Guidelines for data-driven approaches to study transitions in multiscale systems: the case of Lyapunov vectors

The paper formulates evidence-based guidelines: the θ12 alignment between the most unstable and (near-)neutral CLVs can rank the relative stability of metastable regimes and reveal fast transitions when (i) there is a clear time-scale separation and (ii) an invariant neutral direction that is also preserved by the FEM‑BV‑VAR fit; otherwise the method becomes hyper‑parameter sensitive and may fail. These claims are demonstrated on FitzHugh–Nagumo (works, transitions clearly flagged), von Kármán (works but N,n-sensitive), and Lorenz‑63 (fails due to loss of neutral direction in the reconstruction) . The candidate model provides a compatible theoretical justification under explicit assumptions H1–H2 (time-scale separation; neutral preservation) via dominated splitting, 2×2 reductions near folds, and stability/perturbation bounds for reconstructed cocycles and CLV algorithms; it also gives counterexamples when H1/H2 fail. One divergence is practical: the paper documents that too-large N can degrade results due to numerical accumulation (no uniform “N ≥ N0” guidance) , while the model assumes idealized convergence with increasing N (omitting a finite-precision upper bound). Substantively, both are consistent: the paper offers empirical guidance; the model supplies a conditional proof-level rationale.