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Chaos Theory Analysis

Chaos analysis helps distinguish genuinely random behavior from deterministic systems that only look random.

Implemented in openentropy_core::chaos and openentropy_core::analysis.

What it is: A structure/complexity analysis that checks for deterministic dynamics in random-looking data.

Use it for: Research characterization when forensic/entropy checks are not enough to explain behavior.

Input shape: One byte stream (bytes / &[u8]).

Use this when

Section titled “Use this when”
  • You are doing research characterization, not just pass/fail validation.
  • You need to separate random-looking output from structured/chaotic dynamics.
  • You want complexity metrics beyond baseline forensic tests.

Tiers

Section titled “Tiers”
  • Core tier (--chaos): Hurst, Lyapunov, correlation dimension, BiEntropy, epiplexity
  • Extended tier (--chaos-extended): Sample entropy, Approximate entropy, DFA, RQA, rolling/bootstrap Hurst, permutation entropy, Anderson-Darling

Implementation note by module:

  • openentropy_core::chaos: Hurst, Lyapunov, correlation dimension, BiEntropy, epiplexity, Sample entropy, DFA, RQA, rolling Hurst, bootstrap Hurst
  • openentropy_core::analysis: Approximate entropy, permutation entropy, Anderson-Darling

Hurst Exponent

Section titled “Hurst Exponent”

Measures long-range dependence (R/S analysis).

  • H ~= 0.5: random-walk-like
  • H > 0.5: persistent trend behavior
  • H < 0.5: anti-persistent behavior

Lyapunov Exponent

Section titled “Lyapunov Exponent”

Measures sensitivity to initial conditions.

  • lambda ~= 0: no clear deterministic chaos signature
  • lambda > 0: chaotic divergence
  • lambda < 0: convergent behavior

Correlation Dimension

Section titled “Correlation Dimension”

Measures attractor dimensionality.

  • High D2 suggests high-dimensional/random-like behavior
  • Low D2 can indicate deterministic low-dimensional structure

BiEntropy

Section titled “BiEntropy”

Measures entropy persistence through derivative levels of the bitstream.

  • Higher values indicate stronger disorder and less structure

Epiplexity

Section titled “Epiplexity”

Compression-ratio complexity metric.

  • Ratio near 1.0 indicates incompressible/random-like data
  • Lower ratios imply compressible structure

Extended Methods

Section titled “Extended Methods”
  • Sample entropy (sample_entropy): irregularity/complexity estimator (SampEn)
  • Approximate entropy (approximate_entropy): ApEn regularity metric
  • DFA (dfa): long-range correlation estimate via detrended fluctuations
  • RQA (rqa): recurrence structure and determinism metrics
  • Rolling Hurst (rolling_hurst): local H estimate across windows
  • Bootstrap Hurst (bootstrap_hurst): uncertainty intervals and surrogate p-value
  • Permutation entropy (permutation_entropy): ordinal-pattern complexity
  • Anderson-Darling (anderson_darling): distribution conformity test used in extended tier
Section titled “Related”