Strong Trajectorial Ontological Differentiation: A novel approach to unravel phase-space structures

The identification of invariant objects and Lagrangian coherent structures is a cornerstone of dynamical systems. As a consequence, several diagnostic indicators have been established over time, such as the fast Lyapunov indicator, the finite-time Lyapunov exponent, and Lagrangian descriptors, among others. In this work, we introduce the Strong Trajectorial Ontological Differentiation (STOD) as a novel tool to identify phase-space structures. Unlike other indicators, STOD does not rely on the study of the tangent flow; instead, it identifies phase-space structures by comparing trajectories through a component-wise cancellation process inspired on the Ontological Differentiation (OD) that was originally developed for lexical networks [P. Garc\'ia-Cuadrillero, F. Revuelta, J. A. Capit\'an, Phys. Rev. E 113, 014305 (2026)]. By applying a reversed-time version of STOD (FinSTOD) to five paradigmatic autonomous and non-autonomous systems of increasing complexity, we show the excellent performance of this indicator in the identification of phase-space structures, adding a new useful tool to the chaotic toolbox.