In the context of ordinary differential equations the term „homoclinic
snaking“ describes a certain continuation scenario of homoclinic orbits
within a neighbourhood of a heteroclinic cycle between an equilibrium and a
periodic orbit. Often the considered ordinary differential equations
characterise the steady states of partial differential equations. Mostly
these differential equations are reversible and Hamiltonian. In the Diploma
thesis we consider two parameter families of ordinary differential
equations in R 3, which are neither reversible, nor Hamiltonian. We suppose
that there is a heteroclinic cycle between an equilibrium E and a
hyperbolic periodic orbit gamma. Furthermore we formulate assumptions on
the behavior of the intersections of the stable and unstable manifolds of E
and gamma. Under these assumptions we investigate the continuation of
1-homoclinic orbits to E (these are orbits which “run“ one time along the
original heteroclinic cycle). For such orbits we prove „homoclinic
snaking“. We show that the „snaking“ behavior is determined by the
bifurcation of the heteroclinic connections between E and gamma.