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An Analytical Study on the Multi-critical Behaviour and Related Bifurcation Phenomena for Relativistic Black Hole Accretion
We apply the theory of algebraic polynomials to analytically study the
transonic properties of general relativistic hydrodynamic axisymmetric
accretion onto non-rotating astrophysical black holes. For such accretion
phenomena, the conserved specific energy of the flow, which turns out to be one
of the two first integrals of motion in the system studied, can be expressed as
a 8 degree polynomial of the critical point of the flow configuration.
We then construct the corresponding Sturm's chain algorithm to calculate the
number of real roots lying within the astrophysically relevant domain of
. This allows, for the first time in literature, to {\it
analytically} find out the maximum number of physically acceptable solution an
accretion flow with certain geometric configuration, space-time metric, and
equation of state can have, and thus to investigate its multi-critical
properties {\it completely analytically}, for accretion flow in which the
location of the critical points can not be computed without taking recourse to
the numerical scheme. This work can further be generalized to analytically
calculate the maximal number of equilibrium points certain autonomous dynamical
system can have in general. We also demonstrate how the transition from a
mono-critical to multi-critical (or vice versa) flow configuration can be
realized through the saddle-centre bifurcation phenomena using certain
techniques of the catastrophe theory.Comment: 19 pages, 2 eps figures, to appear in "General Relativity and
Gravitation