There are two main ingredients in this paper coming, with equal
weight, from mathematics [of solving nonlinear ordinary differential
equations of the form $y' = f(xy)$ using truly tricky asymptotic and
hyperasymptotic expansions in a consequent and systematic manner]
and from general relativity physics [typically, of a gravitational
radiation by a massive compact object (such as a neutron star)
orbiting around a supermassive object (such as a black hole) and
forming the so called extreme-mass-ratio ``inspiral'' (EMRI)]. In a
way inspired by Ref. [2] (where $f(xy)$ was chosen as $\cos xy$) and
by Refs. [13] and [14] (where $f(xy)$ was chosen as $\tan 2 xy$) the
authors study the EMRI resonances as described by $f(xy)=1+k\,\cos
xy$ (frequency resonances) or by $f(xy)=1+k\,\cos \int y \,dx$
(phase resonances), both in the alternative dynamical regimes
controlled by parameter $k-1$. In both cases the asymptotic
approximation analytic predictions are shown to compare well with
the brute-force numerical solutions. Possible realistic amendments
of $f(xy)$ are also discussed.


MR2798217 Gair, Jonathan; Yunes, Nicolás; Bender, Carl M. Resonances
in extreme mass-ratio inspirals: asymptotic and hyperasymptotic
analysis. J. Math. Phys. 53 (2012), no. 3, 032503, 20 pp. 81Uxx