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

- Guifeng Deng
^{1}Email author, - Qiuying Lu
^{2}and - Nianzu Liu
^{1}

**2013**:193

https://doi.org/10.1186/1687-1847-2013-193

© Deng et al.; licensee Springer 2013

**Received: **28 February 2013

**Accepted: **18 June 2013

**Published: **1 July 2013

## Abstract

This paper studies the difference equation ${x}_{n+3}{x}_{n}=a+{x}_{n+1}+{x}_{n+2}+\gamma {x}_{n}^{2}$, 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 $\gamma =0$ 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 $0<\gamma <1$ the sequences generated by the perturbed third-order Lyness equation are globally asymptotically stable. Moreover, we show that if $\gamma \ge 1$ 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.

## Keywords

## 1 Introduction

*et al.*[5] gave a complete description of global dynamics of the second-order Lyness equation with $a>0$ by the interpretation of the iteration of the map

*F*induced by this recurrence on the Lyness’ cubic which passes through the initial points $({x}_{1},{x}_{0})$. Cima

*et al.*[6] applied an approach different to the ones in Refs. [5, 7] to study equation (1). That is, their main tool is the study of an ordinary differential equation associated to equation (1). They proved that for each $a\ne 1$ the periods of the sequences generated by equation (1) can be almost all natural numbers, depending on the initial points $({x}_{0},{x}_{1})$. In recent years various generalized Lyness difference equations, including the Lyness difference equation with variable coefficients, the order

*k*Lyness difference equation and the perturbed Lyness-type order

*k*difference equation, have been extensively studied [8–14]. It is well known that there are no convergent nontrivial solutions for the Lyness difference equation. In [8], Li found the convergent solutions of the Lyness difference equation with variable coefficients

where $\{{a}_{n}\}$ is a monotonic non-increasing positive sequence, which demonstrates the essential difference between Lyness difference equation with constant coefficients and Lyness difference equation with variable coefficients.

where $a,\gamma \in (0,+\mathrm{\infty})$ and the initial values ${x}_{0},{x}_{1},{x}_{2}\in (0,+\mathrm{\infty})$. It is clear that the quadratic term ${x}_{n}^{2}$ is a small perturbation for small positive number *γ*. The invariant curve of difference equations often plays a critical role in studying the stability behavior of their solutions; see [5, 15, 16]. Kocic *et al.* [17] have applied KAM theory to prove the stability of the solutions of the Lyness equation. Meanwhile, Lyapunov functions have been found in this area by several papers; see [12, 15, 16]. Here, using an extension of the method introduced in [12], which itself generalizes an idea of Merino [18], we obtain the preservation of global asymptotic stability for the third-order Lyness difference equation under quadratic perturbations.

Our main result is the following.

**Theorem 1.1**(1)

*If*$0<\gamma <1$,

*then the positive equilibrium point*${l}_{\gamma}$

*of equation*(2)

*is globally asymptotically stable*,

*where*

(2) *If* $\gamma \ge 1$, *then the sequence* ${\{{x}_{n}\}}_{n=0}^{\mathrm{\infty}}$ *generated by equation* (2) *converges to* +∞.

Cima *et al.* [6] proved that for a given $a>0$ there exist periodic sequences ${\{{x}_{n}\}}_{n=0}^{\mathrm{\infty}}$ generated by equation (1) which have almost all long periods and that for a full measure set of initial conditions the sequences ${\{{x}_{n}\}}_{n=0}^{\mathrm{\infty}}$ are dense in either one or two disjoint bounded intervals of *R*. In summary, the sequences ${\{{x}_{n}\}}_{n=0}^{\mathrm{\infty}}$ are periodic or strictly oscillatory, and the equilibrium point ${l}_{0}$ is locally stable, with ${l}_{0}=1+\sqrt{1+a}$. That is, for every $\u03f5>0$, there is a $\delta >0$ so that, for any positive initial values ${x}_{0}$, ${x}_{1}$ and ${x}_{2}$ with $|{x}_{i}-{l}_{0}|<\delta $ for $i=0,1,2$, one has $|{x}_{n}-{l}_{0}|<\u03f5$ for all $n\ge 0$. According to Theorem 1.1, the sequences ${\{{x}_{n}\}}_{n=0}^{\mathrm{\infty}}$ of solutions of equation (2) are converging to ${l}_{\gamma}$ if $0<\gamma <1$. So, the qualitative nature of the solutions of equation (2) changes when *γ* vanishes. Note that the sequences ${\{{x}_{n}\}}_{n=0}^{\mathrm{\infty}}$ are increasing to +∞ if $\gamma \ge 1$. So, the behavior of the solutions of equation (2) is completely different from the case $0<\gamma <1$. Thus, the values 0 and 1 are two bifurcation points for the difference equation (2) containing the parameter *γ*.

