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

In this paper, we prove existence of symmetric homoclinic orbits for the suspension bridge equation \(u""+\beta u" + e^u-1=0\) for all parameter values \(\beta \in [0.5,1.9]\). For each \(\beta\), a parameterization of the stable manifold is computed and the symmetric homoclinic orbit...

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Bibliographic Details
Published inarXiv.org
Main Authors Jan Bouwe van den Berg, Breden, Maxime, Lessard, Jean-Philippe, Murray, Maxime
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 23.02.2017
Subjects
Boundary value problems
Chebyshev approximation
Orbits
Parameterization
Polynomials
Suspension bridges
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Summary:In this paper, we prove existence of symmetric homoclinic orbits for the suspension bridge equation \(u""+\beta u" + e^u-1=0\) for all parameter values \(\beta \in [0.5,1.9]\). For each \(\beta\), 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.
ISSN:2331-8422