On systems governed by two alternating vector fields (Q1325178)

The purpose of this paper is to investigate the properties of the \(2p\)- periodic non-autonomous system of ODE's \(dx/dt= v(x)+ \gamma_ p(t)[w(x)- v(x)]\), where \(v\) and \(w\) are smooth vector fields on \(\mathbb{R}^ n\), related by an involutive diffeomorphism \(G\) [i.e., \(G^ 2=\text{id}\) and \(w= G_ *(v)]\), and \(\gamma_ p(t)= 0\) and 1 when \(0\leq t< p\) and \(p\leq t< 2p\), respectively [moreover, \(\gamma_ p(t)\) is \(2p\)-periodic]. A natural example of such a system is the model of a tabular catalytic reactor with flow reversal. The \(G\)-symmetric properties of periodic orbits are proved in Theorems 1 and 2 in this paper. Theorem 3 proves the existence of periodic orbits when \(G\) has an isolated fixed-point and \(p\) is small enough, and Theorem 4 considers the measure property of bounded invariant sets for some special systems.