This work studies the dynamics of geodesic motion within a curved spacetime
around a Schwarzschild black hole, perturbed by a gravitational field of a far
axisymmetric distribution of mass enclosing the system. This spacetime can
serve as a versatile model for a diverse range of astrophysical scenarios and,
in particular, for extreme mass ratio inspirals as in our work. We show that
the system is non-integrable by employing Poincar\'e surface of section and
rotation numbers. By utilising the rotation numbers, the widths of resonances
are calculated, which are then used in establishing the relation between the
underlying perturbation parameter driving the system from integrability and the
quadrupole parameter characterising the perturbed metric. This relation allows
us to estimate the phase shift caused by the resonance during an inspiral.Comment: 11 pages, 3 figures, 1 table, Proceedings of RAGtime 23-25. Edited by
Z. Stuchl\'ik, G. T\"or\"ok and V. Karas. Institute of Physics in Opav