Bifurcation of Homoclinic Orbits and Bifurcation from the Essential Spectrum

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Abstract
Bifurcation for nonlinear eigenvalue problems involving a second-order ordinary differential equation on the line is considered. Solutions are required to vanish at infinity in both directions and so correspond to homoclinic orbits. When posed in function spaces, the problem concerns bifurcation from the continuous spectrum. The present approach is based on a resealing that reduces the problem to that of continuing a nontrivial homoclinic orbit in a context where the perturbations are not periodic and are not smooth with respect to uniform convergence. Nonetheless, the nondegeneracy required for continuationamounts to finding simple zeros of a function analogous to Melnikov’s function.

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