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In this paper, we use the language of operads to study open dynamicalsystems. More specifically, we study the algebraic nature of assembling complexdynamical systems from an interconnection of simpler ones. The syntacticarchitecture of such interconnections is encoded using the visual language ofwiring diagrams. We define the symmetric monoidal category W, from which we mayconstruct an operad O(W), whose objects are black boxes with input and outputports, and whose morphisms are wiring diagrams, thus prescribing the algebraicrules for interconnection. We then define two W-algebras, G and L, whichassociate semantic content to the structures in W. Respectively, theycorrespond to general and to linear systems of differential equations, in whichan internal state is controlled by inputs and produces outputs. As an example,we use these algebras to formalize the classical problem of systems of tanksinterconnected by pipes, and hence make explicit the algebraic relationshipsamong systems at different levels of granularity.

Algebras of Open Dynamical Systems on the Operad of Wiring Diagrams