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The recent gravitational wave observations by the LIGO/Virgo collaborationhave allowed the first tests of General Relativity in the extreme gravityregime, when comparable-mass black holes and neutron stars collide. Futurespace-based detectors, such as the Laser Interferometer Space Antenna, willallow tests of Einstein's theory with gravitational waves emitted when a smallblack hole falls into a supermassive one in an extreme mass-ratio inspiral. Oneparticular test that is tailor-made for such inspirals is the search for chaosin extreme gravity. We here study whether chaos is present in the motion oftest particles around spinning black holes of parity-violating modifiedgravity, focusing in particular on dynamical Chern-Simons gravity. We develop aresummation strategy that restores all spin terms in the General Relativitylimit, while retaining up to fifth-order-in-spin terms in the dynamicalChern-Simons corrections to the Kerr metric. We then calculate Poincar\'esurfaces of section and rotation numbers of a wide family of geodesics of thisresummed metric. We find no evidence for geodesic chaos, with at mostdeformations of the resonant torii that shrink as terms of higher-order in spinare included in the dynamical Chern-Simons corrections to the Kerr metric. Ournumerical findings suggest that the geodesics of the as-of-yet unknown exactsolution for spinning black holes in this theory may be integrable, and thatthere may thus exist a fourth integral of motion associated with this exactsolution. The studies presented here begin to lay the foundations for chaotictests of General Relativity with the observation of extreme mass ratioinspirals with the Laser Interferometer Space Antenna.

The exact dynamical Chern Simons metric for a spinning black hole possesses a fourth constant of motion: A Dynamical-Systems-Based Conjecture