Bounded polynomial vector fields
HTML articles powered by AMS MathViewer
- by Anna Cima and Jaume Llibre PDF
- Trans. Amer. Math. Soc. 318 (1990), 557-579 Request permission
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
-
A. A. Andronov, E. A. Leontovich, I. I. Gordon and A. L. Maier, Qualitative theory of second order dynamic systems, Wiley, 1973.
A. Cima, Indices of polynomial vector fields with applications, Ph.D. Thesis, Universitat Autònoma de Barcelona, 1987.
- 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), no. 3, 372–399. MR 884276, DOI 10.1016/0022-0396(87)90133-1
- R. J. Dickson and L. M. Perko, Bounded quadratic systems in the plane, J. Differential Equations 7 (1970), 251–273. MR 252787, DOI 10.1016/0022-0396(70)90110-5
- Enrique A. González Velasco, Generic properties of polynomial vector fields at infinity, Trans. Amer. Math. Soc. 143 (1969), 201–222. MR 252788, DOI 10.1090/S0002-9947-1969-0252788-8
- John Guckenheimer and Philip Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, Applied Mathematical Sciences, vol. 42, Springer-Verlag, New York, 1983. MR 709768, DOI 10.1007/978-1-4612-1140-2
- Morris W. Hirsch and Stephen Smale, Differential equations, dynamical systems, and linear algebra, Pure and Applied Mathematics, Vol. 60, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1974. MR 0486784
- Solomon Lefschetz, Differential equations: Geometric theory, 2nd ed., Pure and Applied Mathematics, Vol. VI, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1963. MR 0153903
- John W. Milnor, Topology from the differentiable viewpoint, University Press of Virginia, Charlottesville, Va., 1965. Based on notes by David W. Weaver. MR 0226651
- Jorge Sotomayor, Curvas definidas por equações diferenciais no plano, Instituto de Matemática Pura e Aplicada, Conselho Nacional de Desenvolvimento Científico e Tecnológico, Rio de Janeiro, 1981 (Portuguese). 13$^\textrm {o}$ Colóquio Brasileiro de Matemática. [13th Brazilian Mathematics Colloquium]. MR 716683
Additional Information
- © Copyright 1990 American Mathematical Society
- 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