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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.


References [Enhancements On Off] (What's this?)

  • [ALGM] A. A. Andronov, E. A. Leontovich, I. I. Gordon and A. L. Maier, Qualitative theory of second order dynamic systems, Wiley, 1973.
  • [C] A. Cima, Indices of polynomial vector fields with applications, Ph.D. Thesis, Universitat Autònoma de Barcelona, 1987.
  • [CGL] B. Coll, A. Gasull and J. Llibre, Some theorems on the existence, uniqueness and nonexistence of limit cycles for quadratic systems, J. Differential Equations 67 (1987), 372-399. MR 884276 (88c:34038)
  • [DP] R. J. Dickson and L. M. Perko, Bounded quadratic systems in the plane, J. Differential Equations 7 (1970), 251-273. MR 0252787 (40:6004)
  • [G] E. A. V. Gonzales, Generic properties of polynomial vector fields at infinity, Trans. Amer. Math. Soc. 143 (1969), 201-222. MR 0252788 (40:6005)
  • [GH] J. Guckenheimer and P. Holmes, Nonlinear oscillations, dynamical systems and bifurcation of vector fields, Springer-Verlag, 1983. MR 709768 (85f:58002)
  • [HS] M. Hirsch and S. Smale, Differential equations, dynamical systems and linear algebra, Academic Press, 1974. MR 0486784 (58:6484)
  • [L] S. Lefschetz, Differential equations: Geometric theory, Interscience, 1962. MR 0153903 (27:3864)
  • [M] J. Milnor, Topology from a differentiable viewpoint, University Press of Virginia, Charlottesville, Virginia, 1965. MR 0226651 (37:2239)
  • [S] J. Sotomayor, Curvas definidas por equacoes diferenciais no plano, Inst. Mat. Pura e Aplicada, Rio de Janeiro, 1981. MR 716683 (84m:34004)

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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

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