Multiple timescales and the parametrisation method in geometric singular perturbation theory

We present a novel method for computing slow manifolds and their fast fibre bundles in geometric singular perturbation problems. This coordinate-independent method is inspired by the parametrisation method introduced by Cabre, Fontich and de la Llave. By iteratively solving a so-called conjugacy equation, our method simultaneously computes parametrisations of slow manifolds and fast fibre bundles, as well as the dynamics on these objects, to arbitrarily high degrees of accuracy. We show the power of this top-down method for the study of systems with multiple (i.e. three or more) timescales. In particular, we highlight the emergence of hidden timescales and show how our method can uncover these surprising multiple timescale structures. We also apply our parametrisation method to several reaction network problems.

Automated summary

The novel method presented in this study computes slow manifolds and fast fibre bundles in geometric singular perturbation problems with high degrees of accuracy, making it suitable for systems with multiple timescales. This top-down approach highlights the emergence of hidden timescales and can uncover surprising multiple timescale structures. It has been successfully applied to various reaction network problems.