  When an EMRI in a perturbed integrable gravitational field, such as a
deformed Kerr black hole, undergoes a prolonged resonance, the frequencies that
engage in resonance retain a fixed rational ratio, despite experiencing
adiabatic changes due to radiation reaction. In the past this plateau effect in
the evolution of the ratio of frequencies has been investigated by studying the
orbital evolution through kludge models, which provide approximate average
losses of energy and angular momentum experienced by a test particle in this
field. By employing a Newtonian gravitational field that closely resembles a
pure Kerr or a perturbed Kerr relativistic field, we demonstrate that the
actual adiabatic evolution of an orbit driven by an artificial ``self-force''
results in more prolonged periods of resonance crossings compared to those
obtained by imposing a predetermined rate of energy and angular momentum change
throughout the orbital progression.
