We classify simple heteroclinic networks for τ-equivariant system in R4 with finite τ ⊂ O(4), proceeding as follows: We define a graph associated with a given τ ⊂ O(n) and identify all so-called simple graphs associated with subgroups of O(4). Then, knowing the graph associated with a given τ, we determine the types of heteroclinic networks that the group admits. Our study is restricted to networks that are maximal in the sense that they have the highest possible number of connections-any non-maximal network can then be derived by deleting one or more connections. Finally, for networks of type A, i.e. admitted by τ ⊂ SO(4), we give necessary and sufficient conditions for fragmentary and essential asymptotic stability. The conditions involve the notion of principal subcycles which are comprised of connections in the direction of strongest expansion. (For other simple heteroclinic networks the conditions for stability are known.) The results are illustrated by a numerical example of a simple heteroclinic network that involves two subcycles that can be essentially asymptotically stable simultaneously.
We classify simple heteroclinic networks for τ-equivariant system in R4 with finite τ ⊂ O(4), proceeding as follows: We define a graph associated with a given τ ⊂ O(n) and identify all so-called simple graphs associated with subgroups of O(4). Then, knowing the graph associated with a given τ, we determine the types of heteroclinic networks that the group admits. Our study is restricted to networks that are maximal in the sense that they have the highest possible number of connections-any non-maximal network can then be derived by deleting one or more connections. Finally, for networks of type A, i.e. admitted by τ ⊂ SO(4), we give necessary and sufficient conditions for fragmentary and essential asymptotic stability. The conditions involve the notion of principal subcycles which are comprised of connections in the direction of strongest expansion. (For other simple heteroclinic networks the conditions for stability are known.) The results are illustrated by a numerical example of a simple heteroclinic network that involves two subcycles that can be essentially asymptotically stable simultaneously.