Inherent Frequency and Spatial Decomposition of the Lorenz Chaotic Attractor

TL;DR

This paper introduces a novel method to analyze Lorenz chaos by decomposing it into spatial scrolls and frequency components, revealing an inherent frequency linked to system parameters and clarifying chaos manifestation.

Contribution

It presents a new approach to decompose Lorenz attractors into spatial and frequency domains, identifying an inherent frequency that characterizes chaos.

Findings

Lorenz attractor has an inherent frequency determined by system parameters.

Chaos mainly occurs during transitions between scrolls, not within them.

The method applies to other double-scroll chaotic systems like Chua's attractor.

Abstract

This letter suggests a new way to investigate 3-D chaos in spatial and frequency domains simultaneously. After spatially decomposing the Lorenz attractor into two separate scrolls with peaked spectra and a 1-D discrete-time zero-crossing series with a wide-band spectrum, it is found that the Lorenz chaotic attractor has an inherent frequency uniquely determined by the three system parameters. This result implies that chaos in the Lorenz attractor is mainly exhibited when the trajectory crosses from one scroll to another, not within the two scrolls. This is also true for some other double-scroll Lorenz-like chaotic attractors, such as Chua's attractor. Some possible applications of the inherent frequency and the spatial decomposition are also discussed.