Logistic map: xₙ₊₁ = r·xₙ(1−xₙ) — pseudo-orbits are shadowed by true orbits
Anosov-Bowen (1975): for hyperbolic systems, every δ-pseudo-orbit (each step has error < δ) is ε-shadowed by a true orbit (ε ~ δ/gap).
Consequence: numerical trajectories (with roundoff δ) are valid representations of true dynamics, even though individual trajectories diverge exponentially.
The logistic map at r≈3.9 is chaotic: Lyapunov exponent λ > 0, distances grow as e^(λt). Yet the true orbit always stays within ε of the noisy orbit — for a while.