  Dynamical systems---by which we mean machines that take time-varying input,
change their state, and produce output---can be wired together to form more
complex systems. Previous work has shown how to allow collections of machines
to reconfigure their wiring diagram dynamically, based on their collective
state. This notion was called "mode dependence", and while the framework was
compositional (forming an operad of re-wiring diagrams and algebra of
mode-dependent dynamical systems on it), the formulation itself was more
"creative" than it was natural.
  In this paper we show that the theory of mode-dependent dynamical systems can
be more naturally recast within the category Poly of polynomial functors. This
category is almost superlatively abundant in its structure: for example, it has
\emph{four} interacting monoidal structures $(+,\times,\otimes,\circ)$, two of
which ($\times,\otimes$) are monoidal closed, and the comonoids for $\circ$ are
precisely categories in the usual sense. We discuss how the various structures
in Poly show up in the theory of dynamical systems. We also show that the usual
coalgebraic formalism for dynamical systems takes place within Poly. Indeed one
can see coalgebras as special dynamical systems---ones that do not record their
history---formally analogous to contractible groupoids as special categories.
