Breakup process and geometrical structure of high-dimensional KAM tori

The breakup of KAM tori in four-dimensional symplectic maps is studied. KAM tori are constructed explicitly by conjugation theory, and the successive steps leading to their breakup is analytically determined for two cases in which rotation vectors are cubic and quadratic. The present paper is motivated by the observation of Boltt and Meiss (Physica D66(1993), 282), but our result differs significantly from their conjecture, because it suggests that the rational approximation does not work in cases with many degrees of freedom. The succession to the breakup depends on the perturbations which determine the dominant type of small divisors in the conjugation function.