Bounds for bounded motion around a perturbed fixed point

We consider a dissipative map of the plane with a bounded perturbation term. This perturba-tion represents e.g. an extra time dependent term, a coupling to another system or noise. Theunperturbed map has a spiral attracting fixed point. We derive an analytical/numerical method todetermine the effect of the additional term on the phase portrait of the original map, as a functionof the bound 6 on the perturbation. This method yields a value 6c such that for e$ < 6c the orbitsabout the attractor are certainly bounded. In that case we obtain a largest region in which all orbitsremain 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.(Received: May 6, 1988; revised: June 20, 1988)