Estimates on the number of limit cycles of a generalized Abel equation. (English) Zbl 1372.34075
Consider the differential equation
\[
{dx\over dt}= x^m \sum^n_{k=0} a_k(t)x^k\tag{\(*\)}
\]
with \(m\in\mathbb{Z}\) under the conditions
(i) \(a_k:\mathbb{R}\to\mathbb{R}\) are continuous and \(T\)-periodic functions for \(k=1,\dots, n\);
(ii) \(\sum^n_{k=1} |a_k(t)|< a_0(t)\,\forall t\).
Under additional conditions on the functions \(a_k\) the authors prove that \((*)\) has at most one, two or three positive and negative isolated \(T\)-periodic solutions.
(i) \(a_k:\mathbb{R}\to\mathbb{R}\) are continuous and \(T\)-periodic functions for \(k=1,\dots, n\);
(ii) \(\sum^n_{k=1} |a_k(t)|< a_0(t)\,\forall t\).
Under additional conditions on the functions \(a_k\) the authors prove that \((*)\) has at most one, two or three positive and negative isolated \(T\)-periodic solutions.
Reviewer: Klaus R. Schneider (Berlin)
MSC:
| 34C25 | Periodic solutions to 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 |
