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 modeling crawling-type locomotion in three dimensions. The symmetry group SE(2) is the set of rigid transformations of the horizontal (ground) plane. Given a time-periodic perturbation, the system will admit a relative limit cycle whereupon each period is related to the previous by a fixed translation and rotation along the ground. This toy model identifies how symmetry reduction and dissipation can conspire to create robust behavior in crawling, and possibly walking, locomotion. (An erratum is attached.)

MSC codes

  1. symmetry
  2. relative periodic orbits
  3. biomechanics
  4. geometric mechanics

MSC codes

  1. 37D99
  2. 37C20
  3. 37C27
  4. 37J15
  5. 70Q05
  6. 92C10

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Information & Authors


Published In

cover image SIAM Journal on Applied Dynamical Systems
SIAM Journal on Applied Dynamical Systems
Pages: 24 - 59
ISSN (online): 1536-0040


Submitted: 29 May 2014
Accepted: 12 November 2015
Published online: 20 January 2016

MSC codes

  1. symmetry
  2. relative periodic orbits
  3. biomechanics
  4. geometric mechanics

MSC codes

  1. 37D99
  2. 37C20
  3. 37C27
  4. 37J15
  5. 70Q05
  6. 92C10



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