Analyzing the topological structure of composite dynamical systems

TL;DR

This paper introduces a mathematical framework using sheaves to analyze the structure and interactions of complex dynamical systems, demonstrated on a Bering Sea food web, linking structural models with data consistency and inference.

Contribution

It develops a formal connection between dynamical structural equation models and sheaves, enabling systematic analysis of subsystem interactions in complex systems.

Findings

Sheaf models can test data consistency and infer missing information.

The framework applies to ecological food webs, exemplified on the Bering Sea system.

Provides a unified approach to studying interconnected dynamical subsystems.

Abstract

This chapter explores dynamical structural equation models (DSEMs) and their nonlinear generalizations into sheaves of dynamical systems. It demonstrates these two disciplines on part of the food web in the Bering Sea. The translation from DSEMs to sheaves passes through a formal construction borrowed from electronics called a netlist that specifies how data route through a system. A sheaf can be considered a formal hypothesis about how variables interact, that then specifies how observations can be tested for consistency, how missing data can be inferred, and how uncertainty about the observations can be quantified. Sheaf modeling provides a coherent mathematical framework for studying the interaction of various dynamical subsystems that together determine a larger system.