On nulls of perturbed Fredholm operators and degenerate homoclinic bifurcations (Q813811)

By an explicit construction in the finite-dimensional null space and co-kernel of a Fredholm operator, it is shown that for an index zero Fredholm operator with \(n\)-dimensional kernel there exist perturbations which possess a \(k\)-dimensional kernel for any \(k\in\{0,1,2,\ldots,n-1\}\). The result is then applied to the linearization along a degenerate homoclinic orbit. However, the bounded solutions found for the perturbed linearized equation do not seem to have an immediate meaning for the existence of homoclinic orbits in the original nonlinear equation.