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The nonalgebraic character of the manifold of differential equations with rational right-hand sides and with multiple limit cycles
Yu. S. Ilyashenko
Abstract:
Let $\mathrm A^R_n$ denote the coefficient space of the equations $\frac{dy}{dx}=\frac{P_n(x,y)}{Q_n(x,y)}$, $(x,y)\in R^2$, where $P_n$ and $Q_n$ are polynomials of degree $n\geqslant2$, and let $M_k$ denote the set of equations $\alpha\in\mathrm A^R_n$ that have limit cycles of multiplicity not less than $k$. For $2\leqslant k\leqslant\frac{n(n+1)}2$ the set $M_k$ is not empty. A proof is given for the
Theorem. The set $M_k$ does not form a semialgebraic manifold.
Bibliography: 4 titles.
Received: 02.03.1970
Citation:
Yu. S. Ilyashenko, “The nonalgebraic character of the manifold of differential equations with rational right-hand sides and with multiple limit cycles”, Mat. Sb. (N.S.), 83(125):3(11) (1970), 452–455; Math. USSR-Sb., 12:3 (1970), 453–457
Linking options:
https://www.mathnet.ru/eng/sm3521https://doi.org/10.1070/SM1970v012n03ABEH000930 https://www.mathnet.ru/eng/sm/v125/i3/p452
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Abstract page: | 468 | Russian version PDF: | 82 | English version PDF: | 2 | References: | 69 | First page: | 3 |
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