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

Bifurcations from an attracting heteroclinic cycle under periodic forcing