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In this paper, we prove existence of symmetric homoclinic orbits for thesuspension 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 stablemanifold is computed and the symmetric homoclinic orbits are obtained bysolving a projected boundary value problem using Chebyshev series. The proof iscomputer-assisted and combines the uniform contraction theorem and the radiipolynomial approach, which provides an efficient means of determining a set,centered at a numerical approximation of a solution, on which a Newton-likeoperator is a contraction.

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