By considering delta function sextupole, octupole, and decapole perturbations and using di!erence action-angle variable equations, we "nd some useful analytical formulae for the estimation of the dynamic apertures of circular accelerators due to single sextupole, single octupole, single decapole (single 2m pole in general); and their combined e!ects are derived based on the Chirikov criterion of the onset of stochastic motions. Comparisons with numerical simulations are made, and the agreement is quite satisfactory.