  We study bifurcation behavior in periodic perturbations of two-dimensional
symmetric systems exhibiting codimension-two bifurcations with a double
eigenvalue when the frequencies of the perturbation terms are small. We
transform the periodically perturbed system to a simpler one which is a
periodic perturbation of the normal form for codimension-two bifurcations with
a double zero eigenvalue and symmetry, and apply the subharmonic and homoclinic
Melnikov methods to analyze bifurcations occurring in the system. In
particular, we show that there exist transverse homoclinic or heteroclinic
orbits, which yield chaotic dynamics, in wide parameter regions. These results
can be applied to three or higher-dimensional systems and even to
infinite-dimensional systems with the assistance of center manifold reduction
and the invariant manifold theory. We illustrate our theory for a pendulum
subjected to position and velocity feedback control when the desired position
is periodic in time. We also give numerical computations by the computer tool
AUTO to demonstrate the theoretical results.
