Swimming as a limit cycle

TL;DR

This paper models steady swimming as a limit cycle by analyzing the passive dynamics of a body in fluid, demonstrating that periodic shape changes can produce stable, swimming-like motions.

Contribution

It introduces a class of dissipative systems representing passive body-fluid dynamics and proves the existence of stable periodic motions as limit cycles in these systems.

Findings

Existence of a hyperbolically stable fixed point for dead fish in stagnant water

Periodic shape changes lead to stable limit cycles approximating swimming

Stable trajectories correspond to regular swimming motions

Abstract

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.