Exploratory experiment: Convergence Score on the Logistic Map
What is computed
The convergence score is the ratio of late-window drift to early-window drift:
\[ S = 1 - \frac{\text{drift}_{\text{late}}}{\text{drift}_{\text{early}}}, \]where drift is the mean absolute change of the iterate over a sliding window. It is evaluated on orbits of the logistic map \( x_{n+1} = r \, x_n (1 - x_n) \) for \( r \) from 2.5 to 4.0, and the Lyapunov exponent \( \lambda \) is computed on the same orbits from the log-derivative sum.
Recorded values
| 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 and higher | \( 3.45 < r < 3.57 \) | \( S \approx 0.8 \text{--} 1.0 \) | \( \lambda < 0 \) |
| Near \( r \approx 3.5699 \) | — | \( S \) drops sharply | \( \lambda \approx 0 \) |
| \( r > 3.57 \) | — | \( S \approx 0.0 \) | \( \lambda > 0 \) |
Where the orbit settles onto a fixed point or a periodic orbit the late-window drift is small and \( S \) is close to 1; where the orbit does not settle within the window the two drifts are comparable and \( S \) is close to 0. In this sweep the drop in \( S \) occurs near the period-doubling accumulation point.
Side by side with the Lyapunov exponent
| Property | S | \( \lambda \) |
|---|---|---|
| Input | A time series | The map and its derivative |
| Computation | Windowed means only | Log-derivative sum |
| Depends on | Window length and placement | Orbit length |
\( S \) depends on the window: an orbit with a long transient or a long period reads as unsettled at short windows. The runs above use a single window length, and the values are reported for that choice.
Experiment Files
exp_chaos_boundary.sx — \( S \) across the logistic-map parameter sweep
exp_s_vs_lyapunov.sx — \( S \) and \( \lambda \) recorded on the same orbits