Rational limit cycles for a class of generalized Abel’s polynomial differential equations. (English) Zbl 1538.34156
Summary: In this manuscript, we deal with a class of generalized Abel’s ordinary polynomial differential equations of the form
\[
\frac{dy}{dt}=F(t)y^2+G(t)y^n,
\]
where \(F(t)\), \(G(t)\) are real polynomials with \(G(t)\neq 0\) and \(n>3\). We prove that these Abel differential equations have non-trivial rational limit cycles. We also discuss the relation between the existence of the non-trivial rational limit cycles and the degrees of real polynomials \(F(t)\), \(G(t)\).
MSC:
| 34C25 | Periodic solutions to ordinary differential equations |
| 34C05 | Topological structure of integral curves, singular points, limit cycles of ordinary differential equations |
| 34C07 | Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert’s 16th problem and ramifications) for ordinary differential equations |
| 37C60 | Nonautonomous smooth dynamical systems |
