FlowClass.jl: Classifying Dynamical Systems by Structural Properties in Julia

Abstract

FlowClass.jl is a Julia package for classifying continuous-time dynamical systems into a hierarchy of structural classes: Gradient, Gradient-like, Morse-Smale, Structurally Stable, and General. Given a vector field \(\mathbf{F}(\mathbf{x})\) defining the system \(\mathrm{d}\mathbf{x}/\mathrm{d}t = \mathbf{F}(\mathbf{x})\), the package performs a battery of computational tests -- Jacobian symmetry analysis, curl magnitude estimation, fixed point detection and stability classification, periodic orbit detection, and stable/unstable manifold computation -- to determine where the system sits within the classification hierarchy. This classification has direct implications for qualitative behaviour: gradient systems cannot oscillate, Morse-Smale systems are structurally stable in less than 3 dimensions, and general systems may exhibit chaos. Much of classical developmental theory going back to Waddington's epigenetic landscape rests on an implicit assumption of gradient dynamics. The package is designed with applications in systems and developmental biology in mind, particularly the analysis of gene regulatory networks and cell fate decision models in the context of Waddington's epigenetic landscape. It provides tools to assess whether a landscape metaphor is appropriate for a given dynamical model, and to quantify the magnitude of non-gradient (curl) dynamics.