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.for some Y (j) > 0 and Z (j) : R + → R + : r → Z (j) (r). The goal of the radii polynomial approach is to provide an efficient way to prove that an operator is a uniform contraction over a subset of X. This subset consists of small balls around the line of centers, provided by the linear interpolation between two numerical approximations of solutions at different parameter values.

This paper develops Chebyshev-Taylor spectral methods for studying stable/unstable manifolds attached to periodic solutions of differential equations. The work exploits the parameterization method -a general functional analytic framework for studying invariant manifolds. Useful features of the parameterization method include the fact that it can follow folds in the embedding, recovers the dynamics on the manifold through a simple conjugacy, and admits a natural notion of a-posteriori error analysis. Our approach begins by deriving a recursive system of linear differential equations describing the Taylor coefficients of the invariant manifold. We represent periodic solutions of these equations as solutions of coupled systems of boundary value problems. We discuss the implementation and performance of the method for the Lorenz system, and for the planar circular restricted three and four body problems. We also illustrate the use of the method as a tool for computing cycle-to-cycle connecting orbits.

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