Time depending processes in physics, chemistry and biology are often
described as ordinary differential equations. Thereby the study of
heteroclinic and homoclinic cycles is of growing interest, since they were
recognized as source of non-trivial dynamics. In this thesis we construct
examples of homoclinic cycles. A homoclinic trajectory is a solution of a
differential equation that converges to the same equilibrium for positive
and negative time. A homoclinic cycle is a set of homoclinic trajectories
to the same equilibrium. Especially we consider relative homoclinic cycles
of codimension-one that is homoclinic cycles that appear in a one parameter
family of vector fields that possess a discrete symmetry. The constructed
vector fields are equivariant with respect to the dihedral group D_m, the
symmetry group of the regular m-gon in the plane. The homoclinic
trajectories approach the hyperbolic equilibrium along leading directions.
The leading eigenvalues are real. Furthermore we have constructed robust
homoclinic orbits.