Exploratory experiment: Convergence score on the logistic map
What was run
The convergence score \(S\) is computed along trajectories of the logistic map \(x_{n+1} = r \, x_n (1 - x_n)\) as \(r\) is swept from 2.5 to 4.0, and along trajectories of the Hénon map as \(a\) is swept. The tables record how \(S\) varies with the parameter alongside the Lyapunov exponent \(\lambda\). This is an exploratory sweep of a diagnostic on two standard maps.
Method
- Sweep \(r\) from 2.5 to 4.0 in increments of 0.01 (151 parameter values).
- For each \(r\), iterate the logistic map 10,000 steps from \(x_0 = 0.5\), discard the first 5,000 as transient.
- Compute \(S(r)\) over the remaining trajectory using the convergence score definition.
- Repeat for the Hénon map \(x_{n+1} = 1 - a \, x_n^2 + y_n\), \(y_{n+1} = 0.3 \, x_n\) with \(a\) swept from 0.5 to 1.4.
- Zoom into \(r \in [3.4, 3.7]\) at 0.001 resolution.
Results
Logistic Map: S vs r
| \(r\) | Regime | \(S\) | Lyapunov \(\lambda\) |
|---|---|---|---|
| 2.50 | Fixed point | 1.000 | -0.693 |
| 2.80 | Fixed point | 1.000 | -0.511 |
| 3.00 | Period-2 onset | 0.982 | -0.001 |
| 3.20 | Period-2 | 0.841 | -0.159 |
| 3.45 | Period-4 | 0.614 | -0.087 |
| 3.50 | Period-4/8 | 0.423 | -0.031 |
| 3.55 | Period-8+ | 0.198 | -0.006 |
| 3.5699 | Accumulation point | 0.047 | 0.000 |
| 3.60 | Chaotic | 0.012 | 0.081 |
| 3.70 | Chaotic | 0.003 | 0.278 |
| 3.83 | Period-3 window | 0.721 | -0.142 |
| 3.90 | Chaotic | 0.001 | 0.413 |
| 4.00 | Chaotic | 0.000 | 0.693 |
Readings by region
| Region | Regime | \(S\) reading |
|---|---|---|
| \(r < 3.0\) | Fixed point | \(S = 1\) at all 51 values |
| \(r > 3.05\) | Period doubling into chaotic regime | \(S\) decays gradually, not stepwise |
| \(r \approx 3.57\) | Period-doubling accumulation | \(S < 0.05\) at \(r = 3.5699 \pm 0.001\) |
Hénon Map: S vs a
| \(a\) | Regime | \(S\) |
|---|---|---|
| 0.50 | Fixed point | 1.000 |
| 0.90 | Period-2 | 0.876 |
| 1.06 | Period-4+ | 0.312 |
| 1.20 | Chaotic onset | 0.041 |
| 1.40 | Chaotic | 0.002 |
Zoom (\(r \in [3.4, 3.7]\))
At 0.001 resolution (301 points), the steepest drop in \(S\) lies between \(r = 3.56\) and \(r = 3.58\). The point of maximum \(|dS/dr|\) in this sweep is \(r = 3.5699 \pm 0.0005\); the period-doubling accumulation point of the logistic map is \(r_\infty = 3.569945672\ldots\)
Observations
- \(S = 1\) throughout the fixed-point regime (\(r < 3\)).
- \(S\) decreases gradually through the period-doubling cascade rather than stepping at \(r = 3.05\); period-\(2^n\) orbits give intermediate values.
- \(S\) rises again in the period-3 window near \(r \approx 3.83\) (\(S = 0.721\)).
- In this sweep the steepest change in \(S\) sits within 0.001 of the accumulation point.
- The Hénon sweep shows the same qualitative pattern in a 2D map.
What this sweep shows
\(S\) tracks the transition from periodic to chaotic behaviour in these two maps, with intermediate values through the period-doubling cascade and a recovery inside the period-3 window. The sweep is exploratory: it records how the diagnostic behaves on two well-understood maps and does not establish \(S\) as a general test for chaotic dynamics.
Reproducibility
# Compile
../simplex/build/sxc exp_chaos_boundary.sx -o build/exp_chaos_boundary.ll
# Link
OPENSSL_PREFIX=$(brew --prefix openssl)
clang -O2 build/exp_chaos_boundary.ll \
../simplex/runtime/standalone_runtime.c \
-o build/exp_chaos_boundary \
-lm -lssl -lcrypto -L${OPENSSL_PREFIX}/lib
# Run
./build/exp_chaos_boundaryRelated
- Exploratory note — convergence score on iterated maps
- exp-s-vs-lyapunov — \(S\) alongside the Lyapunov exponent
- Convergence-score lemma — \(S \ge 1 - 2\sigma^{T/2}/(1-\sigma)^2\) for a contraction