{"paper":{"title":"Bifurcation values for a family of planar vector fields of degree five","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"A. Gasull, H. Giacomini, J. D. Garc\\'ia-Salda\\~na","submitted_at":"2012-02-09T09:06:20Z","abstract_excerpt":"We study the number of limit cycles and the bifurcation diagram in the Poincar\\'{e} sphere of a one-parameter family of planar differential equations of degree five $\\dot {\\bf x}=X_b({\\bf x})$ which has been already considered in previous papers. We prove that there is a value $b^*>0$ such that the limit cycle exists only when $b\\in(0,b^*)$ and that it is unique and hyperbolic by using a rational Dulac function. Moreover we provide an interval of length 27/1000 where $b^*$ lies. As far as we know the tools used to determine this interval are new and are based on the construction of algebraic c"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1202.1919","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}