LYAPUNOV_EXPONENTS
QUANTIFYING_CHAOS // MODULE_10
SYSTEM_PARAMETERS
2.500
4.000
1000
500
λ = lim(n→∞) (1/n) Σ ln|r(1-2xᵢ)|
LIVE_RENDER // D3.JS
DATA_LOG: LYAPUNOV_ANALYSIS
The Lyapunov exponent provides the most fundamental quantitative measure of chaos, characterizing the average rate at which nearby trajectories diverge in phase space. A positive exponent indicates exponential divergence of nearby trajectories and serves as the primary diagnostic for chaos.
Lyapunov Time Examples
- • Weather systems: ~5 days
- • Chaotic circuits: ~1 ms
- • Solar system: 4-5 million years
Kaplan-Yorke Dimension
DL = j + (λ₁ + ... + λj) / |λj+1|
Relates Lyapunov exponents to fractal dimension