Breakup of invariant tori for the four-dimensional semi-standard map

We compute the domain of existence of two-dimensional invariant tori with fixed frequency vectors for a fourdimensional, complex, symplectic map. The map is a generalization of the semi-standard map studied by Greene and Percival; it has three parameters, a, and a, representing the strength of the kicks in each degree of freedom, and E, the coupling. The domain of existence of a torus in (a,, a*) is shown to be complete and log-convex for fixed k = ! /a,a,. Explicit bounds on the domain for fixed k are obtained. Numerical results show that quadratic irrationals can be more robust than the cubic irrational, “the spiral mean”.