  Steady swimming can be characterized as both periodic and stable. These
characteristics are the very definition of limit cycles, and so we ask "Can we
view swimming as a limit cycle?" In this paper we will find that the answer is
"yes". We will define a class of dissipative systems which correspond to the
passive dynamics of a body immersed in a Navier-Stokes fluid (i.e. the dynamics
of a dead fish). Upon performing reduction by symmetry we will find a
hyperbolically stable fixed point which corresponds to the stability of a dead
fish in stagnant water. Given a periodic force on the shape of the body we will
invoke the persistence theorem to assert the existence of a loop which
approximately satisfies the exact equations of motion. If we lift this loop
with a phase reconstruction formula we will find that the lifted loops are not
loops, but stable trajectories which represent regular periodic motion
reminiscent of swimming.
