Periodicity of a second-order switched difference system over integers

In this paper, a second-order switched difference system which consists of two linear difference equations with a switching rule is proposed to study. Specifically, the periodicity of a particular case is addressed, deriving the appropriate rational values for parameter r which possess periodic integer solutions. By the transformation method, the particular second-order difference system is transformed into a first-order switched system. And, we prove that: (1) this system possesses periodic integer solutions of prime period two if and only if r=-1/2; (2) any rational r except for the integers arises periodic integer solutions of prime period three; (3) periodic integer solutions of prime period four exist if and only if r=-1/2; (4) this system possesses no periodic solutions of prime period five. We also prove that if r>0 and the system has periodic integer solutions of prime period k⩾6, then the only possible values of r are reciprocals of integers.