Construction of codimension one homoclinic cycles

摘要

We give an explicit construction of families of D_m-equivariant polynomial vector fields in R~4 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.