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Asymptotic stability at infinity for differentiable vector fields of the plane
Authors:Carlos Gutierrez  Roland Rabanal
Affiliation:Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, Caixa Postal 668, 13560-970, São Carlos, SP, Brazil
Abstract:Let View the MathML source be a differentiable (but not necessarily C1) vector field, where σ>0 and View the MathML source. Denote by R(z) the real part of zC. If for some ?>0 and for all View the MathML source, no eigenvalue of DpX belongs to View the MathML source, then: (a) for all View the MathML source, there is a unique positive semi-trajectory of X starting at p; (b) it is associated to X, a well-defined number I(X) of the extended real line [−∞,∞) (called the index of X at infinity) such that for some constant vector vR2 the following is satisfied: if I(X) is less than zero (respectively greater or equal to zero), then the point at infinity ∞ of the Riemann sphere R2∪{∞} is a repellor (respectively an attractor) of the vector field X+v.
Keywords:Planar vector fields   Asymptotic stability   Markus-Yamabe conjecture   Injectivity
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