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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.