History-Dependent Dynamical Invariants in the Lorenz System

TL;DR

The paper shows that the dissipative Lorenz system can possess a non-local, history-dependent invariant $C_1$, constructed by augmenting the phase space with an auxiliary variable and enforcing $dC_1/dt=0$. The invariant takes the form $C_1(t)=xy(z+1)+\tfrac{1}{2}x^2(z-\rho)-\tfrac{\beta z^2}{2}+\sigma(x-y)u$, with $\dot u=F(x,y,z)u+G(x,y,z)$ and a regularized variable $v=(x-y)u$ to avoid singularities. Numerical tests on both periodic orbits and chaotic trajectories confirm the constancy of $C_1$, and the authors interpret each UPO as a distinct constant value of the invariant, akin to a skeletal memory of the chaotic attractor. This work reframes non-integrability by showing that non-local order can persist alongside dissipation and chaos, and motivates further exploration of invariants, symmetries, and potential physical interpretations of the auxiliary variable. Overall, the study reveals a hidden, non-local structure in the Lorenz attractor and opens pathways for data-driven discovery of invariants in dissipative systems.

Abstract

Contrary to the established view of the Lorenz system as an archetype of dissipative chaos lacking conserved quantities, this work rigorously demonstrates the existence of a novel class of history-dependent dynamical invariants. Through a constructive method that augments the phase space, we derive a non-local invariant whose value remains constant along any trajectory. Its history-dependence arises from an integral term that accumulates the orbit's past, thereby ensuring its conservation. The invariant's constancy is verified with high-precision numerical simulations for both periodic and chaotic orbits. This finding reveals a hidden structure within the attractor and affords a new physical interpretation where unstable periodic orbits (UPOs) correspond to specific values of this conserved quantity. The result redefines the notion of non-integrability in dissipative systems, showing that non-local order can coexist with chaotic behavior.

Figures (3)

• Figure 1: The column with subindex 1 shows a stable periodic orbit for $\sigma=10, \rho=350, \beta=8/3$, displaying the time series of $x, y, z,$ and also of $u$ and $v$. The column with subindex 2 shows the same analysis for an unstable periodic orbit (UPO) with the classical parameters $\sigma=10, \rho=28, \beta=8/3$. This calculation was performed using multiprecision arithmetic at 150 digits for the continuous lines, while the data represented by dots were calculated at machine precision.
• Figure 2: Standard deviation of the regularized form of $C_1$ as a function of the number of digits of precision. Panel ($a$) corresponds to the stable orbit ($\sigma=10, \rho=350, \beta=8/3$), and panel ($b$) shows the case for the unstable orbit ($\sigma=10, \rho=28, \beta=8/3$). In each case the system was integrated for its corresponding period.
• Figure 3: A trajectory starting at a random point ($x_0=-16.164\ldots,\, y_0=-10.847\ldots,\, z_0=42.088\ldots$) on the attractor is shown in the upper figure. For the lower figures, ($b$), shows the dispersion of $C_1$ for an average value of $7033.8699495\ldots\;$ as evaluated along the trajectory (note the reference line for the maximal dispersion value). Here we use $\sigma=10, \rho=28, \beta=8/3$. On the right ($c$), for comparison, is shown the same calculation for the harmonic oscillator, with average energy $E=7000$. The calculations to evaluate $C_1$ and $E$, and their respective dispersions, used 50 digits of precision.