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| Natura: | Preprint |
| Pubblicazione: |
2025
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| Accesso online: | https://arxiv.org/abs/2512.20390 |
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| _version_ | 1866914457085542400 |
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| author | Toledo, B. A. |
| author_facet | Toledo, B. A. |
| contents | 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 Poincaré section analogue. The spike amplitude predicts switching latency via $Δt = t_{\min} + C\mathcal{A}^{-n}$ with $R^2 > 0.95$ across all parameter combinations tested. At canonical parameters $(σ, ρ, β) = (10, 28, 8/3)$, $n = 2.14 \pm 0.17$ with $R^2 = 0.93$ for individual events; the exponent increases with $β$ and decreases with $ρ$, while the $σ$-dependence is non-monotonic. The latency distribution reveals a topological gap of width $Δt_{\mathrm{gap}} \approx 0.68 \pm 0.01$ for $ρ$ 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-Bénard convection context, the auxiliary variables correspond to integrated heat-flux anomalies. Conservation is verified to $O(10^{-36})$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_20390 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Exact Conservation Laws of the Lorenz Attractor: Classification and Deterministic Prediction of Lobe-Switching Events Toledo, B. A. Chaotic Dynamics 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 Poincaré section analogue. The spike amplitude predicts switching latency via $Δt = t_{\min} + C\mathcal{A}^{-n}$ with $R^2 > 0.95$ across all parameter combinations tested. At canonical parameters $(σ, ρ, β) = (10, 28, 8/3)$, $n = 2.14 \pm 0.17$ with $R^2 = 0.93$ for individual events; the exponent increases with $β$ and decreases with $ρ$, while the $σ$-dependence is non-monotonic. The latency distribution reveals a topological gap of width $Δt_{\mathrm{gap}} \approx 0.68 \pm 0.01$ for $ρ$ 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-Bénard convection context, the auxiliary variables correspond to integrated heat-flux anomalies. Conservation is verified to $O(10^{-36})$. |
| title | Exact Conservation Laws of the Lorenz Attractor: Classification and Deterministic Prediction of Lobe-Switching Events |
| topic | Chaotic Dynamics |
| url | https://arxiv.org/abs/2512.20390 |