{
  "id": "1506.06613",
  "version": "v1",
  "published": "2015-06-22T14:07:32.000Z",
  "updated": "2015-06-22T14:07:32.000Z",
  "title": "Contraction After Small Transients",
  "authors": [
    "Michael Margaliot",
    "Eduardo D. Sontag",
    "Tamir Tuller"
  ],
  "comment": "arXiv admin note: substantial text overlap with arXiv:1406.1474",
  "categories": [
    "math.DS"
  ],
  "abstract": "Contraction theory is a powerful tool for proving asymptotic properties of nonlinear dynamical systems including convergence to an attractor and entrainment to a periodic excitation. We consider three generalizations of contraction with respect to a norm that allow contraction to take place after small transients in time and/or amplitude. These generalized contractive systems (GCSs) are useful for several reasons. First, we show that there exist simple and checkable conditions guaranteeing that a system is a GCS, and demonstrate their usefulness using several models from systems biology. Second, allowing small transients does not destroy the important asymptotic properties of contractive systems like convergence to a unique equilibrium point, if it exists, and entrainment to a periodic excitation. Third, in some cases as we change the parameters in a contractive system it becomes a GCS just before it looses contractivity with respect to a norm. In this respect, generalized contractivity is the analogue of marginal stability in Lyapunov stability theory.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-06-22T14:07:32.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "small transients",
      "contraction",
      "contractive system",
      "periodic excitation",
      "lyapunov stability theory"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150606613M"
    }
  }
}