Persistence collapse transitions and observable admissibility in nonlinear dynamical systems: a preregistered empirical benchmark

Abstract

Abstract Detecting transitions between dynamical regimes in nonlinear systems requires both a reproducible measurement protocol and an appropriately chosen observable. We introduce a preregistered empirical benchmark for persistence collapse transitions —the sharp thresholds at which a trajectory ensemble that has been maintaining a measured quantity within a block-mean viability corridor abruptly loses this capacity as a control parameter varies. Applied across seven canonical nonlinear systems including the standard map, tent map, Lorenz-63 attractor, Arnold cat map, and baker map, the benchmark produces a central empirical result: within this finite-time block-mean persistence protocol, transition detection depended strongly on whether the chosen observable tracked the degree of freedom constrained by the relevant invariant or attractor structure. In the standard map, the momentum observable detects a finite-time persistence threshold at $$K^* \approx 1.507$$ , well above the Chirikov last-torus estimate $$K_c \approx 0.9716$$ ; the detected threshold marks the onset of sufficient post-critical transport to drive ensemble corridor exit rather than the mathematical destruction of the last invariant torus. The position observable detects no transition across $$K \in [0, 2]$$ . An action proxy observable ( $$J = p^2/2\pi$$ ) finds a threshold at $$K^* \approx 1.508$$ with normalised spread $$r = 0.0005$$ , supporting the interpretation that the detection signal is shared across momentum-space observables under this protocol. Post-review bootstrap confidence intervals and a local refined control grid support the stability of the standard map momentum and action thresholds under resampling and grid refinement. A targeted one-factor sensitivity analysis shows momentum and action thresholds are stable across corridor width, block size, ensemble size, trajectory horizon, and independent sampled-initial-condition seed variation; Lorenz noise shows moderate protocol sensitivity. The adversarial null is an iid uniform process; the null validation result is scoped to iid processes and does not extend to correlated stochastic alternatives. All confirmatory results derive from a locked preregistration (300 trajectories per cell, seed 12345, 16760 summary rows, 0 standards failures).

The authors' abstract, as published at the source. Scientific Reports, 2026 · DOI ↗