Periodic solutions to two-dimensional forced systems: The Massera theorem and its extension
The author deals with two-dimensional systems of the form \[ \dot x= F(t,x),\tag{1} \] where \(F:\mathbb{R}^3\to \mathbb{R}^2\) is \(T\)-periodic in \(t\), \(T>0\). After a short survey of results concerning the existence of \(qT\)-periodic solutions to (1) (or higher-dimensional systems) with an integer \(q\geq 1\), the author presents a proof of the Massera theorem which is based on the Brouwer asymptotic fixed point theorem. Further, he extends the Massera theorem in the following way. He assumes that \(F\) is continuous and locally Lipschitz in the second variable and that all solutions to (1) can be continued from \(t= 0\) to \(t= T\). If there exists a bounded solution to (1) that is bounded away from a given \(T\)-periodic solution \(\zeta\), then there exists a \(qT\)-periodic solution (other than \(\zeta\)) for some integer \(q\geq 1\) or all solutions to (1) that are bounded and bounded away from \(\zeta\) have a well-defined irrational rotation number about \(\zeta\).
scientific article; zbMATH DE number 1210117