Introduction

From a mathematical perspective, problems in astrophysical accretion fall under the general class of nonlinear dynamics. This occasions no surprise, because accretion, after all, describes the dynamics of a compressible astrophysical fluid, and the fundamental governing equations of such a problem are nonlinear in nature. One of the general issues that is addressed in accretion studies is the physics of the accretor itself, whose gravitational attraction sustains the global inflow process. This is especially important if the accretor is a black hole, which by its very definition is never amenable to any direct physical observation, and, therefore, its properties can only be known by the gravitational influence it exerts on the neighbouring structure of space-time.

If the accretor is a black hole, the in-falling matter has to reach the event horizon on a relativistic scale of velocity, and arguments in favour of this contention have, by now, gained widespread currency. Given physically sensible boundary conditions, this can only imply that at one stage the flow of matter will become transonic, i.e. its flow speed will grow from subsonic values to supersonic values, and in doing so, it will at some point match the speed of acoustic propagation in the fluid. This transition can happen continuously, as in smooth transonic solutions, or discontinuously, as in shocks. In accretion studies both possibilities have been subjected to concerted investigation.

Frequently it happens that the transonic feature is exhibited more than once in the phase portrait of stationary solutions, i.e. the flow will be multi-transonic. This is particularly true in axisymmetric rotational flows, as opposed to steady spherically symmetric flows, where, as it is well known, physical transonicity occurs only once. Physical transonic solutions can be represented mathematically as critical solutions in the phase plane of the flow, i.e. they are associated with critical points -- alternatively known as fixed points or equilibrium points. These solutions may even pass through critical points (as, for instance, a flow through a saddle point).

In this situation much information about various physical properties of accretion processes could be gleaned if these critical points are analyzed carefully, which is the central objective of our work in this field.