Coarse-Graining for Coupled Oscillators: A case study in discovering low-dimensional dynamics

Quantitative science has produced successful models of the
world at scales spanning dozens of orders of magnitude in space, time,
and energy. While in principle, the most finely detailed models are
sufficient to reproduce the predictions of any less-detailed model,
there is clearly scientific value in deriving and studying models that
do not resolve details that are not directly relevant to the question at
hand. The procedure of systematically reducing the degrees of freedom of
a model goes by many names depending on context (renormalization group
in statistical physics) but can be generally be referred to as "model
reduction". In this talk I will discuss progress towards general
principles of model reduction obtained by studying synchronization of
phase oscillators on a modular network. This system is an ideal test-bed
because it exhibits a controllable transition from high- to
low-dimensional behavior; moreover, the resulting low-dimensional
behavior is well-described by coarse-grained equations of a known form.