  The secular evolution of the purely general relativistic low angular momentum
accretion flow around a spinning black hole is shown to exhibit hysteresis
effects. This confirms that a stationary shock is an integral part of such an
accretion disc in the Kerr metric. The equations describing the space gradient
of the dynamical flow velocity of the accreting matter have been shown to be
equivalent to a first order autonomous dynamical systems. Fixed point analysis
ensures that such flow must be multi-transonic for certain astrophysically
relevant initial boundary conditions. Contrary to the existing consensus in the
literature, the critical points and the sonic points are proved not to be
isomorphic in general. Homoclinic orbits for the flow flow possessing multiple
critical points select the critical point with the higher entropy accretion
rate, confirming that the entropy accretion rate is the degeneracy removing
agent in the system. However, heteroclinic orbits are also observed for some
special situation, where both the saddle type critical points of the flow
configuration possesses identical entropy accretion rate. Topologies with
heteroclinic orbits are thus the only allowed non removable degenerate
solutions for accretion flow with multiple critical points, and are shown to be
structurally unstable. Depending on suitable initial boundary conditions, a
homoclinic trajectory can be combined with a standard non homoclinic orbit
through an energy preserving Rankine-Hugoniot type of stationary shock. An
effective Lyapunov index has been proposed to analytically confirm why certain
class of transonic flow can not accommodate shock solutions even if it produces
multiple critical points. (Abridged)
