  Most aquatic vertebrates swim by lateral flapping of their bodies and caudal
fins. While much effort has been devoted to understanding the flapping
kinematics and its influence on the swimming efficiency, little is known about
the stability (or lack of) of periodic swimming. It is believed that stability
limits maneuverability and body designs/flapping motions that are adapted for
stable swimming are not suitable for high maneuverability and vice versa. In
this paper, we consider a simplified model of a planar elliptic body undergoing
prescribed periodic heaving and pitching in potential flow. We show that
periodic locomotion can be achieved due to the resulting hydrodynamic forces,
and its value depends on several parameters including the aspect ratio of the
body, the amplitudes and phases of the prescribed flapping. We obtain
closed-form solutions for the locomotion and efficiency for small flapping
amplitudes, and numerical results for finite flapping amplitudes. We then study
the stability of the (finite amplitude flapping) periodic locomotion using
Floquet theory. We find that stability depends nonlinearly on all parameters.
Interesting trends of switching between stable and unstable motions emerge and
evolve as we continuously vary the parameter values. This suggests that, for
live organisms that control their flapping motion, maneuverability and
stability need not be thought of as disjoint properties, rather the organism
may manipulate its motion in favor of one or the other depending on the task at
hand.
