• Corpus ID: 119127912

# An algebra of open continuous time dynamical systems and networks

@article{Lerman2016AnAO,
title={An algebra of open continuous time dynamical systems and networks},
author={Eugene Lerman and David I. Spivak},
journal={arXiv: Dynamical Systems},
year={2016}
}
• Published 2 February 2016
• Mathematics
• arXiv: Dynamical Systems
Many systems of interest in science and engineering are made up of interacting subsystems. These subsystems, in turn, could be made up of collections of smaller interacting subsystems and so on. In a series of papers David Spivak with collaborators formalized these kinds of structures (systems of systems) as algebras over presentable colored operads. It is also very useful to consider maps between dynamical systems. This amounts to viewing dynamical systems as objects in an appropriate…
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## References

SHOWING 1-10 OF 15 REFERENCES
Algebras of Open Dynamical Systems on the Operad of Wiring Diagrams
• Mathematics, Computer Science
• 2014
This paper uses the language of operads to study the algebraic nature of assembling complex dynamical systems from an interconnection of simpler ones, and defines two W-algebras, G and L, which associate semantic content to the structures in W.
Nonlinear dynamics of networks: the groupoid formalism
• Mathematics
• 2006
A formal theory of symmetries of networks of coupled dynamical systems, stated in terms of the group of permutations of the nodes that preserve the network topology, has existed for some time.
The Behavioral Approach to Open and Interconnected Systems
• J. Willems
• Computer Science
IEEE Control Systems
• 2007
This article has presented an approach to the mathematical description of dynamical systems, and described a methodology for modeling interconnected systems, called tearing, zooming, and linking, that is much better adapted to the physics of interconnected systems than input/output-modeling procedures such as Simulink.
Dynamics on Networks of Manifolds
• Mathematics
• 2015
It is proved that the appropriate maps of graphs called graph brations give rise to maps of dynamical systems, which gives rise to invariant subsystems and injective graph fibrationsGive rise to projections of dynamicals systems.
Symmetry Groupoids and Patterns of Synchrony in Coupled Cell Networks
• Mathematics
SIAM J. Appl. Dyn. Syst.
• 2003
A coupled cell system is a network of dynamical systems, or "cells," coupled together. Such systems can be represented schematically by a directed graph whose nodes correspond to cells and whose
The operad of temporal wiring diagrams: formalizing a graphical language for discrete-time processes
• Computer Science
ArXiv
• 2013
This work investigates the hierarchical structure of processes using the mathematical theory of operads, and defines an operad of black boxes and directed wiring diagrams of processes (which it is called propagators, after Radul and Sussman), which are useful for modeling dynamic flows of information.
Framed bicategories and monoidal fibrations
In some bicategories, the 1-cells are morphisms' between the 0-cells, such as functors between categories, but in others they are objects' over the 0-cells, such as bimodules, spans, distributors,
The operad of wiring diagrams: formalizing a graphical language for databases, recursion, and plug-and-play circuits
It is shown that wiring diagrams form the morphisms of an operad $\mcT$, capturing this self-similarity, and is moved on to show how plug-and-play devices and also recursion can be formulated in the operadic framework as well.
Constructing symmetric monoidal bicategories
We present a method of constructing symmetric monoidal bicategories from symmetric monoidal double categories that satisfy a lifting condition. Such symmetric monoidal double categories frequently
"The ghost in the machine".
Koestler examines the notion that the parts of the human brain-structure which account for reason and emotion are not fully coordinated. This kind of deficiency may explain the paranoia, violence,