Operadic Modeling of Dynamical Systems: Mathematics and Computation

Abstract

Dynamical systems are ubiquitous in science and engineering as models of phenomena that evolve over time. Although complex dynamical systems tend to have important modular structure, conventional modeling approaches suppress this structure. Building on recent work in applied category theory, we show how deterministic dynamical systems, discrete and continuous, can be composed in a hierarchical style. In mathematical terms, we reformulate some existing operads of wiring diagrams and introduce new ones, using the general formalism of C-sets (copresheaves). We then establish dynamical systems as algebras of these operads. In a computational vein, we show that Euler's method is functorial for undirected systems, extending a previous result for directed systems. All of the ideas in this paper are implemented as practical software using Catlab and the AlgebraicJulia ecosystem, written in the Julia programming language for scientific computing.

In Proceedings ACT 2021, arXiv:2211.01102

References in corpus (14)

• A Compositional Framework for Reaction Networks
• The operad of wiring diagrams: formalizing a graphical language for databases, recursion, and plug-and-play circuits
• A Compositional Framework for Markov Processes
• Monoidal Grothendieck construction
• Operadic Modeling of Dynamical Systems: Mathematics and Computation
• The steady states of coupled dynamical systems compose according to matrix arithmetic
• Networks of hybrid open systems
• Double Categories of Open Dynamical Systems (Extended Abstract)
• An Algebra of Resource Sharing Machines
• Generalized Lens Categories via functors ${\mathcal{C}}^{\mathrm{op}}\to \mathsf{Cat}$
• Open Systems in Classical Mechanics
• Compositionality of the Runge-Kutta Method
• Pixel Arrays: A fast and elementary method for solving nonlinear systems
• Runge-Kutta and Networks

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