Global Dynamics of Ordinary Differential Equations: Wall Labelings, Conley Complexes, and Ramp Systems
Year: 2024
Type: preprint
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.
Source: arXiv (Cornell University)
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Awards: Dynamical Systems: New Developments in Theory and Application, Applications of Computational and Topological Methods to Dynamical Systems, G2405702604, On a Combinatorial Characterization of Dynamical Invariant Sets of Regulatory Networks, G295657101, G3064008058, Tripods+X:Res: Collaborative Research: Identification of Gene Regulatory Network Function from Data, G4713059963, G5270150132, G6517621824, Cultivation of new methods for studying spatio-temporal rhythm dynamics through collaboration of theory and experiment, G668241585, HDR TRIPODS: Data Science Principles of the Human-Machine Convergence, Applications to the topological computation for coupled network dynamical systems of Large degrees of freedom, Tripods+X:Res: Collaborative Research: Identification of Gene Regulatory Network Function from Data, G9000050326, G9000205806, G9000209852
OpenAlex ID: W4405469707