  By considering delta function sextupole, octupole, and decapole perturbations
and using difference action-angle variable equations, we find some useful
analytical formulae for the estimation of the dynamic apertures of circular
accelerators due to single sextupole, single octupole, single decapole (single
2$m$ pole in general). Their combined effects 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. These formulae
have been applied to determine the beam-beam limited dynamic aperture in a
circular collider.
