Construction of codimension one homoclinic cycles

Abstract

We give an explicit construction of families of Dm-equivariant polynomial vector fields in possessing a codimension one homoclinic cycle. The homoclinic cycle consists of m homoclinic trajectories all connected to the equilibrium at the origin. The constructed vector fields can provide a setting for a (numerical) bifurcation study of these homoclinic cycles, in particular for m equal to a multiple of 4, where the bifurcations form an open problem.