On Periodic Orbits in Discrete-time Cascade Systems

Question. Does system (1.1) have periodic orbits when system (1.2) has periodic orbits? Recently, there have been a lot of researches in the literature on the periodicity of discrete-time dynamical systems [1, 3–8, 10, 11]. However, to the authors’ knowledge the above question has not received investigations, therefore in this paper we study the above question and obtain a fundamental result. On the basis of the fundamental result in Section 2 we further discuss the number and the minimum period of periodic orbits of system (1.1) when system (1.2) has periodic orbits. In Section 3, we present a result on the stability of periodic orbits.