Exact Conservation Laws of the Lorenz Attractor: Classification and Deterministic Prediction of Lobe-Switching Events

TL;DR

This paper introduces algebraic conservation laws for the Lorenz attractor that enable deterministic prediction of lobe-switching events with high accuracy, revealing new invariants tied to the flow's nullclines.

Contribution

The work systematically identifies and classifies eighteen invariants, demonstrating their effectiveness in predicting lobe switches and analyzing their robustness under stochastic perturbations.

Findings

Achieves 99.2% sensitivity and 0.3% false-positive rate in predicting lobe switches.

Establishes a predictive relationship between spike amplitude and switching latency with high correlation.

Reveals a topological gap in latency distribution explained by the Shilnikov passage map.

Abstract

Predicting when a chaotic trajectory will switch between the lobes of the Lorenz attractor is a long-standing challenge in nonlinear dynamics. This work shows that algebraic conservation laws, constructed by augmenting phase space with history-accumulating auxiliary variables, provide a deterministic solution. Systematic enumeration identifies eighteen valid invariants in three classes, each tied to a nullcline of the Lorenz flow, while six candidates fail, proving that the dynamics constrains which conservation laws are admissible. One class generates sharp spikes synchronized with lobe-switching events, achieving $99.2\%$ sensitivity with $0.3\%$ false-positive rate ($\mathrm{AUC} = 0.9995$) as a continuous Poincar\'e section analogue. The spike amplitude predicts switching latency via $\Delta t = t_{\min} + C\mathcal{A}^{-n}$ with $R^2 > 0.95$ across all parameter combinations tested. At canonical parameters $(\sigma, \rho, \beta) = (10, 28, 8/3)$, $n = 2.14 \pm 0.17$ with $R^2 = 0.93$ for individual events; the exponent increases with $\beta$ and decreases with $\rho$, while the $\sigma$-dependence is non-monotonic. The latency distribution reveals a topological gap of width $\Delta t_{\mathrm{gap}} \approx 0.68 \pm 0.01$ for $\rho$ sufficiently above the onset of chaos, explained by the Shilnikov passage map. Under stochastic perturbations, lobe-sensitive invariants are ${\sim}\,10^3$ times more robust than their smooth counterparts. In the Rayleigh-B\'enard convection context, the auxiliary variables correspond to integrated heat-flux anomalies. Conservation is verified to $O(10^{-36})$.