Abstract. 
This talk will have two parts. In the first part, I will introduce the
'degrade and fire' dynamics of finitely many coupled oscillators. The
dynamics arise as a model of synchronization phenomena observed in
experiments on gene expression in colonies of bacteria. The model is
sufficiently simple that the dynamics can be fully described at any
coupling strength. By way of a suitably-defined return map, I will
present results on the long-time behavior of these systems.
This will motivate the second part of the talk, when I will describe
efforts to generalize this model to a dynamics on continuum-many
oscillators (i.e. indexed by a real parameter), concentrating on the
weak-coupling case, where more is known. I will demonstrate that many
properties of the discrete dynamics carry over to the continuum model.
In this vein, I will show how the discrete dynamics 'embeds' into the
continuum version, and how the discrete dynamics adequately 'shadows'
the continuum dynamics. Joint work with Bastien Fernandez.