## 2 Lyapunov function

*F*in ${R}_{+}^{3}$, where ${R}_{+}^{3}=\{(x,y,z)|x>0,y>0,z>0\}$ and ${l}_{\gamma}=\frac{1+\sqrt{1+a(1-\gamma )}}{1-\gamma}$. It is well known [6, 15] that for $\gamma =0$ equation (2) possesses the following first integral:

*F*maps the level surface ${V}^{-1}(k)$ into itself for every k when $\gamma =0$. In order to study the global convergence properties of the recurrence (2) for $0<\gamma <1$, we introduce a very important function $I(x,y,z)$ given by

*I*, we construct the Lyapunov function $L(x,y,z)$ of equation (2). Set ${k}_{\gamma}=I({l}_{\gamma},{l}_{\gamma},{l}_{\gamma})$, then we show that the Lyapunov function is

*H*at the stationary point ${P}_{\gamma}=({x}_{c},{y}_{c},{z}_{c})=({l}_{\gamma},{l}_{\gamma},{l}_{\gamma})$, where

*I*, we get

*m*th-order principal minor of the Hessian

*H*. Then we obtain

Let ${I}_{k}=\{(x,y,z)|I(x,y,z)=k,(x,y,z)\in {R}_{+}^{3}\}$ be the level surface of *I* for $k>{k}_{\gamma}$. It is well known that the function $V(x,y,z)$ has a minimum at $(\tilde{x},\tilde{x},\tilde{x})$ in ${R}_{+}^{3}$, where $\tilde{x}=1+\sqrt{1+a}$. In order to study the third-order Lyness equation, Cima *et al.* [6] proved the following result.

**Lemma 2.1** *Let* ${L}_{k}=\{(x,y,z)|V(x,y,z)=k,(x,y,z)\in {R}_{+}^{3}\}$, *and put* ${k}_{c}=V(\tilde{x},\tilde{x},\tilde{x})$. *Then set* ${L}_{k}$ *is diffeomorphic to a sphere* ${\mathrm{\Omega}}^{2}$ *for* $k>{k}_{c}$.

By using Lemma 2.1, we have that ${\tilde{L}}_{k}$ is diffeomorphic to a sphere for $0<\gamma <1$. This means that ${I}_{k}$ is also diffeomorphic to a sphere for $k>{k}_{\gamma}$. So, we proved the corollary as follows.

**Proposition 2.2** *For* $k>{k}_{\gamma}$ *the level surface* ${I}_{k}$ *is diffeomorphic to a sphere*.

*et al.*[6] introduced an important curve ℒ that is an invariant, where

*F*, we define a curve $\tilde{\mathcal{L}}$ given by

A straightforward calculation shows that ${l}_{\gamma}$ is the unique solution of ${h}^{\prime}(s)=0$ and that ${lim}_{(1-\gamma )s\to {1}^{+}}h(s)={lim}_{s\to +\mathrm{\infty}}h(s)=+\mathrm{\infty}$. Hence, ${I}_{k}\cap \tilde{\mathcal{L}}$ consists of two points for every $k>{k}_{\gamma}$. It is easy to show that the set ${D}_{k}=\{(x,y,z)|I(x,y,z)\le k,(x,y,z)\in {R}_{+}^{3}\}$ contains the unique critical point $({l}_{\gamma},{l}_{\gamma},{l}_{\gamma})$ for $k>{k}_{\gamma}$. Moreover, ${D}_{k}$ is compact with $k>{k}_{\gamma}$.

