{"paper":{"title":"Some results on homoclinic and heteroclinic connections in planar systems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Armengol Gasull, Hector Giacomini, Joan Torregrosa","submitted_at":"2009-12-17T18:53:35Z","abstract_excerpt":"Consider a family of planar systems depending on two parameters $(n,b)$ and having at most one limit cycle. Assume that the limit cycle disappears at some homoclinic (or heteroclinic) connection when $\\Phi(n,b)=0.$ We present a method that allows to obtain a sequence of explicit algebraic lower and upper bounds for the bifurcation set ${\\Phi(n,b)=0}.$ The method is applied to two quadratic families, one of them is the well-known Bogdanov-Takens system. One of the results that we obtain for this system is the bifurcation curve for small values of $n$, given by $b=\\frac5 7 n^{1/2}+{72/2401}n- {3"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0912.3490","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"}