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

R E S E A R C H

Open Access

A general method for studying quadratic

perturbations of the third-order Lyness

difference equation

Guifeng Deng

1*

_{, Qiuying Lu}

2

_{and Nianzu Liu}

1

*_{Correspondence:} [email protected] 1_{School of Mathematics and}

Information Science, Shanghai Lixin University of Commerce, Shanghai, 201620, P.R. China

Full list of author information is available at the end of the article

Abstract

This paper studies the difference equationxn+3xn=a+xn+1+xn+2+

γ

x2n, whereaand

γ

are arbitrary positive real numbers and the initial valuesx0,x1,x2> 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

Keywords: difference equation; quadratic perturbations; bifurcation point; first integral; Lyapunov function; global asymptotic stability

1 Introduction

Lyness [–] discovered that the solutions of the second-order difference equation xn+xn=xn++ais -periodic fora=  and positive initial conditions while he was working on a problem in Number Theory. The third-order Lyness equation is

xn+xn=a+xn++xn+, a> ,x,x,x> . ()

According to Lyness [–], Todd has discovered every solution of equation () is -periodic ifa= . So, this equation is also known as Todd’s equation. First Zeeman [] and, after and independently, Bastienet al.[] gave a complete description of global dynamics of the second-order Lyness equation witha>  by the interpretation of the iteration of the map Finduced by this recurrence on the Lyness’ cubic which passes through the initial points (x,x). Cimaet al.[] applied an approach different to the ones in Refs. [, ] to study equation (). That is, their main tool is the study of an ordinary differential equation associ-ated to equation (). They proved that for eacha=  the periods of the sequences generated by equation () can be almost all natural numbers, depending on the initial points (x,x). In recent years various generalized Lyness difference equations, including the Lyness dif-ference equation with variable coefficients, the orderkLyness difference equation and the perturbed Lyness-type orderkdifference equation, have been extensively studied [–]. It is well known that there are no convergent nontrivial solutions for the Lyness difference

equation. In [], Li found the convergent solutions of the Lyness difference equation with variable coefficients

xn+=

an+xn+ xn

, x,x> ,

where{an}is a monotonic non-increasing positive sequence, which demonstrates the

es-sential difference between Lyness difference equation with constant coefficients and Ly-ness difference equation with variable coefficients.

In this paper we study the global asymptotic behavior of all solutions of the perturbed third-order Lyness difference equation

xn+=

a+xn++xn++γxn

xn

, n= , , . . . , ()

wherea,γ ∈(, +∞) and the initial valuesx,x,x∈(, +∞). It is clear that the quadratic termx

nis a small perturbation for small positive numberγ. The invariant curve of

dif-ference equations often plays a critical role in studying the stability behavior of their so-lutions; see [, , ]. Kocicet al.[] 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 [, , ]. Here, using an extension of the method intro-duced in [], which itself generalizes an idea of Merino [], 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 . ()If <γ< ,then the positive equilibrium point lγ of equation()is

glob-ally asymptoticglob-ally stable,where

lγ =

 + +a( –γ)  –γ .

()Ifγ ≥,then the sequence{xn}∞n=generated by equation()converges to+∞.

Cimaet al.[] proved that for a givena>  there exist periodic sequences{xn}∞n= gen-erated by equation () which have almost all long periods and that for a full measure set of initial conditions the sequences{xn}∞n=are dense in either one or two disjoint bounded intervals ofR. In summary, the sequences{xn}∞n=are periodic or strictly oscillatory, and the equilibrium pointlis locally stable, withl=  +

√

 +a. That is, for every> , there is aδ>  so that, for any positive initial valuesx,xandxwith|xi–l|<δfori= , , , one has|xn–l|<for alln≥. According to Theorem ., the sequences{xn}∞n=of so-lutions of equation () are converging tolγ if  <γ < . So, the qualitative nature of the

2 Lyapunov function

In this section we assume that  <γ < . We begin by introducing the nonlinear map

F(x,y,z) =

y,z,a+y+z+γx 

x

. ()

Let{xn}∞n=be the sequence generated by equation () with positive initial conditions, then it is not difficult to check thatF(xn,xn+,xn+) = (xn+,xn+,xn+) for alln≥. Moreover, (lγ,lγ,lγ) is a unique fixed point ofF inR+, where R+={(x,y,z)|x> ,y> ,z> }and

lγ=

+√+a(–γ)

–γ . It is well known [, ] that forγ =  equation () possesses the following

first integral:

V(x,y,z) = (x+ )(y+ )(z+ )(a+x+y+z)

xyz .

