Kolmogorov-Sinai Entropy in Nonlinear Dynamical Systems

Povzetek

Kolmogorov-Sinai (KS) entropy quantifies the rate at which a chaotic dynamical system generates information, and
Pesin's theorem relates it to the sum of positive Lyapunov exponents, but practical estimation of KS entropy from
finite, noisy trajectory data remains substantially less reliable than computing Lyapunov exponents directly,
particularly in spatially extended systems. We computed KS entropy via three independent routes, direct
Lyapunov-exponent summation under Pesin's theorem, symbolic-dynamics block entropy, and a
Grassberger-Procaccia correlation-entropy estimator, for a one-dimensional coupled map lattice of logistic maps
with 8 to 128 lattice sites and coupling strengths from 0.0 (uncoupled) to 0.5, across 65 parameter combinations,
each analyzed from time series of 2×10^5 iterations after transient removal. In the uncoupled limit, all three
estimators agreed with the Pesin-theorem value to within 3% at the fully chaotic control parameter (r = 4.0), but
Grassberger-Procaccia correlation-entropy estimates diverged sharply as lattice size increased, underestimating
true KS entropy by 41% at 128 sites and moderate coupling (0.2), a consequence of the curse of dimensionality
afflicting correlation-integral estimation in high-dimensional phase space. Symbolic-dynamics block entropy tracked
the Pesin-theorem value more robustly, remaining within 12% even at 128 sites, but required block lengths that
grew computationally prohibitive (>10^7 distinct symbol blocks) beyond 64 sites, imposing a practical system-size
ceiling on this method distinct from its accuracy limitations. Adding observational noise at signal-to-noise ratios from
40 dB down to 10 dB degraded the correlation-entropy estimator fastest, with error exceeding 50% below 20 dB,
while the Pesin-theorem estimate, computed from the full Jacobian spectrum, remained comparatively robust, with
error below 15% even at 10 dB, since Lyapunov-exponent estimation from noisy data benefited from ensemble
averaging across the lattice. KS entropy itself increased monotonically with coupling strength up to approximately
0.3, peaking at 2.1 bits per iteration at 128 sites, then declined by 18% as coupling strengths approached 0.5 and
the lattice entered a more spatially synchronized, lower-entropy regime. These results identify practical accuracy
and computational trade-offs among common KS entropy estimators and show that estimator choice matters
substantially more in high-dimensional, noisy, or strongly coupled spatially extended chaotic systems than in
low-dimensional benchmarks.