Extreme-mass-ratio inspirals are candidate events for gravitational wave detection in the millihertz band (by detectors like Laser Interferometer Space Antenna (lisa)). 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 inspiralling 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-time scale approach implemented by Flanagan and Hinderer. In both cases, the exact treatment depends on the modeling of the gravitational self-force, which is currently not available. We extend the results from the work of 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.

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