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Discrete Dynamics in Nature and Society
Volume 2017, Article ID 7163809, 21 pages
https://doi.org/10.1155/2017/7163809
Research Article

Nonnegative Periodic Solutions of a Three-Term Recurrence Relation Depending on Two Real Parameters

1Department of Industrial Engineering and Industrial Management, National Tsing Hua University, Hsinchu 30013, Taiwan
2Department of Mathematics, National Tsing Hua University, Hsinchu 30013, Taiwan

Correspondence should be addressed to Sui Sun Cheng; wt.ude.uhtn.htam@gnehcss

Received 30 October 2016; Accepted 7 February 2017; Published 6 September 2017

Academic Editor: Zhan Zhou

Copyright © 2017 Yen Chih Chang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Simple dynamic systems representing time varying states of interconnected neurons may exhibit extremely complex behaviors when bifurcation parameters are switched from one set of values to another. In this paper, motivated by simulation results, we examine the steady states of one such system with bang-bang control and two real parameters. We found that nonnegative and negative periodic states are of special interests since these states are solutions of linear nonhomogeneous three-term recurrence relations. Although the standard approach to analyse such recurrence relations is the method of finding the general solutions by means of variation of parameters, we find novel alternate geometric methods that offer the tracking of solution trajectories in the plane. By means of this geometric approach, we are then able, without much tedious computation, to completely characterize the nonnegative and negative periodic solutions in terms of the bifurcation parameters.