  Oscillons are localised long-lived pulsating states in the three-dimensional
$\phi^4$ theory. We gain insight into the spatio-temporal structure and
bifurcation of the oscillons by studying time-periodic solutions in a ball of a
finite radius. A sequence of weakly localised {\it Bessel waves} -- nonlinear
standing waves with the Bessel-like $r$-dependence -- is shown to extend from
eigenfunctions of the linearised operator. The lowest-frequency Bessel wave
serves as a starting point of a branch of periodic solutions with exponentially
localised cores and small-amplitude tails decaying slowly towards the surface
of the ball. A numerical continuation of this branch gives rise to the
energy-frequency diagram featuring a series of resonant spikes. We show that
the standing waves associated with the resonances are born in the
period-multiplication bifurcations of the Bessel waves with higher frequencies.
The energy-frequency diagram for a sufficiently large ball displays sizeable
intervals of stability against spherically-symmetric perturbations.
