The purpose is not only to study the behaviour of a parametrically excited pendulum, but also to present the topological theory of dynamical systems in order to select in parameter and phase space the desired orbits from a large variety of unstable solutions. Moreover, the authors develop a stabilizing algorithm to control a chaotic trajectory for one or more solutions. The described method is highly recommended instead of any computational or numerical one, because it enables one to choose a particular orbit and provides information about a whole range of other orbits for potential control.