The Kolmogorov-Arnold-Moser theorem was outlined by Kolmogorov (1954) and subsequently proved in the 1960s by Arnol'd (1963) and Moser (1962; Tabor 1989, p. 105). It gives conditions under which chaos is restricted in extent. Moser's 1962 proof was valid for twist maps
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(1)
|
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(2)
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Arnol'd (1963) produced a proof for Hamiltonian systems
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The original theorem required perturbations 
, although this has since been significantly
increased. Arnol'd's proof required 
, and Moser's original proof required 
. Subsequently, Moser's version has been reduced to 
,
then 
,
although counterexamples are known for 
. Conditions for applicability of the KAM theorem are:
1. small perturbations,
2. smooth perturbations, and
3. sufficiently irrational map winding number.
Consider an integrable Hamiltonian system with Hamiltonian 
and an invariant torus 
carrying motion with frequency vector 
. Under the appropriate nondegeneracy and smoothness
assumptions, the KAM theorem guarantees persistence of tori
whose frequency vectors satisfy a suitable Diophantine condition, provided the perturbation
is sufficiently small. Persistence means that the perturbed system has a nearby invariant
torus carrying quasiperiodic
motion with the same frequency vector. The surviving tori
occupy a set of positive measure, and the measure
of its complement tends to zero as the perturbation tends to zero.
For a system with two degrees of freedom, closed orbits on an invariant torus have a rational frequency ratio
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where 
and 
are integers with 
. A sufficient Diophantine condition for persistence is
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|
for all integers 
and 
with 
, where 
is chosen in relation to the perturbation bound.
This condition is a hypothesis of the persistence theorem, not a necessary condition
satisfied by every invariant torus of the perturbed system.
Resonant tori are not protected by this theorem, but are
not necessarily destroyed by every perturbation.
The KAM theorem broke the deadlock of the small divisor problem in classical perturbation theory, and provides the starting point for an understanding of the appearance of
chaos. For an integrable Hamiltonian
system, the standard nondegeneracy condition is that the Hessian
of 
has nonzero determinant,
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In 
degrees of freedom, a standard form of the Diophantine
frequency condition is
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(7)
|
for every nonzero vector of integers 
, with 
.
