{"paper":{"title":"Random Switching between Vector Fields Having a Common Zero","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Edouard Strickler, Michel Bena\\\"im","submitted_at":"2017-02-10T08:06:11Z","abstract_excerpt":"Let $E$ be a finite set, $\\{F^i\\}_{i \\in E}$ a family of vector fields on $\\mathbb{R}^d$ leaving positively invariant a compact set $M$ and having a common zero $p \\in M.$ We consider a piecewise deterministic Markov process $(X,I)$ on $M \\times E$ defined by $\\dot{X}_t = F^{I_t}(X_t)$ where $I$ is a jump process controlled by $X:$ $\\Pr(I_{t+s} = j | (X_u, I_u)_{u \\leq t}) = a_{i j}(X_t) s + o(s)$ for $i \\neq j$ on $\\{I_t = i \\}.$\n  We show that the behavior of $(X,I)$ is mainly determined by the behavior of the linearized process $(Y,J)$ where $\\dot{Y}_t = A^{J_t} Y_t,$ $A^i$ is the Jacobian "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1702.03089","kind":"arxiv","version":3},"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"}