Analysis Overview

Tools for characterizing the dynamical properties of attractors.

The analysis module provides a suite of tools for characterizing chaotic trajectories after they have been integrated.

While the basins module answers the question “Is this attractor hidden?”, the analysis module answers questions like:

  • “How large is the attractor in phase space?”
  • “What are its dominant frequencies?”
  • “Is it truly chaotic, or just a complex limit cycle?”

Available Tools

The analysis tools are broken down into the following categories:

  1. Trajectory Metrics: Fast geometric summaries of the trajectory cloud (bounding box, centroid, spread) and robustness calculations against equilibria.
  2. Spectral Analysis: Fast Fourier Transforms (FFT) to extract peak frequencies and compute spectral entropy.
  3. Poincaré Sections: Reducing the 3D continuous flow into a 2D discrete map by intersecting the trajectory with a plane.
  4. Lyapunov Exponents: (Requires nolds) Calculating the maximal Lyapunov exponent to definitively prove chaotic behavior.
  5. Recurrence: (Requires antropy) Computing the recurrence rate of the trajectory to analyze periodicity.
  6. Stability: Analyzing the Jacobian matrix and applying Matignon’s theorem for fractional-order stability.

Using Analysis in Workflows

Many of these tools are automatically executed during the standard workflows. For example, the unified_chua workflow automatically computes the cloud_summary for any non-divergent trajectory and saves it to the JSON output.

To use these tools manually, import them from hidden_attractors.analysis:

from hidden_attractors.analysis import cloud_summary, poincare_section

Explore the sub-pages in this section for detailed API usage and examples of each technique.