We study one parameter families of vector fields that are defined on three dimensional
manifolds and whose nonwandering sets are structurally stable.
As families, these families may not be structurally stable; heteroclinic
bifurcations that give rise to moduli can occur.
Some but not all moduli are related to the geometry of stable
and unstable manifolds. We study a notion of stability, weaker
then structural stability, in which geometry and dynamics on stable
and unstable manifolds are reflected.
We classify the families from the above mentioned class of families
that are stable in this