**Remark 2.3** According to the transformation (5), we have $I(x,y,z)={V}_{a(1-\gamma )}(u,v,w)$. In [13], the authors obtained a classical property of the function ${V}_{a(1-\gamma )}(u,v,w)$: it tends to +∞ at the infinity point of the locally compact space $\{(u,v,w)\mid u>0,v>0,w>0\}$. Then the property of $I(x,y,z)$ is the same as this one of ${V}_{a(1-\gamma )}(u,v,w)$. This is a direct proof about the property of the set ${D}_{k}$ for $k>{k}_{\gamma}$. Furthermore, the sets ${D}_{k}$ for $k>{k}_{\gamma}$ form a system of compact neighborhood of the fixed point $({l}_{\gamma},{l}_{\gamma},{l}_{\gamma})$.

In order to simplify the proof of our main result, we need the following result.

**Proposition 2.4**

*Set*$S=\{(x,y,z)|(1-\gamma ){x}^{2}=a+y+z,(x,y,z)\in {R}_{+}^{3}\}$

*and*$\mathrm{\Gamma}=\{(x,y,z)|(1-\gamma ){x}^{2}=a+y+z,(1-\gamma ){y}^{2}=a+x+z,(x,y,z)\in {R}_{+}^{3}\}$.

*If*$0<\gamma <1$,

*then*

- (1)
$I(F(x,y,z))<I(x,y,z)$

*for all points*$(x,y,z)\in {R}_{+}^{3}\setminus S$,*and*$I(F(x,y,z))=I(x,y,z)$*for*$(x,y,z)\in S$. - (2)
$I({F}^{2}(x,y,z))<I(x,y,z)$

*for every point*$(x,y,z)\in S\setminus \mathrm{\Gamma}$,*and*$I({F}^{2}(x,y,z))=I(x,y,z)$*for*$(x,y,z)\in \mathrm{\Gamma}$. - (3)
$I({F}^{3}(x,y,z))<I(x,y,z)$

*for all*$(x,y,z)\in \mathrm{\Gamma}$*except the fixed point*$({l}_{\gamma},{l}_{\gamma},{l}_{\gamma})$.

*Proof*Note that

*γ*satisfies the relation $0<\gamma <1$. Using (3) and (4), we have

Therefore, from the above expression, we obtained the assertion (1).

*S*, then the equality $(1-\gamma ){x}^{2}=a+y+z$ is fulfilled. From this, we conclude that

If $0<\gamma <1$, then the expression $I(x,y,z)-I(F(F(x,y,z)))$ equals zero if and only if $a+x+z=(1-\gamma ){y}^{2}$. Note that the coordinates of the point $(x,y,z)$ satisfy the equality $a+y+z=(1-\gamma ){x}^{2}$. From these equations, we proved the assertion (2).

Obviously, the expression $I(x,y,z)-I({F}^{3}(x,y,z))$ equals zero if and only if $a+x+y=(1-\gamma ){z}^{2}$ for $0<\gamma <1$. Note that the point $(x,y,z)$ belongs to Γ. It is easy to see that there exists a unique positive solution $(x,y,z)=({l}_{\gamma},{l}_{\gamma},{l}_{\gamma})$ such that $I(x,y,z)=I({F}^{3}(x,y,z))$ (see first from the three equations $(1-\gamma ){x}^{2}=a+y+z$, $(1-\gamma ){y}^{2}=a+x+z$ and $(1-\gamma ){z}^{2}=a+x+y$ that $x=y=z$). Then we proved the assertion (3). □

**Remark 2.5** It is clear that the inequality $L(F(x,y,z))\le L(x,y,z)$ holds for $0<\gamma <1$. Then $L(x,y,z)$ is called a Lyapunov function [16]. In summary, we deduce that $I({F}^{3}(x,y,z))<I(x,y,z)$ holds for $(x,y,z)\in {R}_{+}^{3}$ except the unique fixed point $({l}_{\gamma},{l}_{\gamma},{l}_{\gamma})$. Let the map *F* in the case $\gamma =0$ be denoted by $F{|}_{\gamma =0}$. The level surface $V(x,y,z)=k$ is an invariant [15, 16] for $F{|}_{\gamma =0}$. We also call the function $V(x,y,z)$ the first integral of the map $F{|}_{\gamma =0}$. Using the function $V(x,y,z)$, we obtain the Lyapunov function $I(x,y,z)$ for equation (2).

