  Extreme mass ratio in-spirals (EMRIs) are candidate events for gravitational
wave detection in the millihertz range (by detectors like LISA and eLISA).
These events involve a stellar-mass black hole, or a similar compact object,
descending in the gravitational field of a supermassive black hole, eventually
merging with it. Properties of the in-spiralling trajectory away from resonance
are well known and have been studied extensively, however little is known about
the behaviour of these binary systems at resonance, when the radial and lateral
frequencies of the orbit become commensurate. We describe the two existing
models, the instantaneous frequency approach used by Gair, Bender, and Yunes,
and the standard two timescales approach implemented by Flanagan and Hinderer.
In both cases, the exact treatment depends on the modelling of the
gravitational self-force, which is currently not available. We extend the
results in Gair, Bender and Yunes to higher order in the on-resonance flux
modification, and argue that the instantaneous frequency approach is also a
valid treatment of the resonance problem. The non-linear differential equations
which arise in treating resonances are interesting from a mathematical view
point. We present our algorithm for perturbative solutions and the results to
third order in the infinitesimal parameter, and discuss the scope of this
approach.
