C. Hinsley

27 September 2025

I’ve come across a lot of little-known tools for analyzing dynamical systems, both physical and theoretical. Since many of these tools are of significant practical utility and can be used in conjunction with one another, I am writing this list to give these tools more exposure. Each of these tools is something I have used and can vouch for — tools that sound nice but often fail in practice are omitted. Numerical methods listed are data-driven and do not require heavy parameter tuning or machine-learning techniques.

Attractor chaos quantification

Tools labeled “local” pertain only to individual orbits within attractors (though they may often be used on a finite set of critical orbits to obtain global information), while “global” labels indicate tools that give information about the entire attractor.

• Partition entropy rate approximation via Lempel-Ziv ‘86 complexity (local)
• Power spectral density (local)
• SRB measures & Young towers (global)
• Topological entropy from time-weighted Poincaré return maps (cf. Milnor-Thurston kneading theory; global)

Attractor classification

• Carpet property (Hittmeyer et al., used to recognize blenders)
• Covariant Lyapunov vectors (Ginelli et al., can be used to classify pseudohyperbolic or quasistochastic, can be useful for determining templates, can be useful for study of vector bundles over attractors)
• Kneading diagrams (can be used to determine wildness in some cases)
• Lyapunov exponents (hyperchaos; cf. ChaosTools.jl)
• Templates/templexes (extracting algebraic invariants from attractors: knots/links, branch torsion/writhing, and more from templates; homologies from templexes; cf. The Topology of Chaos by Gilmore & Lefranc and Knots and Links in Three-Dimensional Flows by Ghrist, Holmes, & Sullivan for an introduction to templates)

Attractor dimension estimation

• Correlation dimension (As far as I know, this is the only general and robust way to find the attractor dimension for physical systems)
• False Nearest Neighbors (used with time-delay embedding)
• Kaplan-Yorke/Lyapunov dimension (Kaplan-Yorke conjecture)