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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Bounded polynomial vector fields


Authors: Anna Cima and Jaume Llibre
Journal: Trans. Amer. Math. Soc. 318 (1990), 557-579
MSC: Primary 58F14; Secondary 34C40, 58F12, 58F25
DOI: https://doi.org/10.1090/S0002-9947-1990-0998352-5
MathSciNet review: 998352
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Abstract: We prove that, for generic bounded polynomial vector fields in $ {{\mathbf{R}}^n}$ with isolated critical points, the sum of the indices at all their critical points is $ {( - 1)^n}$. We characterize the local phase portrait of the isolated critical points at infinity for any bounded polynomial vector field in $ {{\mathbf{R}}^2}$. We apply this characterization to show that there are exactly seventeen different behaviours at infinity for bounded cubic polynomial vector fields in the plane.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1990-0998352-5
Keywords: Bounded vector field, index, blow-up
Article copyright: © Copyright 1990 American Mathematical Society