{"paper":{"title":"Meso-scale obstructions to stability of 1D center manifolds for networks of coupled differential equations with symmetric Jacobian","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Anne-Ly Do, Jeremias Epperlein, Stefan Siegmund, Thilo gross","submitted_at":"2012-07-16T17:26:26Z","abstract_excerpt":"A linear system $\\dot x = Ax$, $A \\in \\mathbb{R}^{n \\times n}$, $x \\in \\mathbb{R}^n$, with $\\mathrm{rk} A = n-1$, has a one-dimensional center manifold $E^c = \\{v \\in \\mathbb{R}^n : Av=0\\}$. If a differential equation $\\dot x = f(x)$ has a one-dimensional center manifold $W^c$ at an equilibrium $x^*$ then $E^c$ is tangential to $W^c$ with $A = Df(x^*)$ and for stability of $W^c$ it is necessary that $A$ has no spectrum in $\\mathbb{C}^+$, i.e.\\ if $A$ is symmetric, it has to be negative semi-definite.\n  We establish a graph theoretical approach to characterize semi-definiteness. Using spanning "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1207.3736","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"}