Ergodic Theory and Visualization I: Mesochronic Plots for Visualization of Ergodic Partition and Invariant Sets

TL;DR

This paper introduces Mesochronic Plots and Scatter Plots for visualizing invariant sets and ergodic partitions in phase space, using time averages of observables, with applications to complex dynamical maps.

Contribution

It develops algorithms for computing ergodic partitions and introduces Mesochronic Plots and Scatter Plots as new visualization tools for high-dimensional dynamical systems.

Findings

Effective visualization of invariant structures in high-dimensional phase spaces.

Application to standard and higher-dimensional maps demonstrating method utility.

Insights into resonance merging and ergodicity conjectures in complex maps.

Abstract

We present a computational study of a visualization method for invariant sets based on ergodic partition theory, first proposed in [1,2]. The algorithms for computation of the time averages of observables on phase space are developed and used to provide an approximation of the ergodic partition of the phase space. We term the graphical representation of this approximation - based on time averages of observables - a Mesochronic Plot (from Greek: \textit{meso} - mean, \textit{chronos} - time}. The method is useful for identifying low-dimensional projections (e.g. two-dimensional slices) of invariant structures in phase spaces of dimensionality bigger than two. We also introduce the concept of the ergodic quotient space, obtained by assigning a point to every ergodic set, and provide an embedding method whose graphical representation we call the Mesochronic Scatter Plot (MSP). We use the Chirikov standard map as a well-known and dynamically rich example in order to illustrate the implementation of our methods. In addition, we expose applications to other higher dimensional maps such as the Fro\'eschle map for which we utilize our methods to analyze merging of resonances and, the three-dimensional Extended standard map for which we study the conjecture on its ergodicity [3]. We extend the study in our next paper [4] by investigating the visualization of periodic sets using harmonic time averages. Both of these methods are related to eigenspace structure of the Koopman operator.