A general method for studying quadratic perturbations of the third-order Lyness difference equation
© Deng et al.; licensee Springer 2013
Received: 28 February 2013
Accepted: 18 June 2013
Published: 1 July 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.
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.
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.
(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
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 .
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.
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.
for all points , and for .
for every point , and for .
for all except the fixed point .
Therefore, from the above expression, we obtained the assertion (1).
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).
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 .
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 .
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.
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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