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In this note we take a dynamical systems approach to the equations of motionof a free test particle moving around a spherically symmetric static dilatonblack hole, written in the Einstein frame. The equations of motion are obtainedusing the Euler-Lagrange formalism. Using the first integrals of motion, wereach the conclusion that the free test particles are moving in a plane, named\emph{plane of motion}. In it we analyze the existence and nature of theequilibrium points and compare the behavior of free test particles near theequilibrium points using the dynamics systems approach. The study revealed thatin the exact phase-plane exist distinct regions of motion, separated through acurve named \emph{separatrix}. In the end we obtained a relation between theparameters describing the black hole and the free test particle that holds on aparabolic separatrix.

Phase-plane analysis of the timelike geodesics around a spherically symmetric static dilaton black hole