dorsal/arxiv
View SchemaResonant Bifurcations
| Authors | Cicogna G |
|---|---|
| Categories | |
| ArXiv ID | solv-int/9911003 |
| URL | https://arxiv.org/abs/solv-int/9911003 |
Abstract
We consider dynamical systems depending on one or more real parameters, and assuming that, for some ``critical'' value of the parameters, the eigenvalues of the linear part are resonant, we discuss the existence -- under suitable hypotheses -- of a general class of bifurcating solutions in correspondence to this resonance. These bifurcating solutions include, as particular cases, the usual stationary and Hopf bifurcations. The main idea is to transform the given dynamical system into normal form (in the sense of Poincar\'e-Dulac), and to impose that the normalizing transformation is convergent, using the convergence conditions in the form given by A. Bruno. Some specially interesting situations, including the cases of multiple-periodic solutions, and of degenerate eigenvalues in the presence of symmetry, are also discussed with some detail.
{
"annotation_id": "8cbb778a-30c7-4c00-8791-9648fec93488",
"date_created": "2026-03-02T18:02:51.576000Z",
"date_modified": "2026-03-02T18:02:51.576000Z",
"file_hash": "c1134b2b7806f45d9f7f7baa3e5e9a88c284813b991c2da5409002d98cd0620f",
"private": false,
"record": {
"abstract": "We consider dynamical systems depending on one or more real parameters, and\nassuming that, for some ``critical\u0027\u0027 value of the parameters, the eigenvalues\nof the linear part are resonant, we discuss the existence -- under suitable\nhypotheses -- of a general class of bifurcating solutions in correspondence to\nthis resonance. These bifurcating solutions include, as particular cases, the\nusual stationary and Hopf bifurcations. The main idea is to transform the given\ndynamical system into normal form (in the sense of Poincar\\\u0027e-Dulac), and to\nimpose that the normalizing transformation is convergent, using the convergence\nconditions in the form given by A. Bruno. Some specially interesting\nsituations, including the cases of multiple-periodic solutions, and of\ndegenerate eigenvalues in the presence of symmetry, are also discussed with\nsome detail.",
"arxiv_id": "solv-int/9911003",
"authors": [
"Cicogna G"
],
"categories": [
"solv-int",
"nlin.SI"
],
"title": "Resonant Bifurcations",
"url": "https://arxiv.org/abs/solv-int/9911003"
},
"schema_id": "dorsal/arxiv",
"source": {
"execution_id": "069ce0b4-6b4e-42d3-a9ff-8ff74c662d19",
"id": "arXiv Dataset IDs",
"type": "Model",
"variant": "snapshot-2026-03-01",
"version": "0.1.0"
},
"user_id": 1000002
}