THE FOLD / SPAWN / NULL ISLAND / THE LYAPUNOV
THE LYAPUNOV
two futures from one place
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Two states start a hair apart and the gap grows by a constant factor every step. The Lyapunov exponent is the logarithm of that factor, and its sign is the whole diagnostic: negative means trajectories merge and the system forgets its initial condition, positive means they separate and the system amplifies it. For the logistic map at r = 4 the exponent is not merely positive but exactly ln 2 — one bit of the starting value is destroyed per step, and after 50 steps a double-precision number has no information left in it at all.
LIT verified live: the time average of ln|f′| along an orbit gives 0.693159, and integrating the same quantity against the exact invariant density 1/(π√(x(1−x))) gives 0.693148 — against ln 2 = 0.693147. At r = 3.5, where a stable four-cycle exists, the exponent is −0.872507. At the period-doubling accumulation it is −0.001163, essentially zero. And a gap of 10−12 grows to 4.4×10−6 in 25 steps, a measured rate of 0.682103.
LIT verified live: the time average of ln|f′| along an orbit gives 0.693159, and integrating the same quantity against the exact invariant density 1/(π√(x(1−x))) gives 0.693148 — against ln 2 = 0.693147. At r = 3.5, where a stable four-cycle exists, the exponent is −0.872507. At the period-doubling accumulation it is −0.001163, essentially zero. And a gap of 10−12 grows to 4.4×10−6 in 25 steps, a measured rate of 0.682103.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at NULL ISLAND — two futures leaving from the same coordinates.
AVAN (AI) computed the exponent twice, by unrelated routes, and the reason is specific rather than decorative. Floating-point orbits of x → 4x(1−x) are known to degrade: the map destroys a bit per step, so after about 50 iterations a double holds nothing of the true orbit, and a long time-average is summing over a trajectory the computer partly invented. The space average has no orbit in it at all — it integrates ln|f′| against the closed-form invariant density — so agreement between the two is meaningful in a way that either alone would not be. The measured separation rate comes out 0.682103 rather than 0.693147, about 1.6% low, and that is the same effect showing its face: the gap saturates once it reaches order 1, and the fit is pulled down by the last points.
AVAN (AI) computed the exponent twice, by unrelated routes, and the reason is specific rather than decorative. Floating-point orbits of x → 4x(1−x) are known to degrade: the map destroys a bit per step, so after about 50 iterations a double holds nothing of the true orbit, and a long time-average is summing over a trajectory the computer partly invented. The space average has no orbit in it at all — it integrates ln|f′| against the closed-form invariant density — so agreement between the two is meaningful in a way that either alone would not be. The measured separation rate comes out 0.682103 rather than 0.693147, about 1.6% low, and that is the same effect showing its face: the gap saturates once it reaches order 1, and the fit is pulled down by the last points.
3 ONE DIMENSION
The exponent across r. Below zero the system forgets; above it, it amplifies.
4 TWO DIMENSIONS · INTERACTIVE
Two orbits from almost the same place. Watch them come apart.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: a bundle of orbits from one neighbourhood, spreading.
AVAN’s addition (the inverse-companion): the forward reading is “chaos amplifies small differences.” The inverse is that nothing is being amplified — information is being read out. At r = 4 the map is conjugate to doubling an angle, and doubling in binary is a shift: each step discards the leading bit and promotes the next. The “unpredictable” behaviour was written in the initial condition’s low-order digits from the start, and the system is simply reciting them. Read backwards, ln 2 is not a rate of creation but a rate of exposure, and a chaotic system is less a generator of randomness than a very fast reader of one.
LIT the time average of ln|f'| along an orbit gives 0.693159, and integrating the same quantity against the exact invariant density 1/(pi sqrt(x(1-x))) gives 0.693148, against ln 2 = 0.693147; at r = 3.5, where a stable four-cycle exists, the exponent is -0.872507; at the period-doubling accumulation it is -0.001163, essentially zero; and a gap of 1e-12 grows to 4.4e-6 in 25 steps, a measured rate of 0.682103
FIG The exponent was computed TWICE by unrelated routes, for a specific reason. Floating-point orbits of x -> 4x(1-x) are known to degrade: the map destroys a bit per step, so after about 50 iterations a double holds nothing of the true orbit and a long time-average is summing over a trajectory the computer partly invented. The space average has no orbit in it at all - it integrates against the closed-form invariant density - so agreement between the two is meaningful in a way either alone would not be. The measured separation rate comes out 0.682103 rather than 0.693147, about 1.6% low, and that is the same effect: the gap saturates at order 1 and the fit is pulled down by the last points.
FIG The exponent was computed TWICE by unrelated routes, for a specific reason. Floating-point orbits of x -> 4x(1-x) are known to degrade: the map destroys a bit per step, so after about 50 iterations a double holds nothing of the true orbit and a long time-average is summing over a trajectory the computer partly invented. The space average has no orbit in it at all - it integrates against the closed-form invariant density - so agreement between the two is meaningful in a way either alone would not be. The measured separation rate comes out 0.682103 rather than 0.693147, about 1.6% low, and that is the same effect: the gap saturates at order 1 and the fit is pulled down by the last points.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of NULL ISLAND · David Lee Wise (ROOT0), with AVAN