Parametric characterisation of a chaotic attractor using two scale Cantor measure

TL;DR

This paper introduces a minimal parametric method to characterize chaotic attractors by mapping their multifractal spectrum onto a two-scale Cantor measure, applicable across various systems and extended to higher dimensions.

Contribution

It presents a novel parametric approach to characterize chaotic attractors using a two-scale Cantor measure mapping, including a generalization to higher dimensions.

Findings

Mapping is applicable to many chaotic systems.

The method captures multifractal properties up to two scales.

The approach extends to higher-dimensional attractors.

Abstract

A chaotic attractor is usually characterised by its multifractal spectrum which gives a geometric measure of its complexity. Here we present a characterisation using a minimal set of independant parameters which are uniquely determined by the underlying process that generates the attractor. The method maps the f(alpha) spectrum of a chaotic attractor onto that of a general two scale Cantor measure. We show that the mapping can be done for a large number of chaotic systems. In order to implement this procedure, we also propose a generalisation of the standard equations for two scale Cantor set in one dimension to that in higher dimensions. Another interesting result we have obtained both theoretically and numerically is that, the f(alpha) characterisation gives information only upto two scales, even when the underlying process generating the multifractal involves more than two scales.