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Algebras of open dynamical systems on the operad of wiring diagrams

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Dmitry Vagner, David I. Spivak, and Eugene Lerman

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 $OW$, 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.

Keywords:
Operads, Monoidal Categories, Wiring Diagrams, Dynamical Systems

2010 MSC:
93A13

*Theory and Applications of Categories,*
Vol. 30, 2015,
No. 51, pp 1793-1822.

Published 2015-12-03.

http://www.tac.mta.ca/tac/volumes/30/51/30-51.pdf

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