Invariant measures

An invariant measure is preserved by a system. This tool compares a finite orbit with selected known references; it does not prove invariance, ergodicity or mixing. Choose a system, retained sample count, initial state and parameters, then Compute. The first iterations are discarded before collecting N samples. Histogram settings change the number of equal-width bins. Editing removes the previous result. Calculation runs in a cancellable worker; progress counts generated steps including discarded ones.

Blue bars are density estimates count/(N·bin width), so their integrated area is1. Unit-interval maps always use[0,1]; the Lorenz x-marginal uses the observed range, widened for a constant series. The final bin includes its right endpoint. Orange steps are exact reference bin masses divided by bin width; this avoids drawing the logistic density’s infinite endpoints. Reference masses are available in the bin table.

Logistic uses x′=rx(1−x),0≤r≤4. Only exactly r=4 has the reference ρ(x)=1/(π√(x(1−x))), mean1/2. Default r=4 is deliberate. Try x₀=.25 (fixed orbit .75) or .5 (eventually0): exceptional orbits differ from this continuous reference. Gauss uses fractional part(1/x), with0→0; ρ(x)=1/(ln2·(1+x)), mean1/ln2−1. Rational or finite-precision trajectories can reach0. Rotation uses fractional part(x+α); uniform density1 and mean1/2 form an invariant reference for any α, without asserting uniqueness for rational α.

Lorenz integrates ẋ=σ(y−x), ẏ=x(ρ−z)−y, ż=xy−βz by RK4 with fixed Δt=.006. Edit all three initial coordinates. Default β=2.6667 is the existing numerical preset. The x-distribution is an empirical marginal; no analytic reference is asserted. Diverged/nonfinite orbits are rejected completely.

The running average uses all retained samples, with population variance. At most600 points are displayed in its curve, including the last one; this is display thinning only. A matching mean is not proof that distributions agree or that an orbit is typical. Sampling, correlations, initial conditions and floating-point periodicity affect results; no confidence interval or ergodicity verdict. Resize redraws cached results. PNG exports both active charts. Valid settings may be saved locally; reload waits for Compute.

CMU · Logistic measure, pp.1–2 · Manchester · Gauss/rotations, §§3.3,8.6