Beyond the Melnikov method: A computer assisted approach

Abstract

We present a Melnikov type approach for establishing transversal intersections of stable/unstable manifolds of perturbed normally hyperbolic invariant manifolds (NHIMs). The method is based on a new geometric proof of the normally hyperbolic invariant manifold theorem, which establishes the existence of a NHIM, together with its associated invariant manifolds and bounds on their first and second derivatives. We do not need to know the explicit formulas for the homoclinic orbits prior to the perturbation. We also do not need to compute any integrals along such homoclinics. All needed bounds are established using rigorous computer assisted numerics. Lastly, and most importantly, the method establishes intersections for an explicit range of parameters, and not only for perturbations that are 'small enough', as is the case in the classical Melnikov approach.

Beyond the Melnikov method II: Multidimensional setting

“…The first concerns the bounds on the center-stable and centerunstable manifolds. The second treats the implicit computations which are needed for (14), (19) and (25)(26).…”

“…The questions that interest us here is how to obtain bounds on u (ε, x) , s (ε, x) from (14), and on cu and cs from (19) and (25)(26). We want to compute bounds on these functions, together with their first and second derivatives.…”

“…All of the three cases: (14), (19) and (25)(26), can be treated in the same way. We will describe this by introducing abstract notation.…”

“…This paper is a sequel to [14], which developed a tool for establishing the splitting of separatrices for an explicit range of perturbation parameters. The paper [14] treated the case of one-dimensional separatrices. In the current work we generalise the results to multidimensional setting.…”

Homoclinic points of 2D and 4D maps via the parametrization method

“…Here however, we are more interested in accurately obtaining these solutions, while permitting the parameters to vary, until they no longer exist. Note that this is also the subject of reference [41], where the authors mention the possibility of applying the parametrization method, along with their technique, to obtain computer-assisted proofs of transversal intersections of manifolds. Here we proceed as described below.…”