  The attractors of a dynamical system govern its typical long-term behaviour.
The presence of many attractors is significant as it means the behaviour is
heavily dependent on the initial conditions. To understand how large numbers of
attractors can coexist, in this thesis we study the occurrence of infinitely
many stable single-round periodic solutions associated with homoclinic
connections in two-dimensional maps. We show this phenomenon has a relatively
high codimension requiring a homoclinic tangency and `global resonance', as has
been described previously in the area-preserving setting. However, unlike in
that setting, local resonant terms also play an important role. To determine
how the phenomenon may manifest in bifurcation diagrams, we also study
perturbations of a globally resonant homoclinic tangency. We find there exist
sequences of saddle-node and period-doubling bifurcations. Interestingly, in
different directions of parameter space, the bifurcation values scale
differently resulting in a complicated shape for the stability region for each
periodic solution. In degenerate directions, the bifurcation values scale
substantially slower as illustrated in an abstract piecewise-smooth $C^{1}$
map.
