  We study time-periodic solutions for the cubic wave equation on an interval
with Dirichlet boundary conditions. We begin by following the perturbative
construction of Vernov and Khrustalev and provide a rigorous derivation of the
fourth-order expansion in small amplitude, which we use to verify the Galerkin
scheme. In the main part, we focus on exploring large solutions numerically. We
find an intricate bifurcation structure of time-periodic solutions forming a
fractal-like pattern and explore it for the first time. Our results suggest
that time-periodic solutions exist for arbitrary frequencies, with appearance
of fine bifurcation structure likely related to the Cantor set families of
solutions described in previous rigorous works.
