Abel-like differential equations with a unique limit cycle
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 \({\mathcal H}= 1\).
scientific article; zbMATH DE number 5903372