  There are few examples of non-autonomous vector fields exhibiting complex
dynamics that may be proven analytically. We analyse a family of periodic
perturbations of a weakly attracting robust heteroclinic network defined on the
two-sphere. We derive the first return map near the heteroclinic cycle for
small amplitude of the perturbing term, and we reduce the analysis of the
non-autonomous system to that of a two-dimensional map on a cylinder.
  Interesting dynamical features arise from a discrete-time Bogdanov-Takens
bifurcation. When the perturbation strength is small the first return map has
an attracting invariant closed curve that is not contractible on the cylinder.
Near the centre of frequency locking there are parameter values with
bistability: the invariant curve coexists with an attracting fixed point.
Increasing the perturbation strength there are periodic solutions that
bifurcate into a closed contractible invariant curve and into a region where
the dynamics is conjugate to a full shift on two symbols.
