{
  "id": "2309.06398",
  "version": "v1",
  "published": "2023-09-12T17:01:50.000Z",
  "updated": "2023-09-12T17:01:50.000Z",
  "title": "Bifurcation and periodic solutions to neuroscience models with a small parameter",
  "authors": [
    "José Oyarce"
  ],
  "categories": [
    "math.DS"
  ],
  "abstract": "The existence of periodic solutions is proven for some neuroscience models with a small parameter. Moreover, the stability of such solutions is investigated, as well. The results are based on a theoretical research dealing with the functional differential equation with parameters $$ \\dot{x}(t)=L(\\tau) x_t + \\varepsilon f(t, x_t), $$ where $L: \\mathbb{R}_+\\rightarrow \\mathcal{L}(C; \\mathbb{R})$ and $f: \\mathbb{R} \\times C \\rightarrow \\mathbb{R}$ are, respectively, linear and nonlinear operators, and $\\varepsilon>0$ is a small enough parameter. The theoretical results are applied to a Parkinson's disease model, where the obtained conclusions are illustrated by numerical simulations.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-09-12T17:01:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "34K13",
      "34K18",
      "34C20",
      "46N60",
      "92C20"
    ],
    "keywords": [
      "periodic solutions",
      "neuroscience models",
      "small parameter",
      "bifurcation",
      "parkinsons disease model"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}