In other words,Fmaps the level surfaceV–(k) into itself for every k whenγ = . In order to study the global convergence properties of the recurrence () for  <γ < , we introduce a very important functionI(x,y,z) given by

I(x,y,z) =[( –γ)x+ ][( –γ)y+ ][( –γ)z+ ](a+x+y+z)

( –γ)_xyz . ()

In fact, the functionI(x,y,z) is the same asV(x,y,z) forγ = . Using the functionI, we construct the Lyapunov function L(x,y,z) of equation (). Setkγ =I(lγ,lγ,lγ), then we

show that the Lyapunov function is

L(x,y,z) =I(x,y,z) –I(lγ,lγ,lγ) =I(x,y,z) –kγ.

Let us determine some properties of the functionI(x,y,z). Note that if ≤γ<  , we get the relations

lim

x→+I(x,y,z) =_ylim_→+I(x,y,z) =_zlim_→+I(x,y,z) = +∞,

lim

x_+y_→+∞I(x,y,z) =_x_+zlim_→+∞I(x,y,z) =_y_+zlim_→+∞I(x,y,z) = +∞

and

lim

x_+y_+z_→+∞I(x,y,z) = +∞.

It is easy to see that the functionI(x,y,z) has continuous second partial derivatives inR +. Thus, the possible critical points are the stationary points obtained by settingIx(x,y,z),

Iy(x,y,z) andIz(x,y,z) equal to zero. Taking partial derivatives and setting them equal to 

gives

Ix(x,y,z) =

[( –γ)y+ ][( –γ)z+ ][( –γ)x–a–y–z] ( –γ)_x_yz = ,

Iy(x,y,z) =

Iz(x,y,z) =

[( –γ)x+ ][( –γ)y+ ][( –γ)z_{–a–x–y]} ( –γ)_xyz = .

The unique positive solution to this system of equations is

xc=yc=zc=lγ =

Computing the second-order derivatives ofI, we get

A=∂

LetHmbe themth-order principal minor of the HessianH. Then we obtain

detH=A> , detH= order to study the third-order Lyness equation, Cimaet al.[] proved the following result.

Lemma . Let Lk={(x,y,z)|V(x,y,z) =k, (x,y,z)∈R+},and put kc=V(x,˜ x,˜ x).˜ Then set

Lkis diffeomorphic to a spherefor k>kc.

To describe the level surfaceI(x,y,z) =kfork>kγ, we apply the invertible linear

trans-formation of variables

which induces an isomorphism between the set Ik and the level surfaceLk={(u,v,w)|

Va(–γ)(u,v,w) =k, (u,v,w)∈R+}, where

Va(–γ)(u,v,w) =

(u+ )(v+ )(w+ )[a( –γ) +u+v+w]

uvw .

By using Lemma ., we have thatLkis diffeomorphic to a sphere for  <γ < . This means

thatIkis also diffeomorphic to a sphere fork>kγ. So, we proved the corollary as follows.

Proposition . For k>kγ the level surface Ikis diffeomorphic to a sphere.

It is easy to see that the setIkis compact fork>kγ. Moreover, Cimaet al.[] introduced

an important curveLthat is an invariant, where

L=

t,t+a t– ,t

t> 

.

Here, according to the mapF, we define a curveLgiven by

L:=

s, s+a ( –γ)s– ,s

s> 

 –γ

⊂R₊.

