  There are few explicit examples in the literature of vector fields exhibiting
complex dynamics that may be proved analytically. This paper reports numerical
experiments performed for an explicit two-parameter family of vector fields
unfolding an attracting heteroclinic network, linking two saddle-foci with $
(\mathbb{SO}(2) \oplus \mathbb{Z}_2)$-symmetry. The vector field is the
restriction to $\mathbb{S}^3$ of a polynomial vector field in $\mathbb{R}^4$.
We investigate global bifurcations due to symmetry-breaking and we detect
strange attractors via a phenomenon called Torus-Breakdown theory. We explain
how an attracting torus gets destroyed by following the changes in the
invariant manifolds of the saddle-foci. Although a complete understanding of
the corresponding bifurcation diagram and the mechanisms underlying the
dynamical changes is still out of reach, using a combination of theoretical
tools and computer simulations, we have uncovered some complex patterns for the
symmetric family under analysis. This also suggests a route to obtain
rotational horseshoes; additionally, we give an attempt to elucidate some of
the bifurcations involved in an Arnold wedge.
