We study bifurcations of homoclinic orbits to hyperbolic saddle equilibria in
a class of four-dimensional systems which may be Hamiltonian or not. Only one
parameter is enough to treat these types of bifurcations in Hamiltonian systems
but two parameters are needed in general systems. We apply a version of
Melnikov’s method due to Gruendler to obtain saddle-node and pitchfork types of
bifurcation results for homoclinic orbits. Furthermore we prove that if these
bifurcations occur, then the variational equations around the homoclinic orbits
are integrable in the meaning of differential Galois theory under the
assumption that the homoclinic orbits lie on analytic invariant manifolds. We
illustrate our theories with an example which arises as stationary states of
coupled real Ginzburg-Landau partial differential equations, and demonstrate
the theoretical results by numerical ones.