Construction of approximate solutions for rigorous numerics of symmetric homoclinic orbits

which is usually obtained by numerical simulations. In this setting, he gives a rigorous numerical method to prove the existence of symmetric homoclinic orbits of (1.1) in a neighborhood of the numerical solution (1.2). We refer to the original paper [5] for the background and motivations of this work. In the method, it is essential to show the following two steps based on the exponential dichotomy property: (i) the existence of orbits on the stable manifold of the origin in a neighborhood of an approximate solution w(t), t ∈ R, which is determined by (1.2), (ii) the intersection of the stable manifold and the S-invariant subspace. It is remarked in the paper that we need to construct a suitable approximate solution w(t) in the sense of C(R), r ≥ 1, since the fundamental matrix solution of the variational equation with respect to w(t) is affected by the hyperbolicity in a neighborhood of the origin and it makes difficult numerical verifications of the above two steps. In this paper, we consider how to practically construct a good approximate solution w(t) for the rigorous numerical method [5] in detail. First of all, it is shown that we