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

Yen Chih Chang,Sui Sun Cheng,Wei Chang Yeh

doi:10.1155/2017/7163809

Yen Chih Chang, Sui Sun Cheng + Show 1 more

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

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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.

Highlights

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
In this paper, motivated by simulation results, we examine the steady states of one such system with bang-bang control and two real parameters
Similar models of piecewise constant dynamic systems which exhibit similar behaviors with parameters can be found in many recent investigations; see for examples [4,5,6,7,8,9,10] and the references therein

Summary

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. The points Pk ∈ {(xk, yk)}k∈N generated by the Tracking Procedure with input (x0, y0) move on Γμ,ρ,f where f = Γμ,ρ(x0, y0) and Γμ,ρ,f is a hyperbola (or a degenerate one made up of two lines). Γ0,−2,f is a degenerate parabola which is made up of two parallel lines L0+,2,f and L0−,2,f as defined in (24) and (25), respectively. In view of Corollary 16, we have found the necessary and sufficient condition for {(xk, yk)}k∈N, which is generated by the Tracking Procedure with input (x0, y0) ≠ (μ/(2+ρ), μ/(2+ ρ)) such that f = Γμ,ρ(x0, y0) and |ρ| < 2, to be periodic. We establish the relation between the nonnegative (or negative) solution of (7) and a special solution μρχ = {μρχk}k∈N in the geometric setting

A Special Positive Orbit

Conclusions