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Cyclic and heteroclinic flows near general static spherically symmetric black holes

We investigate the Michel-type accretion onto a static spherically symmetric black hole. Using a Hamiltonian dynamical approach, we show that the standard method employed for tackling the accretion problem has masked some properties of the fluid flow. We determine new analytical solutions that are neither transonic nor supersonic as the fluid approaches the horizon(s); rather, they remain subsonic for all values of the radial coordinate. Moreover, the three velocity vanishes and the pressure diverges on the horizon(s), resulting in a flowout of the fluid under the effect of its own pressure. This is in favor of an earlier prediction that pressure-dominant regions form near the horizon. This result does not depend on the form of the metric and it applies to a neighborhood of any horizon where the time coordinate is timelike. For anti-de Sitter-like $\text{f}(R)$ black holes we discuss the stability of the critical flow and determine separatrix heteroclinic orbits. For de Sitter-like $\text{f}(R)$ black holes, we construct polytropic cyclic, non-homoclinic, physical flows connecting the two horizons. These flows become non-relativistic for Hamiltonian values higher than the critical value allowing for a good estimate of the proper period of the flow.