2006/01/13 by C. Gutierrez, Carlos Gutiérrez, Benito Pires +6
Mathematics · #34D23 (Primary) #Advanced Differential Equations and Dynamical Systems #Dynamical Systems (math.DS) #FOS: Mathematics #Nonlinear Differential Equations Analysis #Nonlinear Partial Differential Equations #math.DS #msc:34D23
paper · pdf · doi:10.48550/arxiv.math/0601341
16 pages, 7 figures
openalex publication_date 2006/01/13 · arxiv created 2006/07/19 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let X:R2\Dr->R2 be a differentiable (but not necessarily C1) vector field, where r>0 and Dr=z∈ R2:|z|≤ r. If for some e>0 and for all p∈ R2\Dr, no eigenvalue of Dp X belongs to (-e,0]∪ z∈\C:R(z)≥ 0, then (a)For all p∈ R2\Dr, there is a unique positive semi--trajectory of X starting at p; (b)I(X), the index of X at infinity, is a well defined number of the extended real line [-∞,∞); (c) There exists a constant vector v∈ R2 such that if I(X) is less than zero (resp. greater or equal to zero), then the point at infinity ∞ of the Riemann sphere R2∪\set∞ is a repellor (resp. an attractor) of the vector field X+v.