Abel-like differential equations with a unique limit cycle. (English) Zbl 1228.34060
Consider the scalar differential equation
\[
{dx\over dt}= \Biggl(\sum^k_{l=1} a_l\sin^{i_l}(t)\cos^{j_l}(t)\Biggr)\, x^n+ b\sin^{i_b}(t)\cos^{j_b}(t) x^m,\tag{\(*\)}
\]
where \(n,m\geq 2\), \(i_l\), \(i_b\), \(j_l\), \(j_b\) are nonnegative integers. An isolated periodic solution of \((*)\) is called a limit cycle of \((*)\). The supremum of all limit cycles of \((*)\) over \((a_1,\dots, a_k, b)\in\mathbb{R}^{k+1}\) is called the Hilbert number \({\mathcal H}\) of \((*)\). The authors derive conditions which characterize the cases \({\mathcal H}= 0\) and \({\mathcal H}= 1\).
Reviewer: Klaus R. Schneider (Berlin)
MSC:
| 34C25 | Periodic solutions to ordinary differential equations |
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