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We subject the stationary solutions of inviscid and axially symmetric rotational accretion to a time-dependent radial perturbation, which includes nonlinearity to any arbitrary order. Regardless of the order of nonlinearity, the equation of the perturbation bears a form that is similar to the metric equation of an analogue acoustic black hole. We bring out the time dependence of the perturbation in the form of a Li\'enard system, by requiring the perturbation to be a standing wave under the second order of nonlinearity. We perform a dynamical systems analysis of the Li\'enard system to reveal a saddle point in real time, whose implication is that instabilities will develop in the accreting system when the perturbation is extended into the nonlinear regime. We also model the perturbation as a high-frequency travelling wave, and carry out a Wentzel-Kramers-Brillouin analysis, treating nonlinearity iteratively as a very feeble effect. Under this approach both the amplitude and the energy flux of the perturbation exhibit growth, with the acoustic horizon segregating the regions of stability and instability.Comment: 15 pages, ReVTeX. Substantially revised with respect to the previous version. One figure and a new section on travelling waves (Sec. VI) have been added. The bibliography has been revised. arXiv admin note: substantial text overlap with arXiv:1207.107