{
  "id": "1308.4919",
  "version": "v4",
  "published": "2013-07-26T18:39:49.000Z",
  "updated": "2014-10-09T14:36:54.000Z",
  "title": "Transients in the Synchronization of Oscillator Arrays",
  "authors": [
    "Carlos E. Cantos",
    "J. J. P. Veerman"
  ],
  "comment": "11 pages, 4 figures",
  "categories": [
    "math.DS",
    "nlin.CD"
  ],
  "abstract": "The purpose of this note is threefold. First we state a few conjectures that allow us to rigorously derive a theory which is asymptotic in N (the number of agents) that describes transients in large arrays of (identical) linear damped harmonic oscillators in R with completely decentralized nearest neighbor interaction. We then use the theory to establish that in a certain range of the parameters transients grow linearly in the number of agents (and faster outside that range). Finally, in the regime where this linear growth occurs we give the constant of proportionality as a function of the signal velocities (see [3]) in each of the two directions. As corollaries we show that symmetric interactions are far from optimal and that all these results independent of (reasonable) boundary conditions.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2014-03-15T00:24:59.000Z",
      "title": "Transients in the Synchronization of Oscillator Networks",
      "abstract": "We develop a general quantitative theory for the growth of transients in large arrays of (identical) linear oscillators in $\\R$ with completely decentralized nearest neighbor interaction. Transients grow at least linearly in the number of agents. If that is the case, then the constant of proportionality is given by a combination of the signal velocities in each of the two directions. The attenuation of the transients is determined by the squared ratio of these velocities. The optimal case has non-symmetric interaction. We show that these results do not depend on the boundary condition imposed.",
      "journal": null,
      "doi": null
    },
    {
      "version": "v4",
      "updated": "2014-10-09T14:36:54.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37L15"
    ],
    "keywords": [
      "oscillator networks",
      "synchronization",
      "decentralized nearest neighbor interaction",
      "non-symmetric interaction",
      "optimal case"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1308.4919C"
    }
  }
}