In 2006, Dafermos-Holzegel conjectured that the Anti-de Sitter spacetime is unstable solution to the Einstein equations under reflective boundary conditions at the conformal infinity for generic initial data. Recently, Moschidis established a rigorous instability proof. Moreover, Rostworowski-Maliborski enhanced this conjecture by providing strong numerical evidence indicating the existence of "special" initial data leading to time-periodic solutions for the Einstein-Klein-Gordon system in spherical symmetry which are in fact stable. Motivated by these, we construct families of arbitrary small time-periodic solutions to several toy models providing a rigorous proof of the numerical constructions above in a simpler setting. The models we consider include the conformal cubic wave equation and the spherically-symmetric Yang-Mills equation on the Einstein cylinder and our proof relies on modifications of a theorem of Bambursi-Paleari for which the main assumption is the existence of a seed solution, given by a non-degenerate zero of a non-linear operator associated with the resonant system. In the Yang-Mills case, the original version of the theorem of Bambusi-Paleari is not applicable because the non-linearity of smallest degree is non-resonant. The resonant terms are then provided by the next order non-linear terms with an extra correction due to back-reaction terms of the smallest degree non-linearity and we prove an analogous theorem in this setting. Finally, we also consider the massive wave equation in the fixed Anti-de Sitter with a cubic non-linearity, construct families of arbitrary small time-periodic solutions from "special" initial data and show that these are non-linear stable for exponentially long times.