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Low angular momentum, general relativistic, axially symmetric accretion ofhydrodynamic fluid onto Schwarzschild black holes may undergo more than onecritical transition. To obtain the stationary integral solutions correspondingto such multi-critical accretion flow, one needs to employ numerical solutionsof the corresponding fluid dynamics equations. In the present work, we developa {\it{completely analytical}} solution scheme which may be used to findseveral trans-critical flow behaviours of aforementioned accretion, withoutexplicitly solving the flow equations numerically. We study all possiblegeometric configurations of the flow profile, governed by all possiblethermodynamic equations of state. We use Sturm's chain algorithm to find outhow many physically acceptable critical points the accretion flow can have, anddiscuss the transition from the mono to the multi-critical flow profile, andrelated bifurcation phenomena. We thus illustrate, completely analytically, theapplication of certain aspects of the dynamical systems theory in the field oflarge scale astrophysical flow under the influence of strong gravity. Our workmay possibly be generalized to calculate the maximal number of equilibriumpoints certain autonomous dynamical systems can have in general.

Multi-criticality and related bifurcation in accretion discs around non-rotating black holes -- an analytical study