Bounds for bounded motion around a perturbed fixed point

We consider a dissipative map of the plane with a bounded perturbation term. This perturbation represents e.g. an extra time dependent term, a coupling to another system or noise. The unperturbed map has a spiral attracting fixed point. We derive an analytical/numerical method to determine the effect of the additional term on the phase portrait of the original map, as a function of the bound on the perturbation. This method yields a value _c such that for< _c the orbits about the attractor are certainly bounded. In that case we obtain a largest region in which all orbits remain bounded and a smallest region in which these bounded orbits are captured after some time (the analogue of basin and attractor respectively).The analysis is based on the Lyapunov function which exists for the unperturbed map.