Topological Signatures of History-Dependent Invariants in the Lorenz System: Deterministic Probes of Lobe Dynamics

TL;DR

The paper tackles how history-dependent invariants can classify and reveal topology in the Lorenz attractor by constructing an extended phase space with memory via an auxiliary variable. A permutation-based orthogonality method yields 18 invariants organized into three regularization classes, with Class III containing the bilinear term $xy$ and acting as a topological probe of lobe-switching. High-precision numerical validation confirms exact conservation to $O(10^{-36})$ and demonstrates that the differential signal $\Delta S=Q_{\mathrm{III}}-Q_{\mathrm{I}}$ provides a predictive precursor for separatrix crossings whose latency scales with a power law in spike amplitude and linearly with the stable eigenvalue $\beta$. The work connects to Mori-Zwanzig memory, template theory, and symbolic dynamics, offering robust tools for trajectory classification, orbit fingerprinting, and topology-aware prediction in chaotic dissipative systems.

Abstract

We present a systematic classification of history-dependent dynamical invariants in the Lorenz system, identifying eighteen distinct quantities ($K_1$--$K_{18}$) organized into three regularization classes. These are defined by polynomials vanishing on flow nullclines: $p_{\mathrm{I}} = y - x$, $p_{\mathrm{II}} = y + x(z-ρ)$, and $p_{\mathrm{III}} = xy - βz$. Statistical analysis of $N = 4 \times 10^6$ trajectories reveals that Class~III invariants, containing the bilinear product $xy$, function as geometric probes of lobe-switching. In the chaotic regime ($ρ= 28$), Class~III exhibits elevated kurtosis ($κ_{\mathrm{III}} \approx 15.8$ vs $κ_{\mathrm{I}} \approx 8.2$) and heavy tails, a divergence absent in the pre-chaotic phase ($ρ= 23$). Crucially, the differential response $ΔS = Q_{\mathrm{III}} - Q_{\mathrm{I}}$ predicts separatrix crossings with a horizon governed by a shifted power law $Δt = t_{\min} + C\mathcal{A}^{-n}$. Parametric analysis shows $n$ scales linearly with $β$ ($m \approx 0.196$), where $n \approx 1.0$ at $β=8/3$ reflects relaxation along the stable manifold. High-precision validation (80-digit arithmetic) confirms conservation to $O(10^{-36})$, establishing Class~III invariants as encoders of the attractor's symbolic dynamics.

Figures (2)

• Figure 1: Geometric structure of the regularization classes. (a) Three-dimensional representation of the level surfaces of the invariant polynomials in the Lorenz phase space: $P_5 = c_1$ (Class I, blue), $P_{11} = c_2$ (Class II, green), and $P_{13} = c_3$ (Class III, red), where the constants are anchored to an unstable periodic orbit. A representative chaotic trajectory (dark curve) illustrates the system's evolution relative to these conserved manifolds. The black dot marks a point on the UPO. (b) Two-dimensional cross-section in local coordinates $(\eta, \xi)$ on an optimized Poincaré plane showing the topological relationship between the three regularization classes: $K_5 = c_1$ (Class I, blue solid), $K_{11} = c_2$ (Class II, green dashed), and $K_{13} = c_3$ (Class III, red dot-dashed). The intersection point corresponds to configurations where all three polynomial constraints are satisfied simultaneously on the UPO. Parameters: $\sigma = 10$, $\rho = 28$, $\beta = 8/3$.
• Figure 2: Statistical signatures and topological precursor detection. Probability density functions of standardized evolution functions $Q_{\mathrm{I}}$ (Class I, solid blue) and $Q_{\mathrm{III}}$ (Class III, dashed red) on semi-logarithmic scale, with Z-score normalization. (a) Pre-chaotic regime ($\rho = 23$): distributions are statistically similar ($\kappa \approx 28.2$ for both), establishing that no intrinsic bias exists in the Class III formulation. (b) Chaotic regime ($\rho = 28$): Class III develops heavier tails ($\kappa_{\mathrm{III}} \approx 15.8$ versus $\kappa_{\mathrm{I}} \approx 8.2$) extending beyond $\pm 5\sigma$, indicating enhanced intermittency at topological transitions. Both classes exhibit pronounced negative skewness ($S_{\mathrm{I}} \approx -1.66$, $S_{\mathrm{III}} \approx -2.73$). (c) Scaling law for the topological precursor. Latency $\Delta t$ (time from precursor signal to separatrix crossing) versus spike amplitude $\mathcal{A}$ of the differential signal $\Delta S = Q_{13} - Q_5$. Gray points: individual events; black circles: logarithmically binned averages; green curve: fitted shifted power law $\Delta t = t_{\min} + C\mathcal{A}^{-n}$ with $t_{\min} \approx 0.14$ and $n \approx 1.0$ at $\beta = 8/3$. Parametric analysis reveals that $n$ scales linearly with $\beta$ (slope $m \approx 0.196$, $R^2 = 0.94$), establishing that predictive capability is governed by relaxation along the stable manifold. Statistics computed from $N = 4 \times 10^6$ trajectory samples.