  We investigate a singularly perturbed, non-convex variational problem arising
in materials science with a combination of geometrical and numerical methods.
Our starting point is a work by Stefan M\"uller, where it is proven that the
solutions of the variational problem are periodic and exhibit a complicated
multi-scale structure. In order to get more insight into the rich solution
structure, we transform the corresponding Euler-Lagrange equation into a
Hamiltonian system of first order ODEs and then use geometric singular
perturbation theory to study its periodic solutions. Based on the geometric
analysis we construct an initial periodic orbit to start numerical continuation
of periodic orbits with respect to the key parameters. This allows us to
observe the influence of the parameters on the behavior of the orbits and to
study their interplay in the minimization process. Our results confirm previous
analytical results such as the asymptotics of the period of minimizers
predicted by M\"uller. Furthermore, we find several new structures in the
entire space of admissible periodic orbits.
