A general method for studying quadratic perturbations of the third-order Lyness difference equation

Guifeng Deng,Qiuying Lu,Nianzu Liu

doi:10.1186/1687-1847-2013-193

Abstract

This paper studies the difference equation x n + 3 x n =a+ x n + 1 + x n + 2 +γ x n 2 , where a and γ are arbitrary positive real numbers and the initial values x 0 , x 1 , x 2 >0. It is known that for γ=0 the above equation is the third-order Lyness’ one, studied in several papers. Using an extension of the quasi-Lyapunov method, we prove that for 0<γ<1 the sequences generated by the perturbed third-order Lyness equation are globally asymptotically stable. Moreover, we show that if γ≥1 all solutions of it converge to +∞. Therefore, the values 0 and 1 are two bifurcation points for the equation containing the parameter γ.MSC:39A11, 39A20.

Highlights

1 Introduction Lyness [ – ] discovered that the solutions of the second-order difference equation xn+ xn = xn+ + a is -periodic for a = and positive initial conditions while he was working on a problem in Number Theory
First Zeeman [ ] and, after and independently, Bastien et al [ ] gave a complete description of global dynamics of the second-order Lyness equation with a > by the interpretation of the iteration of the map F induced by this recurrence on the Lyness’ cubic which passes through the initial points (x, x )
In this paper we study the global asymptotic behavior of all solutions of the perturbed third-order Lyness difference equation xn+

Summary

Introduction

1 Introduction Lyness [ – ] discovered that the solutions of the second-order difference equation xn+ xn = xn+ + a is -periodic for a = and positive initial conditions while he was working on a problem in Number Theory. They proved that for each a = the periods of the sequences generated by equation ( ) can be almost all natural numbers, depending on the initial points (x , x ). It is well known that there are no convergent nontrivial solutions for the Lyness difference equation.

Results

Conclusion