  We provide conditions guaranteeing that certain classes of robust
heteroclinic networks are asymptotically stable.
  We study the asymptotic stability of ac-networks --- robust heteroclinic
networks that exist in smooth ${\mathbb Z}^n_2$-equivariant dynamical systems
defined in the positive orthant of ${\mathbb R}^n$. Generators of the group
${\mathbb Z}^n_2$ are the transformations that change the sign of one of the
spatial coordinates. The ac-network is a union of hyperbolic equilibria and
connecting trajectories, where all equilibria belong to the coordinate axes
(not more than one equilibrium per axis) with unstable manifolds of dimension
one or two. The classification of ac-networks is carried out by describing all
possible types of associated graphs.
  We prove sufficient conditions for asymptotic stability of ac-networks. The
proof is given as a series of theorems and lemmas that are applicable to the
ac-networks and to more general types of networks. Finally, we apply these
results to discuss the asymptotic stability of several examples of heteroclinic
networks.
