Abstract
We characterize the elements of the set H n of degree n homogeneous polynomial vector fields that are structurally stable with respect to perturbation in H n , both on the plane and on the Poincaré sphere. We use this information to characterize elements of the set W n of smooth vector fields on ĝ2 beginning with terms of order n at (0, 0) that are structurally stable in a neighborhood of (0, 0) under perturbation in W n . We also determine the set of elements of H n that are determining for topological equivalence at (0, 0), in the sense that the topological type of the singularity at (0, 0) is invariant under the addition of higher order terms.
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Shafer, D.S. Weak Singularities of Planar Vector Fields Under Weak Perturbation. Journal of Dynamics and Differential Equations 16, 65–90 (2004). https://doi.org/10.1023/B:JODY.0000041281.28941.6e
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DOI: https://doi.org/10.1023/B:JODY.0000041281.28941.6e