Transverse Bifurcations Occurring on Simple Robust Homoclinic Cycles with Complex Transverse Eigenvalues

This paper investigates codimension-one transverse bifurcations of simple robust homoclinic cycles with a pair of complex transverse eigenvalues in Γ-equivariant systems on R5. These cycles are classified into two classes according to the isotropy subgroup. For one class, it is proved that, when the bifurcation is supercritical, a compact invariant set consisting of invariant cylinders bifurcates from the homoclinic network. Example systems and numerical simulations are provided for both classes.