Locomotion via swimming or walking is characterized by stability and periodicity.  This suggests that locomotion can be characterized as a limit cycle.  My research proves the existence of limit cycles in a symmetry reduced phase spaces.  The relative limit cycle in the unreduced phase space corresponds to orderly periodic motion by a simple Lie group such as SE(3).  These constructions are generalizable, and could open the door for a new paradigm for control of robotic systems, and design of fluid pumps [1,2].

[1] H.O. Jacobs. (2015) The role of SE(d) reduction in understanding mid-Reynolds swimming. In “Geometry, Mechanics, and Dynamics: The Legacy of Jerry Marsden”. (pp 137-166). Fields Institute Communications Series, Springer (2015) (arXiv:1307.4599)

[2] J. Eldering, H.O. Jacobs. The role of symmetry and dissipation in biolocomotion, to appear in SIAM J. Applied Dynamical Systems, 2016 (arXiv:1212.1978)

blowup  snapshots