  The dynamics of a test particle in a non-spinning binary black hole system of
equal masses is numerically investigated. The binary system is modeled in the
context of the pseudo-Newtonian circular restricted three-body problem, such
that the primaries are separated by a fixed distance and move in a circular
orbit around each other. In particular, the Paczy\'{n}ski-Wiita potential is
used for describing the gravitational field of the two non-Newtonian primaries.
The orbital properties of the test particle are determined through the
classification of the initial conditions of the orbits, using several values of
the Jacobi constant, in the Hill's regions of possible motion. The initial
conditions are classified into three main categories: (i) bounded, (ii)
escaping and (iii) displaying close encounters. Using the smaller alignment
index (SALI) chaos indicator, we further classify bounded orbits into regular,
sticky or chaotic. To gain a complete view of the dynamics of the system, we
define grids of initial conditions on different types of two-dimensional
planes. The orbital structure of the configuration plane, along with the
corresponding distributions of the escape and collision/close encounter times,
allow us to observe the transition from the classical Newtonian to the
pseudo-Newtonian regime. Our numerical results reveal a strong dependence of
the properties of the considered basins with the Jacobi constant as well as
with the Schwarzschild radius of the black holes.
