Discrete Dynamics in Nature and Society

Volume 2017, Article ID 7163809, 21 pages

https://doi.org/10.1155/2017/7163809

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

^{1}Department of Industrial Engineering and Industrial Management, National Tsing Hua University, Hsinchu 30013, Taiwan^{2}Department 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.

#### 1. Introduction

Simple dynamic systems representing time varying states of interconnected compartments or “neurons” may exhibit extremely complex behaviors when bifurcation parameters are switched from one set of values to another. An example has been given in several of our previous studies [1] and a slightly modified model of which is described as follows. Let neurons be placed on the vertices of a regular -gon and let the time dependent state values of the neurons be denoted by , for Suppose the rate of change is determined by the resultant effect of a constant multiple of , plus a magnified on-off (or bang bang) state dependent control mechanism as well as its two near neighbors , in the form Then assuming “uniformity” among the neuron interactions, we havewhere is identified with while with for compatibility reasons. Here is the on-off Heaviside step function defined by

When the real parameters and are switched among different real values, it is expected (and verified by simulations) that complex dynamic behaviors will be more than abundant. Some of these behaviors can be explained (see [1–3]) but some not (at least to the best of our knowledge). Similar models of piecewise constant dynamic systems which exhibit similar behaviors with parameters can be found in many recent investigations; see for examples [4–10] and the references therein.

In this note, we discuss the steady state solutions of the above dynamic system (i.e., those that satisfy for all and for each , or ). Then we will face the existence problem of periodic solutions of the following three-term recurrence relation:In [1–3], we are lucky to obtain complete information about the periodic solutions of (4) when and , , or by breaking the solutions into two sequences, a companion and an error (more specifically, for a solution of (4) with and , the companion sequence is which is an even integral sequence so that , where is the greatest even integer that is less than or equal to , and the error sequence is defined by ).

Yet, for other values of and , simulations show complex periodic behaviors beyond our present comprehension, except when the periodic solutions are also “nonnegative” (or “negative”). Such exceptional results, when examined more closely, can be explained.

In this paper, we will devote ourselves to explaining the behaviors of these nonnegative (or negative) and periodic solutions of (7).

First, a real sequence is a solution of (4) if it renders (4) into an identity after substitution. It is nonnegative (or negative) if all its terms are nonnegative (respectively negative). If is a nonnegative solution of (4), then clearly satisfieswhile if is a negative solution of (4), thenHence it seems that we are back to the usual nonhomogeneous linear second-order difference equations studied in elementary theory of difference equations. Since in (5) and (6), we can further restrict ourselves in the sequel to the following difference equation:with arbitrary By means of the techniques of general solutions plus the method of variation of parameters, it may be argued that the existence of nonnegative (respectively negative) periodic solutions can be handled completely. Such an assertion may be true in theory, but the general solution here is - as well as -dependent and hence actual attempts lead to many complications.

That said, in this paper, we will handle our equation from a novel approach and the crux of which is based on representing each pair of two consecutive terms of a solution as a point in the plane and invent a geometric method to track the movements of these points. First, the totality of such pairs is called the orbit of this solution. More precisely, let be a solution of (4). We define the orbit of byand the “positive” orbit byAs examples, let and let and be solutions of (4) with or , respectively. The orbits with and with are depicted in Figures 1 and 2, respectively. These figures clearly suggest that is “-periodic,” that is “-periodic,” and that there are accompanying “distinctive” features which can be exploited further.