Topological fingerprinting of dynamical systems

Poincaré established a framework for understanding the dependence of a dynamical system's properties on its topology. Topological properties offer detailed insights into the fundamental mechanisms — stretching, squeezing, tearing, folding, and twisting — that govern the shaping of a dynamical system's flow in state space. These mechanisms serve as a conduit between the system's dynamics and its topology [Ghil & Sciamarella, NPG, 2023]. A topological analysis based on the templex approach [Charó et al., Chaos, 2022] involves finding a topological representation of the underlying structure of the flow by the construction of a cell complex that approximates its branched manifold and a directed graph on this complex. A pivotal feature of the cell complex that facilitates the characterization of the flow dynamics is the joining locus, upon which all the fundamental mechanisms that sculpt the flow leave a pronounced signature.The local dimension d(x) and the inverse persistence θ(x) of the state x of a dynamical system [Lucarini et al., 2016; Faranda et al., Sci. Rep., 2017] provide information on the rarity and predictability of specific states, respectively. We demonstrate herein that these two measures, d and θ  also provide information about the localization of the joining locus.The present work proposes a new topological method for fingerprinting a system’s nonlinear behavior using the concept of persistent generatexes. This novel approach integrates the strengths of two topological data analysis methods: the templex and persistent homologies. Rather than employing a single cell complex and a digraph to characterize the flow of the system, our approach emphasizes the localization of the joining locus through the calculation of local dimension and the inverse persistence, leading to the construction of a family of nested digraphs. The dynamical paths, namely the nonequivalent ways of travelling through the flow, are found to be the most persistent cycles; here the concept of persistence is used in the sense of the persistent homology approach [Edelsbrunner & Harer, Contemporary mathematics, 2008]. The dynamical paths give us the ‘topological fingerprinting’ of a system’s dynamics.