Stratification of three-dimensional real flows I: Fitting Domains

2022/07/04 by Clementa Alonso-González, Alonso-González, Clementa, Fernando Sanz Sánchez +1
Mathematics · Physics and Astronomy · #34A26 #34A34 #34C08 #34C40 #37C10 #37D05 #37D15 #Advanced Differential Equations and Dynamical Systems #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems

paper · pdf · doi:10.48550/arxiv.2207.01662

openalex publication_date 2022/07/04 · openalex created_date 2022/07/08 · openalex updated_date 2026/07/28

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 Λ, 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 \V_n\ of neighborhoods well adapted for the description of the local dynamics of ξ: the frontier Fr(V_n) is everywhere tangent to ξ except around Fr(V_n)∩Λ, where transvesality is mandatory.

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