Global Attractor of Solutions of a Rational System in the Plane

Miron B Bekker,Martin J Bohner,Hristo D Voulov

doi:10.1155/2015/195247

Abstract

We consider a two-dimensional autonomous system of rational difference equations with three positive parameters. It was conjectured that every positive solution of this system converges to a finite limit. Here we confirm this conjecture, subject to an additional assumption.

Highlights

Rational systems of first-order difference equations in the plane have been receiving increasing attention in the last decade
Along with the results published in [4], several conjectures about some nontrivial cases were posed
In view of (5), the point (x∗, x∗2 − a) is a global attractor for all positive solutions of (4), which confirms the conjecture in case (10) holds

Summary

Introduction

Rational systems of first-order difference equations in the plane have been receiving increasing attention in the last decade. We have confirmed in [2] that [4, Conjecture 2.4 on page 1223] is true. Our goal here is to confirm another conjecture, in the case when α1 ≥ 0, (3). It is clear that the x-component of any positive solution {(xn, yn)}n∈N0 of (4) must satisfy the inequality xnxn+1 > a as well as the difference equation xn+2 = f (xn+1, xn) , n ∈ N0,. Every solution {xn}n∈N0 of (6), with positive initial values x0 and x1 such that x0x1 > a, is positive, satisfies the inequality xxn∗x. NT+h1u>s, tahfeofrixaelldnp∈oinNt0(,xa∗n,dx∗co) nisvearggelosbtaol the equilibrium attractor for all trajectories {Tn(u, V)}n∈N0 with initial point (u, V) ∈ G. In view of (5), the point (x∗, x∗2 − a) is a global attractor for all positive solutions of (4), which confirms the conjecture in case (10) holds

Preliminaries

Main Results

Conflict of Interests