To give an analytic estimate of the stability domain of nonlinear systems, like large accelerators, has been a subjected of intense theoretical studies for many years. Up to now, even for the simple case of the one‐dimensional He´non map, one could not determine the border of the stability domain using analytic tools. On the other hand this estimate can easily be found by tracking through the map. A promising new attempt is presented here, to estimate the dynamic aperture for the He´non map via following the invariant manifolds of its hyperbolic fixed point. The same technique is then applied to a different map and an attempt to generalize the method presented in this paper to generic 1‐D polynomial maps is briefly discussed.