  We present a new representation of solutions of the Benjamin-Ono equation
that are periodic in space and time. Up to an additive constant and a Galilean
transformation, each of these solutions is a previously known, multi-periodic
solution; however, the new representation unifies the subset of such solutions
with a fixed spatial period and a continuously varying temporal period into a
single network of smooth manifolds connected together by an infinite hierarchy
of bifurcations. Our representation explicitly describes the evolution of the
Fourier modes of the solution as well as the particle trajectories in a
meromorphic representation of these solutions; therefore, we have also solved
the problem of finding periodic solutions of the ordinary differential equation
governing these particles, including a description of a bifurcation mechanism
for adding or removing particles without destroying periodicity. We illustrate
the types of bifurcation that occur with several examples, including degenerate
bifurcations not predicted by linearization about traveling waves.
