  Based on the secular theory, we discuss the orbital evolution of stars in a
nuclear star cluster to which a secondary massive black hole is infalling with
vanishing eccentricity. We find that the eccentricities of the stars could show
sharp transitions, depending strongly on their initial conditions. By examining
the phase-space structure of an associated Hamiltonian, we show that these
characteristic behaviors are partly due to a probabilistic bifurcation at a
separatrix crossing, resulting from the retrograde apsidal precession by the
cluster potential. We also show that separatrix crossings are closely related
to realization of a large eccentricity and could be important for astrophysical
phenomena such as tidal disruption events or gravitational wave emissions.
