  A (closed) dynamical system is a notion of how things can be, together with a
notion of how they may change given how they are. The idea and mathematics of
closed dynamical systems has proven incredibly useful in those sciences that
can isolate their object of study from its environment. But many changing
situations in the world cannot be meaningfully isolated from their environment
- a cell will die if it is removed from everything beyond its walls. To study
systems that interact with their environment, and to design such systems in a
modular way, we need a robust theory of open dynamical systems.
  In this extended abstract, we put forward a general definition of open
dynamical system. We define two general sorts of morphisms between these
systems: covariant morphisms which include trajectories, steady states, and
periodic orbits; and contravariant morphisms which allow for plugging variables
of some systems into parameters of other systems. We define an indexed double
category of open dynamical systems indexed by their interface and use a double
Grothendieck construction to construct a double category of open dynamical
systems.
  In our main theorem, we construct covariantly representable indexed double
functors from the indexed double category of dynamical systems to an indexed
double category of spans. This shows that all covariantly representable
structures of dynamical systems - including trajectories, steady states, and
periodic orbits - compose according to the laws of matrix arithmetic.
