  In this paper, we use the language of operads to study open dynamical
systems. More specifically, we study the algebraic nature of assembling complex
dynamical systems from an interconnection of simpler ones. The syntactic
architecture of such interconnections is encoded using the visual language of
wiring diagrams. We define the symmetric monoidal category W, from which we may
construct an operad O(W), whose objects are black boxes with input and output
ports, and whose morphisms are wiring diagrams, thus prescribing the algebraic
rules for interconnection. We then define two W-algebras, G and L, which
associate semantic content to the structures in W. Respectively, they
correspond to general and to linear systems of differential equations, in which
an internal state is controlled by inputs and produces outputs. As an example,
we use these algebras to formalize the classical problem of systems of tanks
interconnected by pipes, and hence make explicit the algebraic relationships
among systems at different levels of granularity.
