A theorem outlined in 1954 by Kolmogorov which was subsequently proved in the 1960s by Arnold and Moser (Tabor 1989,
p. 105). It gives conditions under which Chaos is restricted in extent. Moser's 1962 proof was valid for
Twist Maps

(1) | |||

(2) |

In 1963, Arnold produced a proof for Hamiltonian systems

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- 1. small perturbations,
- 2. smooth perturbations, and
- 3. sufficiently irrational Winding Number.

Moser considered an integrable Hamiltonian function with a Torus and set of frequencies having an
incommensurate frequency vector
(i.e.,
for all Integers ).
Let be perturbed by some periodic function . The KAM theorem states that, if is small enough, then for
almost every there exists an invariant Torus of the perturbed system such that
is ``close to'' . Moreover, the Tori form a set of Positive measures whose
complement has a measure which tends to zero as . A useful paraphrase of the KAM theorem is, ``For sufficiently
small perturbation, almost all Tori (excluding those with rational frequency vectors) are preserved.'' The
theorem thus explicitly excludes Tori with rationally related frequencies, that is, conditions of the
form

(4) |

(5) |

(6) |

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 a Hamiltonian System, the Isoenergetic
Nondegeneracy condition

(7) |

(8) |

**References**

Tabor, M. *Chaos and Integrability in Nonlinear Dynamics: An Introduction.* New York: Wiley, 1989.

© 1996-9

1999-05-26