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 orbits are...

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Bibliographic Details
Main Authors Berg, Jan Bouwe van den, Breden, Maxime, Lessard, Jean-Philippe, Murray, Maxime
Format Journal Article
LanguageEnglish
Published 23.02.2017
Subjects
Mathematics - Analysis of PDEs
Mathematics - Dynamical Systems
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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.
DOI:10.48550/arxiv.1702.07412