  While recent gravitational wave observations by LIGO and Virgo allow for
tests of general relativity in the extreme gravity regime, these observations
are still blind to a large swath of phenomena outside these instruments'
sensitivity curves. Future gravitational-wave detectors, such as LISA, will
enable probes of longer-duration and lower-frequency events. In particular,
LISA will enable the characterization of the non-linear dynamics of extreme
mass-ratio inspirals, when a small compact object falls into a supermassive
black hole. In this paper, we study the motion of test particles around
spinning black holes in two quadratic gravity theories: scalar Gauss-Bonnet and
dynamical Chern-Simons gravity. We show that geodesic trajectories around
rotating black holes in these theories are likely to not have a fourth constant
of the motion. In particular, we show that Poincar\'e sections of the orbital
phase space present chaotic features that will affect the inspiral of small
compact objects into supermassive black holes in these theories. Nevertheless,
the characteristic size of these chaotic features is tiny and their location in
parameter space is very close to the event horizon of the supermassive black
hole. Therefore, the detection of such chaotic features with LISA is likely
very challenging, at best.