## 3 Proof of Theorem 1.1

In this section we prove the main result.

*Proof* Now, we show that the positive equilibrium point ${l}_{\gamma}$ of equation (2) is globally asymptotically stable for $0<\gamma <1$.

*D*is closed, bounded and compact. Since there must be a subsequence ${\{{F}^{3{n}_{l}}({x}_{0},{x}_{1},{x}_{2})\}}_{l=0}^{\mathrm{\infty}}$ such that ${lim}_{l\to \mathrm{\infty}}{F}^{3{n}_{l}}({x}_{0},{x}_{1},{x}_{2})=(\tilde{x},\tilde{y},\tilde{z})$, where $(\tilde{x},\tilde{y},\tilde{z})\in D$. Next, considering another subsequence ${\{{F}^{3}({F}^{3{n}_{l}}({x}_{0},{x}_{1},{x}_{2}))\}}_{l=0}^{\mathrm{\infty}}$ of ${\{{F}^{3n}({x}_{0},{x}_{1},{x}_{2})\}}_{n=0}^{\mathrm{\infty}}$, we obtain that ${lim}_{l\to \mathrm{\infty}}I({F}^{3}({F}^{3{n}_{l}}({x}_{0},{x}_{1},{x}_{2})))=k$. Because $I(x,y,z)$ and $F(x,y,z)$ are continuous functions in ${R}_{+}^{3}$, it is not difficult to deduce that $I(\tilde{x},\tilde{y},\tilde{z})={lim}_{l\to \mathrm{\infty}}I({F}^{3{n}_{l}}({x}_{0},{x}_{1},{x}_{2}))=k$ and

This is a contradiction. So, we have ${lim}_{n\to \mathrm{\infty}}I({F}^{3n}(x,y,z))={k}_{\gamma}$ for all $(x,y,z)\in {R}_{+}^{3}$. Therefore, we have the limits ${lim}_{n\to \mathrm{\infty}}{x}_{3n}={l}_{\gamma}$, ${lim}_{n\to \mathrm{\infty}}{x}_{3n+1}={l}_{\gamma}$ and ${lim}_{n\to \mathrm{\infty}}{x}_{3n+2}={l}_{\gamma}$. That is, the positive equilibrium point ${l}_{\gamma}$ of equation (2) is globally asymptotically stable when $0<\gamma <1$.

*a*and

*γ*are also positive in equation (2). For $\gamma \ge 1$, we have

*m*such that ${lim}_{n\to \mathrm{\infty}}{x}_{3n+m}=b$ for $0\le m\le 2$. We write $3n+m$ in place of

*n*in equation (6). This gives the recurrence relation

Take the limits on both sides of the above inequality and obtain $b\ge \frac{a}{b}+b$. This is a contradiction. Hence, the sequence ${\{{x}_{n}\}}_{n=0}^{\mathrm{\infty}}$ converges to +∞ as $n\to \mathrm{\infty}$ if $\gamma \ge 1$. The proof is completed. □

**Remark 3.1** Indeed, using the extension of the quasi-Lyapounov method, we can obtain the global asymptotic behavior of the second-order Lyness equation ${x}_{n+2}{x}_{n}=a+{x}_{n+1}+\gamma {x}_{n}^{2}$ for certain domains of values for parameters.

## Declarations

### Acknowledgements

We thank the anonymous referees for their careful reading of the manuscript and their helpful comments. This work was supported by the National Natural Science Foundation of China (No: 11101283, No: 11101370), Innovation Program of Shanghai Municipal Education Commission (No: 12YZ173), SRF for Rocs, SEM (No: 114329A4C11604), and the Group of Accounting and Governance Disciplines (No: 10kq03).

