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^⁗+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.