S as Chaos Detector
Theorem Statement
Define the convergence score as the ratio of late-window drift to early-window drift:
\[ S = 1 - \frac{\text{drift}_{\text{late}}}{\text{drift}_{\text{early}}} \]where drift is measured as the mean absolute change over a sliding window. Then for the logistic map \( x_{n+1} = r \, x_n (1 - x_n) \):
- \( S \to 1 \) in the ordered regime (\( r < r_\infty \)) where the system converges to a fixed point or periodic orbit
- \( S \to 0 \) in the chaotic regime (\( r > r_\infty \)) where drift persists indefinitely
- \( S \) detects the Feigenbaum point \( r_\infty \approx 3.5699 \) as the transition boundary
Proof Sketch
In the ordered regime, the system converges to an attractor. Late drift decays exponentially, so \( \text{drift}_{\text{late}} / \text{drift}_{\text{early}} \to 0 \), giving \( S \to 1 \).
In the chaotic regime, the system has positive Lyapunov exponent. Drift does not decay, so \( \text{drift}_{\text{late}} \approx \text{drift}_{\text{early}} \), giving \( S \to 0 \).
The transition occurs at the accumulation of period-doubling bifurcations (the Feigenbaum point), detected by \( S \) without any model of the underlying dynamics.
S-\(\lambda\) Complementarity (Proposition 12.1)
The convergence score \( S \) and the Lyapunov exponent \( \lambda \) are complementary diagnostics:
| Property | S | \( \lambda \) |
|---|---|---|
| Requires model | No (model-free) | Yes (needs derivative) |
| Detects order | \( S = 1 \) | \( \lambda < 0 \) |
| Detects chaos | \( S = 0 \) | \( \lambda > 0 \) |
| Transition point | S drops sharply | \( \lambda \) crosses 0 |
| Computation | Windowed means only | Log-derivative sum |
| Applicable to | Any time series | Known dynamical systems |
Logistic Map Results
| Regime | \( r \) range | S value | \( \lambda \) sign |
|---|---|---|---|
| Fixed point | \( r < 3.0 \) | \( S \approx 1.0 \) | \( \lambda < 0 \) |
| Period-2 | \( 3.0 < r < 3.45 \) | \( S \approx 1.0 \) | \( \lambda < 0 \) |
| Period-4+ | \( 3.45 < r < 3.57 \) | \( S \approx 0.8 \text{--} 1.0 \) | \( \lambda < 0 \) |
| Feigenbaum | \( r \approx 3.5699 \) | \( S \) drops sharply | \( \lambda \approx 0 \) |
| Chaos | \( r > 3.57 \) | \( S \approx 0.0 \) | \( \lambda > 0 \) |
Significance
- Model-free — requires only a time series, no knowledge of the generating process
- Applicable to optimisation — detect when an adaptive system has entered chaotic dynamics (learning rate too high, conflicting objectives, etc.)
- Complementary — use S for black-box systems, \( \lambda \) when the model is known; they agree at all boundary points
Experiment Files
exp_chaos_boundary.sx — S across the logistic map bifurcation diagram, Feigenbaum point detection
exp_s_vs_lyapunov.sx — S-\(\lambda\) complementarity validation across parameter space