an efficient means of determining a set, centered at a numerical approximation of a solution, on which a Newton-like operator is a contraction.",

keywords = "Contraction mapping, Rigorous numerics, Stable manifolds, Suspension bridge equation, Symmetric homoclinic orbits, Traveling waves",

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AU - Lessard, Jean Philippe

AU - Murray, Maxime

PY - 2018/3/1

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N2 - In this paper, we prove existence of symmetric homoclinic orbits for the suspension bridge equation u⁗+βu″+eu−1=0 for all parameter values β∈[0.5,1.9]. For each β a parameterization of the stable manifold is computed and the symmetric homoclinic orbits are obtained by solving a projected boundary value problem using Chebyshev series. The proof is computer-assisted and combines the uniform contraction theorem and the radii polynomial approach, which provides an efficient means of determining a set, centered at a numerical approximation of a solution, on which a Newton-like operator is a contraction.

AB - In this paper, we prove existence of symmetric homoclinic orbits for the suspension bridge equation u⁗+βu″+eu−1=0 for all parameter values β∈[0.5,1.9]. For each β a parameterization of the stable manifold is computed and the symmetric homoclinic orbits are obtained by solving a projected boundary value problem using Chebyshev series. The proof is computer-assisted and combines the uniform contraction theorem and the radii polynomial approach, which provides an efficient means of determining a set, centered at a numerical approximation of a solution, on which a Newton-like operator is a contraction.

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KW - Traveling waves

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Continuation of homoclinic orbits in the suspension bridge equation: A computer-assisted proof

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