2003/03/25 by C. A. Morales, Morales, C. A., Maria José Pacífico +2
Engineering · Mathematics · #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Biology Tumor Growth #Primary 37D30 #Secondary 37D45 #Stability and Controllability of Differential Equations #math.DS #msc:37D30 #msc:37D45
paper · pdf · doi:10.48550/arxiv.math/0303310
17 pages, 3 figures
arxiv created 2003/03/25 · openalex publication_date 2003/03/25 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A recent problem in dynamics is to determinate whether an attractor Λ of a Cr flow X is Cr robust transitive or not. By \em attractor we mean a transitive set to which all positive orbits close to it converge. An attractor is Cr robust transitive (or \em Cr robust for short) if it exhibits a neighborhood U such that the set ∩t>0Yt(U) is transitive for every flow Y Cr close to X. We give sufficient conditions for robustness of attractors based on the following definitions. An attractor is \em singular-hyperbolic if it has singularities (all hyperbolic) and is partially hyperbolic with volume expanding central direction \citeMPP. An attractor is \em Cr critically-robust if it exhibits a neighborhood U such that ∩t>0Yt(U) is in the closure of the closed orbits is every flow Y Cr close to X. We show that on compact 3-manifolds all Cr critically-robust singular-hyperbolic attractors with only one singularity are Cr robust.