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

## TL;DR

This paper proves the existence of symmetric homoclinic orbits in the suspension bridge equation across a range of parameters using a computer-assisted method involving Chebyshev series and contraction theorems.

## Contribution

It introduces a computer-assisted proof technique for homoclinic orbits in a nonlinear differential equation, combining the uniform contraction theorem and radii polynomial approach.

## Key findings

- Existence of symmetric homoclinic orbits for all β in [0.5, 1.9]
- Effective parameterization of stable manifolds
- Validated solutions using a contraction mapping approach

## Abstract

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.