{
  "id": "1307.7143",
  "version": "v3",
  "published": "2013-07-26T18:36:42.000Z",
  "updated": "2014-10-07T18:59:12.000Z",
  "title": "Signal Velocity in Oscillator Networks",
  "authors": [
    "Carlos E. Cantos",
    "J. J. P. Veerman"
  ],
  "comment": "18 pages, 4 figures",
  "categories": [
    "math.DS"
  ],
  "abstract": "We investigate a system of coupled oscillators on the circle, which arises from a simple model for behavior of large numbers of autonomous vehicles. The model considers asymmetric, linear, decentralized dynamics, where the acceleration of each vehicle depends on the relative positions and velocities between itself and a set of local neighbors. We first derive necessary and sufficient conditions for asymptotic stability, then derive expressions for the phase velocity of propagation of disturbances in velocity through this system. We show that the high frequencies exhibit damping, which implies existence of well-defined signal velocities $c_+>0$ and $c_-<0$ such that low frequency disturbances travel through the flock as $f(x-c_+t)$ in the direction of increasing agent numbers and $f(x-c_-t)$ in the other.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-02-19T02:59:13.000Z",
      "abstract": "We investigate decentralized systems of linear coupled oscillators on a circle. First we state and prove necessary and sufficient conditions for nearest neighbor systems to be asymptotically stable. We then investigate these asymptotically stable systems further. We establish that in this case the system behaves like a wave equation in which the high frequencies are damped (due to dispersion). Thus low frequency signals travel through the flock as $f(x-c_+t)$ in the direction of increasing agent numbers and $f(x-c_-t)$ in the other. Here $c_+>0$ and $c_-<0$ are called the \\emph{signal velocities}. The motivation for this work is in more realistic systems of finitely many coupled oscillators on the line, where the same is true away from the boundaries. We can thus study solutions of those systems simply by analyzing what happens near the boundaries.",
      "comment": "11 pages, 4 figures",
      "journal": null,
      "doi": null
    },
    {
      "version": "v3",
      "updated": "2014-10-07T18:59:12.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "34D06"
    ],
    "keywords": [
      "signal velocity",
      "oscillator networks",
      "low frequency signals travel",
      "nearest neighbor systems",
      "system behaves"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1307.7143C"
    }
  }
}