  We subject the steady solutions of a spherically symmetric accretion flow to
a time-dependent radial perturbation. The equation of the perturbation includes
nonlinearity up to any arbitrary order, and bears a form that is very similar
to the metric equation of an analogue acoustic black hole. Casting the
perturbation as a standing wave on subsonic solutions, and maintaining
nonlinearity in it up to the second order, we get the time-dependence of the
perturbation in the form of a Li\'enard system. A dynamical systems analysis of
the Li\'enard system reveals a saddle point in real time, with the implication
that instabilities will develop in the accreting system when the perturbation
is extended into the nonlinear regime. The instability of initial subsonic
states also adversely affects the temporal evolution of the flow towards a
final and stable transonic state.
