The Kolmogorov-Arnold-Moser theorem

As already noted, the strength of the interaction is a crucial parameter the KAM theorem (after Kolmogorov, Arnold and Moser [523, 524, 525] expresses the fact that the invariant tori on which the EBK quantisation (see above) is based are, in the classical system, replaced by very complex fractal structures in which only residues of the tori can survive. Thus, a progressively increasing volume of phase space is lost to chaotic orbits as the strength of the dynamical coupling between the modes is increased. 

The KAM theorem states that, if a nondegenerate unperturbed Hamiltonian is subjected to a sufficiently small, conservative, Hamiltonian perturbation, most of the invariant tori survive, being only slightly deformed, in the phase space of the perturbed system. These invariant tori become densely filled with the trajectories of the perturbed system, which wind around them conditionally-periodically, the number of independent frequencies being equal to the number of degrees of freedom. 

The KAM theorem, it should be noted, has nothing to say about what happens when the strength of the perturbation increases. However, a considerable amount of experience has accumulated from detailed numerical calculations performed for many systems. One can visualise the results by studying Poincare sections if a cut is made across an invariant torus (see fig. 10.3) and a numerical calculation of trajectories is performed over a sufficiently long time, the stable orbits fill the deformed tori densely, and so result in closed curves in the two-dimensional cut, whereas the irregular or chaotic orbits yield a random speckle. 

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Although in a weakly time-dependent flow all resonant tori disappear together with some of the nearly resonant tori around them, the Kolmogorov-Arnold-Moser theorem ensures that infinitely many invariant surfaces survive a small perturbation. For sufficiently small e the remaining invariant surfaces formed by quasiperiodic orbits, so called KAM tori, still occupy a non-zero volume of the phase space. The condition for a torus to survive a given perturbation is that its rotation number should be sufficiently far from any rational number so that the inequality... 

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