Kolmogorov–Arnold–Moser theoremtheorem
In 1954, Andrey Kolmogorov proposed the idea, and Vladimir Arnold and Jürgen Moser rigorously proved in the 1960s that in Hamiltonian systems (where energy is conserved), under small perturbations, so-called invariant tori — multidimensional surfaces on which motion remains regular — are preserved. This is like a layer cake: some layers (tori) can withstand a light shaking, while others crumble into chaos. The theorem established a boundary between order and chaos.
How it works
Thanks to this theorem, engineers can be confident that particles in ring accelerators will not crash into the walls due to minor magnetic field errors, and astronomers understand that exoplanet orbits can exist for long periods.
💡 It is precisely the KAM theorem that explains why Saturn's rings have sharply defined gaps — resonant perturbations from moons destroy the regular orbits of particles in these zones.
|\mathbf{k} \cdot \boldsymbol{\omega}| \ge \frac{\gamma}{|\mathbf{k}|^\tau}
k is an integer vector with components k₁,...,kₙ, ω is the frequency vector, γ > 0 is a small constant depending on the perturbation, τ > n−1 is a parameter defining the Diophantine property; the symbol |k| denotes the sum of absolute values |k₁|+...+|kₙ|.