Global Dynamics of Ordinary Differential Equations: Wall Labelings, Conley Complexes, and Ramp Systems

TL;DR

This paper develops a combinatorial topological framework to analyze the global dynamics of ODEs, especially in gene regulatory networks, by extending existing software to extract equilibria, periodic orbits, and bifurcations.

Contribution

It introduces a novel topological approach that preserves algebraic invariants and extends the DSGRN software for comprehensive global dynamical analysis of ODEs.

Findings

Successfully characterizes equilibria and periodic orbits

Provides bounds and algebraic topological validation

Demonstrates effectiveness through multiple examples

Abstract

We introduce a combinatorial topological framework for characterizing the global dynamics of ordinary differential equations (ODEs). The approach is motivated by the study of gene regulatory networks, which are often modeled by ODEs that are not explicitly derived from first principles. The proposed method involves constructing a combinatorial model from a set of parameters and then embedding the model into a continuous setting in such a way that the algebraic topological invariants are preserved. In this manuscript, we build upon the software Dynamic Signatures Generated by Regulatory Networks (DSGRN), a software package that is used to explore the dynamics generated by a regulatory network. By extending its functionalities, we deduce the global dynamical information of the ODE and extract information regarding equilibria, periodic orbits, connecting orbits and bifurcations. We validate our results through algebraic topological tools and analytical bounds, and the effectiveness of this framework is demonstrated through several examples and possible future directions.