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Transactions of the American Mathematical Society

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

 

 

Topological types of polynomial differential equations


Author: L. Markus
Journal: Trans. Amer. Math. Soc. 171 (1972), 157-178
MSC: Primary 34C99
DOI: https://doi.org/10.1090/S0002-9947-1972-0306634-4
MathSciNet review: 0306634
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Abstract: Consider a first order system of real ordinary differential equations, with polynomial coefficients, having no critical points in the number space $ {R^n}$. Two such differential systems are called topologically equivalent in case there exists a homeomorphism of $ {R^n}$ onto itself carrying the sensed (not parametrized) solutions of the first system onto the solution family of the second system. Let $ {B^n}(m)$ be the cardinal number of topological equivalence classes for systems in $ {R^n}$ with polynomial coefficients of degree at most $ m$. The author proves that $ {B^2}(m)$ is finite and obtains explicit upper and lower bounds in terms of $ m$. Also examples are given to show that $ {B^n}(m)$ is noncountable for $ n \geqslant 3$ and $ m \geqslant 6$.


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DOI: https://doi.org/10.1090/S0002-9947-1972-0306634-4
Keywords: Polynomial differential equations, foliation, topological equivalence, separatrix, orbit space
Article copyright: © Copyright 1972 American Mathematical Society