Alonso-González, Clementa, Sanz Sánchez, Fernando
Stratification of three-dimensional real flows I: Fitting domains
Journal of Differential Equations. 2023, 361: 40-96. https://doi.org/10.1016/j.jde.2023.02.029
URI: http://hdl.handle.net/10045/132799
DOI: 10.1016/j.jde.2023.02.029
ISSN: 0022-0396 (Print)
Abstract: 
Let ξ be an analytic vector field in R3 with an isolated singularity at the origin and having only hyperbolic singular points after a reduction of singularities π : M → R3. The union of the images by π of the local invariant manifolds at those hyperbolic points, denoted by ∧, is composed of trajectories of ξ accumulating to 0 ∈ R3. Assuming that there are no cycles nor polycycles on the divisor of π, together with a Morse-Smale type property and a non-resonance condition on the eigenvalues at these points, in this paper we prove the existence of a fundamental system {Vn} of neighborhoods well adapted for the description of the local dynamics of ξ: the frontier Fr(Vn) is everywhere tangent to ξexcept around Fr(Vn) ∩ ∧, where transversality is mandatory.
Keywords:Real vector fields, Singularities, Foliations, Reduction of singularities, Vector fields dynamics
Elsevier
info:eu-repo/semantics/article