  We discuss a two-dimensional model for the dynamics of axonemal deformations
driven by internally generated forces of molecular motors. Our model consists
of an elastic filament pair connected by active elements. We derive the dynamic
equations for this system in presence of internal forces. In the limit of small
deformations, a perturbative approach allows us to calculate filament shapes
and the tension profile. We demonstrate that periodic filament motion can be
generated via a self-organization of elastic filaments and molecular motors.
Oscillatory motion and the propagation of bending waves can occur for an
initially non-moving state via an instability termed Hopf bifurcation. Close to
this instability, the behavior of the system is shown to be independent of
microscopic details of the axoneme and the force-generating mechanism. The
oscillation frequency however does depend on properties of the molecular
motors. We calculate the oscillation frequency at the bifurcation point and
show that a large frequency range is accessible by varying the axonemal length
between 1 and 50$\mu$m. We calculate the velocity of swimming of a flagellum
and discuss the effects of boundary conditions and externally applied forces on
the axonemal oscillations.