It is easy to check thatLcontains the unique fixed point (lγ,lγ,lγ). Let

h(s) =I|_L=I

s, s+a ( –γ)s– ,s

=[( –γ)s+ ]

_{[( –γ)(s+a) – ]}

( –γ)_{s(s+a)[( –γ)s– ]} , s>   –γ.

A straightforward calculation shows that lγ is the unique solution of h(s) =  and that

lim(–γ)s→+h(s) =lim_s→+∞h(s) = +∞. Hence,I_k∩Lconsists of two points for everyk>k_γ.

It is easy to show that the setDk={(x,y,z)|I(x,y,z)≤k, (x,y,z)∈R+}contains the unique critical point (lγ,lγ,lγ) fork>kγ. Moreover,Dkis compact withk>kγ.

Remark . According to the transformation (), we haveI(x,y,z) =Va(–γ)(u,v,w). In [], the authors obtained a classical property of the functionVa(–γ)(u,v,w): it tends to +∞at the infinity point of the locally compact space{(u,v,w)|u> ,v> ,w> }. Then the property of I(x,y,z) is the same as this one ofVa(–γ)(u,v,w). This is a direct proof about the property of the setDk fork>kγ. Furthermore, the setsDk fork>kγ form a

system of compact neighborhood of the fixed point (lγ,lγ,lγ).

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

Proposition . Set S={(x,y,z)|( –γ)x=a+y+z, (x,y,z)∈R₊}and={(x,y,z)|( –

γ)x_{=a+y+z, ( –γ)y}_{=a+x+z, (x,y,z)∈R}

+}.If <γ < ,then () I(F(x,y,z)) <I(x,y,z)for all points(x,y,z)∈R

+\S,andI(F(x,y,z)) =I(x,y,z)for (x,y,z)∈S.

() I(F_{(x,y,z)) <I(x,y,z)for every point(x,y,z)∈S\,andI(F}_{(x,y,z)) =I(x,y,z)for} (x,y,z)∈.

() I(F_{(x,y,z)) <I(x,y,z)for all(x,y,z)∈except the fixed point(l}

Proof Note thatγ satisfies the relation  <γ < . Using () and (), we have

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

Suppose that (x,y,z)∈, then it is easy to check that

F(x,y,z) =F(y,z,x) =F

z,x,a+z+x+γy 

y

=F(z,x,y) =

x,y,a+x+y+γz 

z

.

Therefore,

L(x,y,z) –LF(x,y,z)

=I(x,y,z) –I

x,y,a+x+y+γz 

z

=[( –γ)x+ ][( –γ)y+ ] ( –γ)_xy

×

[( –γ)z+ ](a+x+y+z)

z –

[( –γ)a+x+y_z+γz+ ](a+x+y+a+x+y_z+γz)

a+x+y+γz z

= z

a+x+y+γz

[( –γ)x+ ][( –γ)y+ ]γ

( –γ)_xy

( –γ)z–a+x+y z



≥.

Obviously, the expressionI(x,y,z)–I(F_{(x,y,z)) equals zero if and only ifa+x+y= (–γ)z} for  <γ < . Note that the point (x,y,z) belongs to. It is easy to see that there exists a unique positive solution (x,y,z) = (lγ,lγ,lγ) such thatI(x,y,z) =I(F(x,y,z)) (see first from

the three equations ( –γ)x_{=a+y+z, ( –γ)y}_{=a+x+zand ( –γ)z}_=a+x+ythat

x=y=z). Then we proved the assertion ().

Remark . It is clear that the inequalityL(F(x,y,z))≤L(x,y,z) holds for  <γ < . Then L(x,y,z) is called a Lyapunov function []. In summary, we deduce thatI(F_{(x,y,z)) <} I(x,y,z) holds for (x,y,z)∈R

+except the unique fixed point (lγ,lγ,lγ). Let the mapF in

the caseγ =  be denoted byF|γ=. The level surfaceV(x,y,z) =kis an invariant [, ] forF|γ=. We also call the functionV(x,y,z) the first integral of the mapF|γ=. Using the functionV(x,y,z), we obtain the Lyapunov functionI(x,y,z) for equation ().

