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We investigate a singularly perturbed, non-convex variational problem arising in material science with a combination of geometrical and numerical methods. Our starting point is a work by Stefan Müller, 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 Müller. Furthermore, we find several new structures in the entire space of admissible periodic orbits. [ABSTRACT FROM AUTHOR]

A detailed geometric analysis of the Goldbeter-Lefever model of glycolytic oscillations is given. In suitably scaled variables the governing equations are a planar system of ordinary differential equations depending singularly on two small parameters ε and δ. In [L. Segel and A. Goldbeter, J. Math. Biol., 32 (1994), pp. 147-160] it was argued that a limit cycle of relaxation type exists for ε « δ « 1. The existence of this limit cycle is proved by analyzing the problem in the spirit of geometric singular perturbation theory. The degeneracies of the limiting problem corresponding to (ε, δ) = (0, 0) are resolved by a novel variant of the blow-up method. It is shown that repeated blowups lead to a clear geometric picture of this fairly complicated two-parameter multiscale problem. [ABSTRACT FROM AUTHOR]