• Corpus ID: 119127912

An algebra of open continuous time dynamical systems and networks

  title={An algebra of open continuous time dynamical systems and networks},
  author={Eugene Lerman and David I. Spivak},
  journal={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… 

Networks of open systems

  • E. Lerman
  • Mathematics
    Journal of Geometry and Physics
  • 2018


Span categories provide a framework for formalizing mathematical models of open systems in classical mechanics. The categories appearing in classical mechanics do not have pullbacks, which requires

Open systems in classical mechanics

Author(s): Baez, John C; Weisbart, David; Yassine, Adam | Abstract: Span categories provide a framework for formalizing mathematical models of open systems in classical mechanics. The categories

Lie 2-algebras of vector fields

We show that the category of vector fields on a geometric stack has the structure of a Lie 2-algebra. This proves a conjecture of R.~Hepworth. The construction uses a Lie groupoid that presents the

Open Petri nets

  • J. BaezJ. Master
  • Computer Science
    Mathematical Structures in Computer Science
  • 2020
Two forms of semantics for open Petri nets are described using symmetric monoidal double functors out of pen(Petri), including an operational semantics and a reachability semantics that simply says which markings of the outputs can be reached from a given marking of the inputs.

Coarse-Graining Open Markov Processes

Coarse-graining is a standard method of extracting a simple Markov process from a more complicated one by identifying states. Here we extend coarse-graining to open Markov processes. An "open" Markov

Agoric computation: trust and cyber-physical systems

This thesis reviews the logical and mathematical basis of stateless abstractions and takes steps towards the software implementation of agoric modelling as a framework for simulation and verification of the reliability of increasingly complex systems, and reports on experimental results related to a few select applications.

25th Annual Computational Neuroscience Meeting: CNS-2016

Table of contents Functional advantages of cell-type heterogeneity in neural circuits, Dynamics and biomarkers of mental disorders, and Objective criteria for computational neuroscience model selection are presented.

A bicategory of decorated cospans

If $\mathbf{C}$ is a category with pullbacks then there is a bicategory with the same objects as $\mathbf{C}$, spans as morphisms, and maps of spans as 2-morphisms, as shown by Benabou. Fong has



Algebras of Open Dynamical Systems on the Operad of Wiring Diagrams

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.

Dynamics on networks I. Combinatorial categories of modular continuous-time systems

A new framework for the study of complex continuous time dynamical systems based on viewing them as collections of interacting control modules is developed, which enables the groupoid formalism in a coordinate-free setting and to extend it from ordinary differential equations to vector fields on manifolds.

Modular dynamical systems on networks

It is shown that a class of maps between graphs called graph fibrations give rise to maps between dynamical systems on networks, which allows us to produce conjugacy between dynamicals systems out of combinatorial data.

Nonlinear dynamics of networks: the groupoid formalism

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.

Patterns of Synchrony in Coupled Cell Networks with Multiple Arrows

A framework for coupled cell systems that permits a classification of robust synchrony in terms of network architecture and the existence of other robust dynamical patterns using a concept of quotient network is presented.

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

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

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

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,