Birkhoff averages and the breakdown of invariant tori in volume-preserving maps

In this paper, we develop numerical methods based on the weighted Birkhoff average for studying two-dimensional invariant tori volume-preserving maps. The do not rely symmetries, such as time-reversal symmetry, nor approximating by periodic orbits. rate of convergence gives a sharp distinction between chaotic and regular dynamics allows accurate computation rotation vectors Resonant rotational are distinguished computing resonance order vector to given precision. Critical parameter values, where destroyed, computed decrease in average. We apply these three-dimensional generalization Chirikov's standard map: an angle-action map with two angle variables. Computations grids frequency perturbation amplitude allow estimates critical set. also use continuation follow fixed vectors. test three conjectures cubic fields that have been proposed give locally robust tori.