The nonalgebraic character of the manifold of differential equations with rational right-hand sides and with multiple limit cycles

Yu. S. Ilyashenko

Sbornik. Mathematics, Tome 12 (1970) no. 3, pp. 453-457

Résumé

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.