{"paper":{"title":"Infinite Orbit depth and length of Melnikov functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Dmitry Novikov, Jessie Pontigo-Herrera, Laura Ortiz-Bobadilla, Pavao Mardesic","submitted_at":"2019-07-22T23:24:17Z","abstract_excerpt":"In this paper we study polynomial Hamiltonian systems $dF=0$ in the plane and their small perturbations: $dF+\\epsilon\\omega=0$. The first nonzero Melnikov function $M_{\\mu}=M_{\\mu}(F,\\gamma,\\omega)$ of the\n  Poincar\\'e map along a loop $\\gamma$ of $dF=0$ is given by an iterated integral. In a previous work (see arXiv 1703.03837), we bounded the length of the iterated integral $M_\\mu$ by a geometric number $k=k(F,\\gamma)$ which we call orbit depth. We conjectured that the bound is optimal. Here, we give a simple example of a Hamiltonian system $F$ and its orbit $\\gamma$ having infinite orbit de"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1907.09627","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"}