Table 5 The Chaos Decision Tree Algorithm uses permutation entropy, calculated from data that have been de-noised and downsampled (if oversampled), to track the degree of chaos in a system, which might change as the state of the system changes.

From: A simple method for detecting chaos in nature

  Measurement noise level (% of std. dev.)
System 0% 10% 20% 30% 40%
Logistic map52 0.93*** 0.80*** 0.93*** 0.93*** 0.91***
Hénon map61 0.92*** 0.93*** 0.94*** 0.92*** 0.88***
Lorenz system53 0.81*** 0.71*** 0.73*** 0.69*** 0.62***
Cortical model20 0.69*** 0.55*** 0.39*** 0.33*** 0.24***
Neuron integrated circuit27 0.94***     
  1. We here show Spearman correlations between permutation entropy and largest Lyapunov exponents, which measure degree of chaos but which are difficult to estimate from empirical data. Data include four simulated systems and recordings from an integrated circuit in different states. See Methods for how ground-truth largest Lyapunov exponents were calculated for these systems
  2. ***p < 0.001 (two-tailed) after Bonferroni-correcting for multiple comparisons to the same set of ground-truth largest Lyapunov exponents