Measuring chaos in the Lorenz and Rössler models: Fidelity tests for reservoir computing

Abstract

This study is focused on the qualitative and quantitative characterization of chaotic systems with the use of symbolic description. We consider two famous systems: Lorenz and Rössler models with their iconic attractors, and demonstrate that with adequately chosen symbolic partition three measures of complexity, such as the Shannon source entropy, the Lempel-Ziv complexity and the Markov transition matrix, work remarkably well for characterizing the degree of chaoticity, and precise detecting stability windows in the parameter space. The second message of this study is to showcase the utility of symbolic dynamics with the introduction of a fidelity test for reservoir computing for simulating the properties of the chaos in both models' replicas. The results of these measures are validated by the comparison approach based on one-dimensional return maps and the complexity measures.

Figures (12)

• Figure 1: (A) The Lorenz attractor at the classic value $r=29$. The superimposed red dots defined by the $x$-variable critical events are well-aligned on some straight-line intervals transverse to the wings of the Lorenz butterfly in the phase space. (B) The $x$-variable plotted against time. Local maxima and minima marked with red dots are detected to convert the $x$-dynamics into binary sequences using the simple rule: $\{\dot x =0\ |\, x>0\} \to "0"$ and $\{\dot x=0\ |\, x<0\} \to "1"$ in this case.
• Figure 2: (A) Chaotic transient converging to a stable attractor (green), encoded as [$\overline{00001111}$], in the 3D phase space of the Lorenz model near a stability window at $r\simeq$69.67. (B) The $x$-variable time traces passes through a periodic pattern with the Markov transition probabilities $p_{00}=3/4$ and $p_{01}=1/4$.
• Figure 3: (A) Convergence to the stable periodic orbit (green), encoded as [$\overline{000111}$], after a long chaotic transient in the 3D phase space of the Lorenz model at $r=92.5$. Superimposed red dots defined as critical events $\{x'=0|\, x>0$ and $x<0\}$ fill out two hooks on the bending wings of the butterfly in the phase space. (B) The $x$-variable plotted against time reveals the attracting periodic pattern with the Markov transition probabilities $p_{11}=2/3$ and $p_{10}=1/3$.
• Figure 4: (A) Long chaotic transient (grey) towards a stable periodic orbit (green) in the 3D phase space of the Rössler model at $a=0.341$ and $c=4.8$. Black dots indicate the location of $z'=0$ events to generate binary sequences $\{...00011.,.\}$ depending on where the critical events occur below or above some $z$-threshold. (B) Spiking $z$-variable plotted against time becomes regularized to produce a periodic pattern of low complexity.
• Figure 5: Conditional entropy versus the words length for the Lorenz model. (A) Conditional entropy for chaotic sequences with $N=16988$ for $r=30$, and $N=28475$ for $r=75$. (B) Conditional entropy for periodic sequences with length $N=25672$ for $r=59.25$, and $N=32058$ for $r=92.5$. Open circles refers to the conditional entropy $h_m$ estimated from a sequence generated by the genuine Lorenz model. Filled black circles shows $h_m$ estimated from sequences generated by the trained reservoir computer whith the same lengths of binary sequences.