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Extreme mass-ratio inspirals (EMRI) are one of the most sensitive probes ofblack hole spacetimes with gravitational wave measurements. In this work, wesystematically analyze the dynamics of an EMRI system near orbital resonances,assuming the background spacetime is weakly perturbed from Kerr. Using theaction-angle formalism, we have derived an effective resonant Hamiltonian thatdescribes the dynamics of the resonant degree of freedom, for the case that theEMRI motion across the resonance regime. This effective resonant Hamiltoniancan also be used to derive the condition that the trajectory enters/exits aresonant island and the permanent change of action variables across theresonance with the gravitational wave radiation turned on. The orbital chaos,on the other hand, generally leads to transitions between different branches ofrotational orbits with finite changes of the action variables. These findingsare demonstrated with numerical orbital evolutions that are mapped intorepresentations using action-angle variables. This study is one part of theprogram of understanding EMRI dynamics in a generic perturbed Kerr spacetime,which paves the way of using EMRIs to precisely measure the black holespacetime.

Resonant dynamics of extreme mass-ratio inspirals in a perturbed Kerr spacetime