Recurrence Analysis

Quantifying periodicity using Recurrence Quantification Analysis (RQA).

Recurrence Quantification Analysis (RQA) is a method for studying the structure of dynamical systems by looking at how often the trajectory returns to previously visited regions in phase space.

The hidden-attractors-fo library wraps the antropy package to provide fast recurrence metrics.

Note: You must have installed the library with the [analysis] extra to use this feature.

Recurrence Rate (REC)

The fundamental metric of a Recurrence Plot is the Recurrence Rate (REC). It represents the percentage of points in the trajectory that are within a small distance ϵ\epsilon of each other (excluding points that are immediately adjacent in time).

from hidden_attractors.analysis import recurrence_rate
import numpy as np

# Extract the x1 coordinate
x1 = x[:, 0]

# Calculate the recurrence rate
# threshold (epsilon) determines how close points must be to count as a recurrence
rec = recurrence_rate(x1, threshold=0.1)

print(f"Recurrence Rate: {rec:.4f}")

Interpretation

  • High REC (e.g., > 0.5): The system is highly periodic (a limit cycle). The trajectory constantly repeats its path.
  • Low/Moderate REC (e.g., 0.01 - 0.1): The system is chaotic. It occasionally returns to similar states, but the motion is not strictly periodic.
  • Very Low REC (≈0\approx 0): The system is noisy or a transient that never returns to previous states (e.g., a divergent trajectory).