## Authors’ Affiliations

## References

- Lyness RC: Note 1581.
*Math. Gaz.*1942, 26: 62.View ArticleGoogle Scholar - Lyness RC: Note 1847.
*Math. Gaz.*1945, 29: 231. 10.2307/3609268View ArticleGoogle Scholar - Lyness RC: Note 2952.
*Math. Gaz.*1961, 45: 201.View ArticleGoogle Scholar - Zeeman, EC: Geometric unfolding of a difference equation. Unpublished paper. Hertford College, Oxford (1996)Google Scholar
- Bastien G, Rogalski M: Global behaviour of the solutions of Lyness’ difference equations ${u}_{n+2}{u}_{n}={u}_{n+1}+a$.
*J. Differ. Equ. Appl.*2004, 10(11):977-1003. 10.1080/10236190410001728104MathSciNetView ArticleGoogle Scholar *Cima, A, Gasull, A, Mañosa, V: Dynamics of the third order Lyness’ difference equation. arXiv:math.DS/0612407*. e-printatarXiv.org- Beukers F, Cushman R: Zeeman’s monotonicity conjecture.
*J. Differ. Equ.*1998, 143: 191-200. 10.1006/jdeq.1997.3359MathSciNetView ArticleGoogle Scholar - Li XY: A counterexample to Ladas’ conjecture for Lyness equation.
*Chin. Sci. Bull.*1998, 43(16):1788. (in Chinese)Google Scholar - Li XY, Xiao GF: Periodicity and strict oscillation for generalized Lyness equations.
*Appl. Math. Mech.*2000, 21(4):455-460. 10.1007/BF02463768MathSciNetView ArticleGoogle Scholar - Cima A, Gasull A, Mañosa V: On 2- and 3-periodic Lyness difference equations.
*J. Differ. Equ. Appl.*2012, 18(5):849-864. 10.1080/10236198.2010.524212View ArticleGoogle Scholar - Feuer J: Periodic solutions of the Lyness max equation.
*J. Math. Anal. Appl.*2003, 288(2):147-160.MathSciNetView ArticleGoogle Scholar - Bastien G, Rogalski M: Results and problems about solutions of perturbed Lyness’ type order
*k*difference equations in ${R}_{\ast}^{+}{u}_{n+k}({u}_{n}+\lambda )=f({u}_{n+k-1},\dots ,{u}_{n+1})$ , with examples, and test of the efficiency of a quasi-Lyapunov function method.*J. Differ. Equ. Appl.*2012. 10.1080/10236198.2012.748758Google Scholar - Bastien G, Rogalski M: Results and conjectures about order q Lyness’ difference equation ${u}_{n+k}{u}_{n}=a+{u}_{n+q-1}+\cdots +{u}_{n+1}$ in ${R}_{\ast}^{+}$, with a particular study of the case $q=3$.
*Adv. Differ. Equ.*2009., 2009: Article ID 134749Google Scholar - Bastien G, Rogalski M: Global behaviour of solutions of cyclic systems of
*q*order 2 or 3 generalized Lyness’ difference equations and of other more general equations of higher order.*J. Differ. Equ. Appl.*2011, 17(11):1651-1672. 10.1080/10236191003730555MathSciNetView ArticleGoogle Scholar - Gao M, Kato Y: Some invariants for
*k*th-order Lyness equation.*Appl. Math. Lett.*2004, 17(10):1183-1189. 10.1016/j.aml.2003.07.011MathSciNetView ArticleGoogle Scholar - Kulenović MRS: Invariants and related Liapunov functions for difference equations.
*Appl. Math. Lett.*2000, 13(7):1-8. 10.1016/S0893-9659(00)00068-9MathSciNetView ArticleGoogle Scholar - Kocic VL, Ladas G, Tzanetopoulos G, Thomas E: On the stability of Lyness’ equation.
*Dyn. Contin. Discrete Impuls. Syst.*1995, 1: 245-254.MathSciNetGoogle Scholar - Merino O: Global attractivity of the equilibrium of a difference equation: an elementary proof assisted by computer algebra system.
*J. Differ. Equ. Appl.*2011, 17(1):34-41.MathSciNetView ArticleGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.