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 Mat. Sb. (N.S.), 1969, Volume 80(122), Number 3(11), Pages 388–404 (Mi msb3625)

This article is cited in 7 scientific papers (total in 7 papers)

An example of eqations $\frac{dw}{dz}=\frac{P_n(z,w)}{Q_n(z,w)}$ having a countable number of limit cycles and arbitrarily large Petrovskii–Landis genus

Yu. S. Ilyashenko

Abstract: In this work we construct an open set $V$ in the space of coefficients $A_n$ of the equations $\frac{dw}{dz}=\frac{P_n(z,w)}{Q_n(z,w)}$ such that on the solutions of an arbitrary equation $\alpha\in V$ there exist a countable number of homotopically distinct limit cycles. Also, for each natural number $N$ we construct an open set $V_N\subset A_n$ such that an arbitrary equation $\alpha\in V_N$ has a Petrovskii–Landis genus which exceeds $N$.
Bibliography: 9 titles.

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English version:
Mathematics of the USSR-Sbornik, 1969, 9:3, 365–378

Bibliographic databases:

UDC: 517.92
MSC: 34C07, 34C05
Received: 04.02.1969

Citation: Yu. S. Ilyashenko, “An example of eqations $\frac{dw}{dz}=\frac{P_n(z,w)}{Q_n(z,w)}$ having a countable number of limit cycles and arbitrarily large Petrovskii–Landis genus”, Mat. Sb. (N.S.), 80(122):3(11) (1969), 388–404; Math. USSR-Sb., 9:3 (1969), 365–378

Citation in format AMSBIB
\Bibitem{Ily69} \by Yu.~S.~Ilyashenko \paper An example of eqations $\frac{dw}{dz}=\frac{P_n(z,w)}{Q_n(z,w)}$ having a~countable number of limit cycles and arbitrarily large Petrovskii--Landis genus \jour Mat. Sb. (N.S.) \yr 1969 \vol 80(122) \issue 3(11) \pages 388--404 \mathnet{http://mi.mathnet.ru/msb3625} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=259239} \zmath{https://zbmath.org/?q=an:0202.09201|0214.09404} \transl \jour Math. USSR-Sb. \yr 1969 \vol 9 \issue 3 \pages 365--378 \crossref{https://doi.org/10.1070/SM1969v009n03ABEH001288} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Yu. S. Ilyashenko, “The nonalgebraic character of the manifold of differential equations with rational right-hand sides and with multiple limit cycles”, Math. USSR-Sb., 12:3 (1970), 453–457
2. I. A. Pushkar', “Many-Dimensional Generalization of the Il'yashenko Theorem on Abelian Integrals”, Funct. Anal. Appl., 31:2 (1997), 100–108
3. I. A. Pushkar', “Limit cycles generated by perturbations of Hamiltonian systems”, Russian Math. Surveys, 57:5 (2002), 1002–1004
4. Ilyashenko Y., “Centennial History of Hilbert's 16th Problem”, Bull. Amer. Math. Soc., 39:3 (2002), 301–354
5. A. A. Shcherbakov, “Dynamics of Local Groups of Conformal Mappings and Generic Properties of Differential Equations on $\mathbb C^2$”, Proc. Steklov Inst. Math., 254 (2006), 103–120
6. I. A. Khovanskaya (Pushkar'), “Weak Infinitesimal Hilbert's 16th Problem”, Proc. Steklov Inst. Math., 254 (2006), 201–230
7. A. A. Glutsyuk, Yu. S. Ilyashenko, “Restricted version of the infinitesimal Hilbert 16th problem”, Mosc. Math. J., 7:2 (2007), 281–325
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