Ergodic theory and visualization. II. Harmonic mesochronic plots visualize (quasi)periodic sets

TL;DR

This paper introduces a new visualization method for measure-preserving dynamical systems using harmonic time averages, enabling detection of periodic, quasi-periodic, and chaotic regions in phase space, applicable to complex systems.

Contribution

The paper develops a novel harmonic mesochronic plotting technique based on ergodic theory, extending previous methods to higher-dimensional systems for detailed phase space analysis.

Findings

Successfully visualizes periodic and quasi-periodic sets in various maps.

Effectively detects chaotic regions with high precision.

Demonstrates applicability to high-dimensional measure-preserving systems.

Abstract

We present a new method of analysis of measure-preserving dynamical systems, based on frequency analysis and ergodic theory, which extends our earlier work [1]. Our method employs the novel concept of harmonic time average [2], and is realized as a computational algorithms for visualization of periodic and quasi-periodic sets or arbitrary periodicity in the phase space. Besides identifying all periodic sets, our method is useful in detecting chaotic phase space regions with a good precision. The range of method's applicability is illustrated using well-known Chirikov standard map, while its full potential is presented by studying higher-dimensional measure-preserving systems, in particular Froeschl\'e map and extended standard map.