The role of symmetry and dissipation in biolocomotion

Jaap Eldering, Henry O. Jacobs

Published 2012-12-10, updated 2014-09-04Version 5

In this paper we illustrate the potential role which relative limit cycles may play in biolocomotion. We do this by describing, in great detail, an elementary example of reduction of a lightly dissipative system modelling crawling-type locomotion. The symmetry group $(\mathbb{R})$ is the set of translations along a one-dimensional ground. Given a time-periodic perturbation, the system will admit a relative limit cycle whereupon each period is related to the previous by a shift along the ground. Generalization to a two-dimensional ground is described later in the paper with respect to the symmetry group $\mathrm{SE}(2)$. In this case the resulting limit cycles allow the body to turn and translate by a fixed angle with each period of the perturbation. These toy models identify how symmetry reduction and dissipation can conspire to create robust behavior in crawling, and possibly walking, locomotion.