  We present a comprehensive mechanism for the emergence of rotational
horseshoes and strange attractors in a class of two-parameter families of
periodically-perturbed differential equations defining a flow on a
three-dimensional manifold. When both parameters are zero, its flow exhibits an
attracting heteroclinic network associated to two periodic solutions. After
slightly increasing both parameters, while keeping a two-dimensional connection
unaltered, we focus our attention in the case where the two-dimensional
invariant manifolds of the periodic solutions do not intersect. We prove a wide
range of dynamical behaviour, ranging from an attracting quasi-periodic torus
to rotational horseshoes and H\'enon-like strange attractors. We illustrate our
results with an explicit example.
