On systems governed by two alternating vector fields. (English) Zbl 0797.34047

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.


34C25 Periodic solutions to ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
37B99 Topological dynamics
34C30 Manifolds of solutions of ODE (MSC2000)
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