{
  "id": "1212.1978",
  "version": "v5",
  "published": "2012-12-10T06:25:45.000Z",
  "updated": "2014-09-04T09:58:04.000Z",
  "title": "The role of symmetry and dissipation in biolocomotion",
  "authors": [
    "Jaap Eldering",
    "Henry O. Jacobs"
  ],
  "comment": "26 pages, 6 figures, 1 table, comments welcome",
  "categories": [
    "math.DS",
    "physics.bio-ph"
  ],
  "abstract": "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.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2014-01-27T16:10:15.000Z",
      "title": "Walking as a limit cycle through symmetry reduction",
      "abstract": "We construct a simple, yet rigorous mathematical model for walking. Our model exhibits robust limit cycle behavior, a feature commonly observed in biomechanical experiments. As such, it may serve as a case study for more complex models by identifying core mechanisms of walking. The singular nature of contact makes walking a difficult problem to analyze mathematically. To overcome this, we consider a simple triple mass-spring system as a model for a walker and we regularize the ground contact. We reduce the system by translation symmetries and prove the existence of limit cycles in the reduced phase space under small periodic forcings. In the unreduced phase space, the lifted trajectories are stable and relatively periodic, wherein each period is related to the previous by an $x$-translation. In conclusion, we have constructed a model which is complex enough to capture behavior characteristic of walking, yet simple enough to prove rigorous results.",
      "comment": "22 pages, 5 figures, 1 table, comments welcome",
      "journal": null,
      "doi": null
    },
    {
      "version": "v5",
      "updated": "2014-09-04T09:58:04.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53D20",
      "34C15",
      "37M99",
      "92B25",
      "92C10"
    ],
    "keywords": [
      "symmetry reduction",
      "robust limit cycle behavior",
      "phase space",
      "simple triple mass-spring system",
      "small periodic forcings"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1212.1978E"
    }
  }
}