On generic polinomial differential equations of second order on the circle

V. Sh. Roitenberg

Sibirskie èlektronnye matematičeskie izvestiâ, Tome 17 (2020), pp. 2122-2130

Résumé

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 $N$.

Détail

Keywords: differential equation of second order, polynomial right-hand side, cylindrical phase space, rough equation, limit cycle.