Structure analysis of the Lorenz-84 chaotic attractor

TL;DR

This work addresses the challenge of analyzing weakly dissipative chaotic attractors, using Lorenz-84 as a canonical test case. It introduces color tracer mapping and unstable periodic orbit extraction to build and validate a topological template, complemented by a skeleton analysis that reveals a toroidal, period-2–centered structure and a novel multidirectional stretching mechanism. The authors derive a squeezed template and a reduced six-branch template, and validate them by computing theoretical linking numbers that agree with numerical values, thereby providing a robust topological description of a thick, weakly dissipative attractor. The methodology offers a practical framework for understanding complex 3D chaos beyond strongly dissipative templates, with implications for analyzing real-world weakly dissipative systems.

Abstract

The structure of the Lorenz-84 attractor is investigated in this study. Its dynamics belonging to weakly dissipative chaos, classical approaches cannot be used to analyze its structure. The color tracer mapping is introduced for this purpose and used to extract the three-dimensional structure of the attractor. The analysis shows that the attractor is a non trivial case of toroidal chaos: it is organized around a period-2 cavity. Moreover, the structure reveals a new mechanism generating chaos in the attractor: a multidirectional stretching. The attractor structure is then artificially represented on a two-dimensional branched manifold and its validation performed using a set of periodic orbits previously extracted.

Figures (20)

• Figure 1: Topological characterization methodology applied to the Rössler attractor Ross76 using reproducible methodology. $lk()$ refers to linking number between unstable periodic orbits in knot theory.
• Figure 2: Lorenz--84 attractor solution to system \ref{['eq:lorenz84']} for the parameters $a=0.25$, $b=4.0$, $F=8.0$ and $G=1.0$. (a) $(x, y)$ projection of the attractor. (b) $\mathcal{L}$, the rotated Lorenz--84 attractor $(x_r, y_r, z_r)$ solution to system \ref{['eq:lorenz84']} where the flow evolves clockwise in the projection plane ($y_r, z_r$). The grey line is the Poincaré section \ref{['eq:lorenz_02_section']}. (c) Poincaré section.
• Figure 3: (a) Poincaré section \ref{['eq:lorenz_02_section']} of the attractor $\mathcal{L}$. (b) First return map to the Poincaré section (a) using $\rho_n$. (c) First return map to the Poincaré section (a) using $\gamma_n$\ref{['eq:gamma_n']}. Part I (color green) is the first part of \ref{['eq:gamma_n']} while part II (color pink) is the second line of \ref{['eq:gamma_n']}. This colored partition is also visible in (c) clarifying the relative position of periodic points of (b).
• Figure 4: The period--2 orbit (in blue) represented passing through a Poincaré section (in red) of the Lorenz--84 attractor (a) (a part of the Lorenz--84 attractor is also represented in light gray). Zooms on the Poincaré section are also presented (b) at the vicinity of the period--2 orbit crossings (blue circle). The crossing points (in green) around the period--2 orbit belong to the long transient to reach the Lorenz-84 attractor (in red).
• Figure 5: Colored map $\Gamma^{\alpha}$ at three successive iterations using the $\alpha$-palette defined at iteration $n$ as a vertical gradient of colors in a restricted part of the section. Note that a $\beta$-palette is simultaneously used at iteration $n$ with a vertical gradient of gray, in the remaining part of the section.