  There are few explicit examples in the literature of vector fields exhibiting
complex dynamics that may be proved analytically. We construct explicitly a
{two parameter family of vector fields} on the three-dimensional sphere
$\EU^3$, whose flow has a spiralling attractor containing the following: two
hyperbolic equilibria, heteroclinic trajectories connecting them {transversely}
and a non-trivial hyperbolic, invariant and transitive set. The spiralling set
unfolds a heteroclinic network between two symmetric saddle-foci and contains a
sequence of topological horseshoes semiconjugate to full shifts over an
alphabet with more and more symbols, {coexisting with Newhouse phenonema}. The
vector field is the restriction to $\EU^3$ of a polynomial vector field in
$\RR^4$. In this article, we also identify global bifurcations that induce
chaotic dynamics of different types.
