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A general method for studying quadratic perturbations of the third-order Lyness difference equation
Advances in Difference Equations volume 2013, Article number: 193 (2013)
This paper studies the difference equation , where a and γ are arbitrary positive real numbers and the initial values . It is known that for 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 the sequences generated by the perturbed third-order Lyness equation are globally asymptotically stable. Moreover, we show that if all solutions of it converge to +∞. Therefore, the values 0 and 1 are two bifurcation points for the equation containing the parameter γ.
Lyness [1–3] discovered that the solutions of the second-order difference equation is 5-periodic for and positive initial conditions while he was working on a problem in Number Theory. The third-order Lyness equation is
According to Lyness [1–3], Todd has discovered every solution of equation (1) is 8-periodic if . So, this equation is also known as Todd’s equation. First Zeeman  and, after and independently, Bastien et al.  gave a complete description of global dynamics of the second-order Lyness equation with by the interpretation of the iteration of the map F induced by this recurrence on the Lyness’ cubic which passes through the initial points . Cima et al.  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 the periods of the sequences generated by equation (1) can be almost all natural numbers, depending on the initial points . 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 , Li found the convergent solutions of the Lyness difference equation with variable coefficients
where 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.
In this paper we study the global asymptotic behavior of all solutions of the perturbed third-order Lyness difference equation
where and the initial values . It is clear that the quadratic term 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.  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 , 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 1.1 (1) If , then the positive equilibrium point of equation (2) is globally asymptotically stable, where
(2) If , then the sequence generated by equation (2) converges to +∞.
Cima et al.  proved that for a given there exist periodic sequences generated by equation (1) which have almost all long periods and that for a full measure set of initial conditions the sequences are dense in either one or two disjoint bounded intervals of R. In summary, the sequences are periodic or strictly oscillatory, and the equilibrium point is locally stable, with . That is, for every , there is a so that, for any positive initial values , and with for , one has for all . According to Theorem 1.1, the sequences of solutions of equation (2) are converging to if . So, the qualitative nature of the solutions of equation (2) changes when γ vanishes. Note that the sequences are increasing to +∞ if . So, the behavior of the solutions of equation (2) is completely different from the case . Thus, the values 0 and 1 are two bifurcation points for the difference equation (2) containing the parameter γ.
2 Lyapunov function
In this section we assume that . We begin by introducing the nonlinear map
Let be the sequence generated by equation (2) with positive initial conditions, then it is not difficult to check that for all . Moreover, is a unique fixed point of F in , where and . It is well known [6, 15] that for equation (2) possesses the following first integral:
In other words, F maps the level surface into itself for every k when . In order to study the global convergence properties of the recurrence (2) for , we introduce a very important function given by
In fact, the function is the same as for . Using the function I, we construct the Lyapunov function of equation (2). Set , then we show that the Lyapunov function is
Let us determine some properties of the function . Note that if , we get the relations
It is easy to see that the function has continuous second partial derivatives in . Thus, the possible critical points are the stationary points obtained by setting , and equal to zero. Taking partial derivatives and setting them equal to 0 gives
The unique positive solution to this system of equations is
Next, we check the Hessian H at the stationary point , where
Computing the second-order derivatives of I, we get
Let be the m th-order principal minor of the Hessian H. Then we obtain
Thus, the function attains a strict minimum at in , and it has no other critical point. We obtain
Let be the level surface of I for . It is well known that the function has a minimum at in , where . In order to study the third-order Lyness equation, Cima et al.  proved the following result.
Lemma 2.1 Let , and put . Then set is diffeomorphic to a sphere for .
To describe the level surface for , we apply the invertible linear transformation of variables
which induces an isomorphism between the set and the level surface , where
By using Lemma 2.1, we have that is diffeomorphic to a sphere for . This means that is also diffeomorphic to a sphere for . So, we proved the corollary as follows.
Proposition 2.2 For the level surface is diffeomorphic to a sphere.
It is easy to see that the set is compact for . Moreover, Cima et al.  introduced an important curve ℒ that is an invariant, where
Here, according to the map F, we define a curve given by
It is easy to check that contains the unique fixed point . Let
A straightforward calculation shows that is the unique solution of and that . Hence, consists of two points for every . It is easy to show that the set contains the unique critical point for . Moreover, is compact with .
Remark 2.3 According to the transformation (5), we have . In , the authors obtained a classical property of the function : it tends to +∞ at the infinity point of the locally compact space . Then the property of is the same as this one of . This is a direct proof about the property of the set for . Furthermore, the sets for form a system of compact neighborhood of the fixed point .
In order to simplify the proof of our main result, we need the following result.
Proposition 2.4 Set and . If , then
for all points , and for .
for every point , and for .
for all except the fixed point .
Proof Note that γ satisfies the relation . Using (3) and (4), we have
Therefore, from the above expression, we obtained the assertion (1).
For every point belonging to S, then the equality is fulfilled. From this, we conclude that
If , then the expression equals zero if and only if . Note that the coordinates of the point satisfy the equality . From these equations, we proved the assertion (2).
Suppose that , then it is easy to check that
Obviously, the expression equals zero if and only if for . Note that the point belongs to Γ. It is easy to see that there exists a unique positive solution such that (see first from the three equations , and that ). Then we proved the assertion (3). □
Remark 2.5 It is clear that the inequality holds for . Then is called a Lyapunov function . In summary, we deduce that holds for except the unique fixed point . Let the map F in the case be denoted by . The level surface is an invariant [15, 16] for . We also call the function the first integral of the map . Using the function , we obtain the Lyapunov function 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 of equation (2) is globally asymptotically stable for .
Recall that is a global minimum value for the function in (4). By Proposition 2.2, the level surface is diffeomorphic to a sphere for every . Moreover, the level surface continuously shrinks to a point as . Note that the equality is fulfilled for all positive initial conditions, where the sequence is the solution of equation (2). It follows from Proposition 2.4 that for the monotone sequence converges to as . Otherwise, there must be a real number such that for some . It is easy to see that the sequence belongs to a set . Obviously, the set D is closed, bounded and compact. Since there must be a subsequence such that , where . Next, considering another subsequence of , we obtain that . Because and are continuous functions in , it is not difficult to deduce that and
This is a contradiction. So, we have for all . Therefore, we have the limits , and . That is, the positive equilibrium point of equation (2) is globally asymptotically stable when .
Finally, we give the proof of the assertion (2) of Theorem 1.1. Note that the initial values , and are positive real numbers, and the parameters a and γ are also positive in equation (2). For , we have
So, the subsequences , and of solution of equation (2) are monotonically increasing. Then the sequence converges to +∞ as . Otherwise, there must be an integer m such that for . We write in place of n in equation (6). This gives the recurrence relation
Take the limits on both sides of the above inequality and obtain . This is a contradiction. Hence, the sequence converges to +∞ as if . 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 for certain domains of values for parameters.
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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).
The authors declare that they have no competing interests.
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.