  We consider $C^2$ vector fields in the three dimensional sphere with an
attracting heteroclinic cycle between two periodic hyperbolic solutions with
real Floquet multipliers. The proper basin of this attracting set exhibits
historic behavior and from the asymptotic properties of its orbits we obtain a
complete set of invariants under topological conjugacy in a neighborhood of the
cycle. As expected, this set contains the periods of the orbits involved in the
cycle, a combination of their angular speeds, the rates of expansion and
contraction in linearizing neighborhoods of them, besides information regarding
the transition maps and the transition times between these neighborhoods. We
conclude with an application of this result to a class of cycles obtained by
the lifting of an example of R. Bowen.
