Stratification of three-dimensional real flows I: Fitting domains

Abstract

Let ξ be an analytic vector field in R3with 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 πof 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 transversalityis mandatory.