Operadic Modeling of Dynamical Systems: Mathematics and Computation

TL;DR

This paper introduces a novel operadic framework for modeling dynamical systems, capturing their modular structure, and demonstrates its implementation in Julia, enhancing both theoretical understanding and computational tools.

Contribution

It reformulates dynamical systems using operads and C-sets, establishing a hierarchical compositional approach and extending functorial properties of Euler's method.

Findings

Operadic modeling captures modular structure of dynamical systems.

Euler's method is shown to be functorial for undirected systems.

Software implementation in Julia demonstrates practical applicability.

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.