# Geometry and numerical continuation of multiscale orbits in a nonconvex variational problem

## TL;DR

This paper combines geometric and numerical methods to analyze multiscale periodic solutions in a complex non-convex variational problem relevant to materials science.

## Contribution

It transforms the Euler-Lagrange equation into a Hamiltonian system and uses geometric singular perturbation theory for numerical continuation of solutions.

## Key findings

- Confirmed asymptotic behavior of minimizer periods
- Constructed initial periodic orbits for numerical continuation
- Discovered new structures in the space of periodic orbits

## Abstract

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.