Final forms for a three-dimensional vector field under blowing-up. (English) Zbl 0607.58027

We study the final situations which may be obtained for a singular vector field by permissible blowing-ups of the ambient space (in dimension three). The final situations generalize the simple singularities for the two dimensional case, i.e. the linear part of the vector field has two distinct eigenvalues \(a\neq b\neq 0\) with a/b not a strictly positive rational number. In the simple corners, the only integral branches are the two components of the exceptional divisor, and in the other simple points, there is exactly one integral branch not contained in the exceptional divisor. In dimension three we have a similar behaviour, hence a description of the integral branches once a final situation is reached. These final situations are also persistent under permissible blowing-ups, like in the two dimensional case. Technically, we take a logarithmic approach, by marking in each step the exceptional divisor of the transformation and we use numerical invariants in order to control the algorithms for reaching the final forms.


37G99 Local and nonlocal bifurcation theory for dynamical systems
58C25 Differentiable maps on manifolds
58K99 Theory of singularities and catastrophe theory
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