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.
When the perturbation strength is small there is an attracting invariant closed curve not contractible on the cylinder. Near the centre of frequency locking there are parameter values for which this curve coexists with an attracting periodic solution. 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. These last two dynamical features appear at a discrete-time Bogdanov-Takens bifurcation.
Year of publication: 2018