  Motivated by the study of small amplitudes non-linear waves in the
Anti-de-Sitter spacetime and in particular the conjectured existence of
periodic in time solutions to the Einstein equations, we construct families of
arbitrary small time-periodic solutions to the conformal cubic wave equation
and the spherically-symmetric Yang-Mills equations on the Einstein cylinder
$\mathbb{R}\times \mathbb{S}^3$. For the conformal cubic wave equation, we
consider both spherically-symmetric solutions and complexed-valued aspherical
solutions with an ansatz relying on the Hopf fibration of the $3$-sphere. In
all three cases, the equations reduce to $1+1$ semi-linear wave equations. Our
proof relies on a theorem of Bambusi-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. For the problems that
we consider, such seed solutions are simply given by the mode solutions of the
linearized equations. Provided that the Fourier coefficients of the systems can
be computed, the non-degeneracy conditions then amount to solving infinite
dimensional linear systems. Since the eigenfunctions for all three cases
studied are given by Jacobi polynomials, we derive the different Fourier and
resonant systems using linearization and connection formulas as well as
integral transformation of Jacobi polynomials. 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 nonresonant. The resonant terms are
then provided by the next order non-linear terms with an extra correction due
to backreaction terms of the smallest degree non-linearity and we prove an
analogous theorem in this setting.
