Chaos

Deterministic-chaos random number generators for stochastic computing bitstream encoding. Provides alternatives to linear PRNGs (LFSR, Mersenne Twister) with desirable statistical properties for SC arithmetic.

Why Chaotic RNG for Stochastic Computing?

Stochastic computing encodes values as the probability of a 1-bit in a random bitstream. The quality of the random source directly affects arithmetic accuracy. Linear PRNGs have short-range correlations that bias SC multiplication (AND gates). Chaotic maps produce sequences with:

• Broadband spectrum — no periodic structure to alias with SC gate frequencies
• Low short-range autocorrelation — adjacent bits are nearly independent
• Deterministic reproducibility — same seed → same bitstream on hardware
• Minimal state — one float (logistic map) vs 624 words (MT19937)

Available Generators

ChaoticRNG — Logistic Map

The logistic map x_{n+1} = r * x_n * (1 - x_n) at r=4.0 is fully chaotic with Lyapunov exponent ln(2) ≈ 0.693. The invariant density is Beta(0.5, 0.5) on (0, 1) — values cluster near 0 and 1. The generate_bitstream() method applies the inverse CDF (2/π) * arcsin(√x) to uniformize before thresholding.

Parameter	Default	Meaning
r	4.0	Bifurcation parameter. Must be in (3.57, 4.0] for chaos.
x	0.37	Initial condition in (0, 1). Avoid 0, 0.5, 1 (fixed/periodic points).
burn_in	100	Steps to discard before first output.

Analysis methods:

• lyapunov_exponent(n_steps) — Estimate maximal Lyapunov exponent via derivative averaging. At r=4.0 the theoretical value is ln(2) ≈ 0.6931.
• shannon_entropy(n_samples, n_bins) — Estimate Shannon entropy in bits. At r=4.0, values follow Beta(0.5, 0.5) with entropy ~log2(n_bins) - 0.27 bits below uniform.
• autocorrelation(n_samples, max_lag) — Compute autocorrelation up to max_lag. A good chaotic RNG shows near-zero autocorrelation for all lags > 0.

TentMapRNG — Piecewise Linear Alternative

The tent map x_{n+1} = μ * min(x_n, 1 - x_n) is topologically conjugate to the logistic map at r=4 but has uniform invariant density on (0, 1) — no CDF correction needed for SC bitstreams.

Parameter	Default	Meaning
mu	1.9999	Slope parameter. Must be in (1, 2]. Default slightly below 2.0 to avoid float64 degeneracy.
x	0.37	Initial condition in (0, 1).

Reference: Phatak & Rao, "Logistic map as a random number generator", Physical Review E 51(4), 1995.

Usage

from sc_neurocore.chaos import ChaoticRNG, TentMapRNG

# Logistic map — generate SC bitstream
rng = ChaoticRNG(r=4.0, x=0.37)
bitstream = rng.generate_bitstream(p=0.7, length=10000)
print(f"P(1) = {bitstream.mean():.3f}")  # ≈ 0.700

# Quality check
print(f"Lyapunov: {rng.lyapunov_exponent():.4f}")  # ≈ 0.6931
print(f"Entropy: {rng.shannon_entropy():.2f} bits")

# Tent map — uniform output, no CDF correction
tent = TentMapRNG(mu=1.9999, x=0.37)
samples = tent.random(10000)
print(f"Mean: {samples.mean():.3f}")  # ≈ 0.500

# Vectorized parallel maps for bulk generation
bulk = rng.random_vectorized(size=100000, n_maps=8)

::: sc_neurocore.chaos.rng options: show_root_heading: true members: - ChaoticRNG - TentMapRNG