  In the present work a systematic study has been presented in the context of
the existence of libration points, their linear stability, the regions of
motion where the third particle can orbit and the domain of basins of
convergence linked to libration points in the spatial configuration of the
collinear restricted four-body problem with non-spherical primaries (i.e., the
primaries are oblate or prolate spheroid). The parametric evolution of the
positions of the libration points as function of the oblateness and prolateness
parameters of the primaries and the stability of these points in linear sense
are illustrated numerically. Moreover, the numerical investigation shows that
the only libration points which lie on either of the axes are linearly stable
for several combinations of the oblateness parameter and mass parameter whereas
the non-collinear libration points are found linearly unstable, consequently
unstable in nonlinear sense also, for studied value of mass parameter and
oblateness parameter. Moreover, the regions of possible motion are also
depicted, where the infinitesimal mass is free to orbit, as function of
Jacobian constant. In addition, the basins of convergence (BoC) linked to the
libration points are illustrated by using the multivariate version of the
Newton-Raphson (NR) iterative scheme.

Chaotic Dynamics;Earth and Planetary Astrophysics