On polynomial differential equations of the second order on a circle without singular points

In this paper, autonomous differential equations of the second order are considered, the right-hand sides of which are polynomials of degree $n$ with respect to the first derivative with periodic continuously differentiable coefficients, and the corresponding vector fields on the cylindrical phase space. The free term and the leading coefficient of the polynomial is assumed not to vanish, which is equivalent to the absence of singular points of the vector field. Rough equations are considered for which the topological structure of the phase portrait does not change under small perturbations in the class of equations under consideration. It is proved that the equation is rough if and only if all its closed trajectories are hyperbolic. Rough equations form an open and everywhere dense set in the space of the equations under consideration. It is shown that for $n > 4$ an equation of degree $n$ can have arbitrarily many limit cycles. For $n = 4$, the possible number of limit cycles is determined in the case when the free term and the leading coefficient of the equation have opposite signs.