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

Area of Science:

• * Physics
• * Applied Mathematics
• * Computational Science

Background:

• * Extends prior work on visualizing ergodic partitions using Koopman operator theory.
• * Builds upon Mezić and Banaszuk's frequency analysis method for dynamical systems.
• * Leverages the concept of Fourier time average for analyzing system dynamics.

Purpose of the Study:

• * To apply and analyze a visualization method for measure-preserving dynamical systems.
• * To computationally identify periodic, quasi-periodic, and chaotic regions in phase space.
• * To demonstrate the method's utility in higher-dimensional dynamical systems.

Main Methods:

• * Utilizes frequency analysis grounded in Koopman operator theory.
• * Employs Fourier time average for computational analysis.
• * Develops algorithms for visualizing phase space partitions.

Main Results:

• * Successfully visualizes periodic and quasi-periodic sets in phase space.
• * Provides a method for identifying chaotic zones as the complement of periodic partitions.
• * Illustrates applicability using the Chirikov standard map, Froeschlé map, and Extended Standard Map.

Conclusions:

• * The developed visualization method is effective for analyzing measure-preserving dynamical systems.
• * The approach can distinguish between regular and chaotic dynamics.
• * Demonstrates potential for exploring complex, higher-dimensional systems.