Geodesic
On generic polinomial differential equations of second order on the circle
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 17 (2020), pp. 2122-2130.

Voir la notice de l'article provenant de la source Math-Net.Ru

The paper considers second-order differential equations whose right-hand sides are polynomials with respect to the first derivative with periodic continuously differentiable coefficients and corresponding dynamical systems on a cylindrical phase space. The leading coefficient of the polynomial is assumed to be unequal to zero. The concept of a rough equation is introduced – an equation for which the topological structure of the phase portrait does not change when pass to an equation with "close" coefficients. It is proved that the equations for which all singular points and closed trajectories are hyperbolic and there are no trajectories going from saddle to saddle are rough and form an open everywhere dense set in the space of all the considered equations. In addition, we prove that for any natural numbers $N$ and $n>1$, there is a rough equation whose right side is a polynomial of degree $n$, and the number of limit cycles that are not homotopy to zero on the phase cylinder is greater than $N$.
Keywords: differential equation of second order, polynomial right-hand side, cylindrical phase space, rough equation, limit cycle. 
@article{SEMR_2020_17_a114,
     author = {V. Sh. Roitenberg},
     title = {On generic polinomial differential equations of second order on the circle},
     journal = {Sibirskie \`elektronnye matemati\v{c}eskie izvesti\^a},
     pages = {2122--2130},
     publisher = {mathdoc},
     volume = {17},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SEMR_2020_17_a114/}
}
TY  - JOUR
AU  - V. Sh. Roitenberg
TI  - On generic polinomial differential equations of second order on the circle
JO  - Sibirskie èlektronnye matematičeskie izvestiâ
PY  - 2020
SP  - 2122
EP  - 2130
VL  - 17
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/SEMR_2020_17_a114/
LA  - ru
ID  - SEMR_2020_17_a114
ER  - 
%0 Journal Article
%A V. Sh. Roitenberg
%T On generic polinomial differential equations of second order on the circle
%J Sibirskie èlektronnye matematičeskie izvestiâ
%D 2020
%P 2122-2130
%V 17
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SEMR_2020_17_a114/
%G ru
%F SEMR_2020_17_a114
V. Sh. Roitenberg. On generic polinomial differential equations of second order on the circle. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 17 (2020), pp. 2122-2130. http://geodesic.mathdoc.fr/item/SEMR_2020_17_a114/

[1] N.N. Bautin, E.A. Leontovich, Methods and techniques for qualitative study of dynamical systems on the plane, Nauka, M., 1990 | MR | Zbl

[2] E.A. Barbashin, V.A. Tabueva, Dynamical systems with cylindrical phase space, Nauka, M., 1969 | MR | Zbl

[3] V. Sh. Roitenberg, “Limit cycles of Lienard equations with periodic coefficients”, Math. Methods in Ingineering and Technology, 1 (2013), 5–7

[4] V. Sh. Roitenberg, “On limit cycles of a second-order differential equation on the circle that arise from infinity”, Bulletin of the Moscow State Regional University. Ser. Phisics and Mathematics, 2 (2017), 6–15 | DOI

[5] V. Sh. Roitenberg, “On limit cycles of some second order differential equation on the circle”, Sientifical and Technical Bulletin of the Volga Region, 1 (2017), 25–28 | DOI

[6] V.A. Pliss, “The number of periodic solutions of equations whose right-hand member is a polynomial”, Docl. Akad. Nauk SSSR, 127:5 (1959), 965–968 | MR | Zbl

[7] A.L. Neto, “On the number of solutions of the equation for which x(0)=x(1)”, Invent. Math., 59:2 (1980), 67–76 | DOI | MR | Zbl

[8] Ju. Ilyashenko, “Hilbert-type number for Abel equations, growth and zeros of holomorfic functions”, Nonlinearity, 13:4 (2000), 1337–1342 | DOI | MR | Zbl

[9] A. Casull, A. Guillamon, “Limit cycles for generalized Abel equations”, Int. J. Bifurcation Chaos Appl. Sci. Eng., 16:12 (2006), 3737–3745 | DOI | MR | Zbl

[10] M. Caubergh, F. Dumortier, “Hilbert's 16th problem for classical Lienard equations of even degree”, J. Differ. Equations, 244:6 (2008), 1359–1394 | DOI | MR | Zbl

[11] A.A. Andronov, E.A. Leontovich, I.I. Gordon, A.G. Maier, Qualitative theory of second-order dynamical systems, John Wiley Sons, Jerusalem etc, 1973 | MR | Zbl

[12] A.A. Andronov, E.A. Leontovich, I.I. Gordon, A.G. Maier, The theory of bifurcations of dynamical systems on the plane, Nauka, M., 1967 | MR | Zbl

[13] J. Palis, W. Melo, Geometric theory of dynamical systems. An introduction, Springer-Verlag, New York etc, 1982 | MR | Zbl