Robust homoclinic cycles in 4

The conditions of existence of robust homoclinic cycles for G -equivariant vector fields in 4 with G a finite group are investigated. Depending on the action of G , such cycles are either of type A, B or C. We first introduce a notion of minimal admissible group. The existence of robust homoclinic cycles for vector fields which are equivariant by such a group is generic. Then we show that for type A cycles, the number n of equilibria is either even or equal to three. In the case of type B cycles, n can only be equal to two, three or six. Finally, for those of type C, n is either four or eight. Moreover, we provide expressions for the generators of the minimal admissible groups for each of the above mentioned cycles and we show the explicit form of vector fields generating three type A homoclinic cycles with six, eight and 24 equilibria, respectively.