In the paper the authors prove a result which aims to be a partial generalization of the topological linearization theorem for vector fields of Hartman-Grobman. Their theorem says that if a singularity p of a vector field X is hyperbolic, then the vector field X is locally equivalent (in a neighborhood of the singularity p) to the linear part at p of the vector field X (i.e. to dX(p)). In the paper they will present a result which gives a sufficient condition for a vector field X on (R sup 3) to be equivalent at a singularity to the first non-vanishing jet j(k)x(p). The condition is stated in terms of the blowing vector field (x bar) (in terms of spherical coordinates, i.e. on (S sup 2) x (R sup 3)), and essentially means that there are no saddle-connections for (x bar) / (S sup 2) x (0). In a future paper the authors hope to get a result in the direction which completely generalizes Hartman-Groman's result. (Copyright (c) 1986 by Faculty of Mathematics and Informatics, Delft, The Netherlands.)
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