Periodicity of a second-order switched difference system over integers

Wanping Liu

Applied Mathematics and Computation, 2015, vol. 250, issue C, 733-743

Abstract: 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.

Keywords: Switched difference system; Integer solution; Fibonacci sequence; Eventually periodic solution; Matrix equation (search for similar items in EconPapers)
Date: 2015
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Persistent link: https://EconPapers.repec.org/RePEc:eee:apmaco:v:250:y:2015:i:c:p:733-743

DOI: 10.1016/j.amc.2014.11.024

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