3 Proof of Theorem 1.1

In this section we prove the main result.

Proof Now, we show that the positive equilibrium pointlγ of equation () is globally

asymptotically stable for  <γ< .

Recall thatkγ =I(lγ,lγ,lγ) is a global minimum value for the functionI(x,y,z) in ().

By Proposition ., the level surfaceI(x,y,z) =kis diffeomorphic to a sphere for every k>kγ. Moreover, the level surfaceI(x,y,z) =kcontinuously shrinks to a point (lγ,lγ,lγ)

ask→k+

γ. Note that the equalityI(xn,xn+,xn+) =I(Fn(x,x,x)) is fulfilled for all pos-itive initial conditions, where the sequence{xn}∞n=is the solution of equation (). It follows from Proposition . that for (x,y,z)= (lγ,lγ,lγ) the monotone sequence{I(Fn(x,y,z))}∞n= converges to kγ asn→ ∞. Otherwise, there must be a real numberk>kγ such that

limn→∞I(Fn(x,x,x)) =kfor some (x,x,x)∈R+. It is easy to see that the sequence

{Fn_(x

{Fnl_(x

,x,x)}l∞=such thatliml→∞Fnl_(x

,x,x) = (x,y,z), where (x,y,z)∈D. Next, con-sidering another subsequence{F_(Fnl_(x

,x,x))}∞_l=of{Fn(x,x,x)}∞n=, we obtain that

lim_l→∞I(F_(Fnl_(x

,x,x))) =k. BecauseI(x,y,z) andF(x,y,z) are continuous functions inR

+, it is not difficult to deduce thatI(x,y,z) =liml→∞I(Fnl(x,x,x)) =kand

k=lim l→∞I

FFnl_(x ,x,x)

=I

F lim l→∞

Fnl_(x ,x,x)

=IF(x,y,z)<I(x,y,z).

This is a contradiction. So, we havelimn→∞I(Fn(x,y,z)) =kγ for all (x,y,z)∈R+. There-fore, we have the limits limn→∞xn=lγ,limn→∞xn+=lγ andlimn→∞xn+=lγ. That

is, the positive equilibrium pointlγ of equation () is globally asymptotically stable when

 <γ< .

Finally, we give the proof of the assertion () of Theorem .. Note that the initial values x,xandxare positive real numbers, and the parametersaandγ are also positive in equation (). Forγ≥, we have

xn+=

a+xn++xn++γxn

xn

> a xn

+xn>xn. ()

So, the subsequences{xn}∞n=,{xn+}∞n=and{xn+}∞n=of solution{xn}∞n=of equation () are monotonically increasing. Then the sequence{xn}∞n= converges to +∞asn→ ∞. Otherwise, there must be an integermsuch that limn→∞xn+m =b for ≤m≤. We

write n+min place ofnin equation (). This gives the recurrence relation

xn+m+> a xn+m

+xn+m.

Take the limits on both sides of the above inequality and obtainb≥a_b+b. This is a con-tradiction. Hence, the sequence{xn}∞n=converges to +∞asn→ ∞ifγ≥. The proof is

completed.

Remark . Indeed, using the extension of the quasi-Lyapounov method, we can obtain the global asymptotic behavior of the second-order Lyness equationxn+xn=a+xn++γxn

for certain domains of values for parameters.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

GD carried out the study of the third-order difference equation and drafted the manuscript. QL conceived of the study and helped to draft the manuscript. NL was involved in revising it critically for important intellectual content. All authors have read and approved the final manuscript.

Author details

1_{School of Mathematics and Information Science, Shanghai Lixin University of Commerce, Shanghai, 201620, P.R. China.} 2_{Department of Mathematics, Zhejiang Sci-Tech University, Hangzhou, Zhejiang Province 310018, P.R. China.}

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).

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doi:10.1186/1687-1847-2013-193