- Open Access
Dynamics of a system of rational third-order difference equation
© Zhang et al.; licensee Springer 2012
- Received: 5 April 2012
- Accepted: 27 July 2012
- Published: 6 August 2012
In this paper, we study the dynamical behavior of positive solution for a system of a rational third-order difference equation
where , ; .
- difference equation
- local behavior
where p, q are positive integers.
where , and the initial conditions and are arbitrary nonnegative numbers.
where , and the initial conditions ; .
has a unique solution . A point is called an equilibrium point of (4) if , , i.e., for all .
is said to be stable relative to if for every , there exists such that for any initial conditions (), with , , implies , .
is called an attractor relative to if for all (), , .
is called asymptotically stable relative to if it is stable and an attractor.
Unstable if it is not stable.
Theorem 2.1 ()
Assume that, , is a system of difference equations andis the equilibrium point of this system, i.e., . If all eigenvalues of the Jacobian matrix, evaluated atlie inside the open unit disk, thenis locally asymptotically stable. If one of them has a modulus greater than one, thenis unstable.
Theorem 2.2 ()
Consider the system (3), if , , system (3) has equilibrium and . In addition, if , , then system (3) has an equilibrium point , and if , , then system (3) has an equilibrium point . Finally, if and , is the unique equilibrium point.
This completes our inductive proof. □
Corollary 3.1 If, , then by Theorem 3.1 converges exponentially to the equilibrium point.
Then the equilibriumis locally asymptotically stable.
This shows that all the roots of the characteristic equation lie inside unit disk. So the unique equilibrium is locally asymptotically stable. □
the equilibrium is locally unstable,
the positive equilibrium is locally unstable.
- (ii)We can easily obtain that the linearized system of (3) about the equilibrium is(11)
It is clear that not all of , . Therefore, by Theorem 2.2, the positive equilibrium is locally unstable. □
Therefore, (14) is true. This completes the proof of (i). Similarly, we can obtain the proof of (ii). Hence, it is omitted. □
Since the system of the difference equation (3) is the extension of the third-order equation in  in the six-dimensional space. In this paper, we investigated the local behavior of solutions of the system of difference equation (3) using linearization. But as we saw linearization do not say anything about the global behavior and fails when the eigenvalues have modulus one. Some powerful tools such as semiconjugacy and weak contraction in  cannot be used to analyze global behavior of system (3). The global behavior of the system (3) will be next our aim to study.
The authors would like to thank the editor and anonymous reviewers for their helpful comments and valuable suggestions, which have greatly improved the quality of this paper. This work is partially supported by the Scientific Research Foundation of Guizhou Provincial Science and Technology Department (J2096).
- Marwan A: Dynamics of a rational difference equation. Appl. Math. Comput. 2006, 176: 768–774. 10.1016/j.amc.2005.10.024MathSciNetView ArticleGoogle Scholar
- Agop M, Rusu I: El Naschie’s self-organization of the patterns in a plasma discharge: experimental and theoretical results. Chaos Solitons Fractals 2007, 34: 172–186. 10.1016/j.chaos.2006.04.017View ArticleGoogle Scholar
- Cinar C:On the positive solutions of the difference equation . Appl. Math. Comput. 2004, 150: 21–24. 10.1016/S0096-3003(03)00194-2MathSciNetView ArticleGoogle Scholar
- Shojaei M, Saadati R, Adibi H: Stability and periodic character of a rational third order difference equation. Chaos Solitons Fractals 2009, 39: 1203–1209. 10.1016/j.chaos.2007.06.029MathSciNetView ArticleGoogle Scholar
- Cinar C:On the solutions of the difference equation . Appl. Math. Comput. 2004, 158: 793–797. 10.1016/j.amc.2003.08.139MathSciNetView ArticleGoogle Scholar
- Cinar C:On the difference equation . Appl. Math. Comput. 2004, 158: 813–816. 10.1016/j.amc.2003.08.122MathSciNetView ArticleGoogle Scholar
- Cinar C:On the positive solutions of the difference equation . Appl. Math. Comput. 2004, 158: 809–812. 10.1016/j.amc.2003.08.140MathSciNetView ArticleGoogle Scholar
- Papaschinopoulos G, Schinas CJ: On a system of two nonlinear difference equations. J. Math. Anal. Appl. 1998, 219: 415–426. 10.1006/jmaa.1997.5829MathSciNetView ArticleGoogle Scholar
- Clark D, Kulenovic MRS: A coupled system of rational difference equations. Comput. Math. Appl. 2002, 43: 849–867. 10.1016/S0898-1221(01)00326-1MathSciNetView ArticleGoogle Scholar
- Clark D, Kulenovic MRS, Selgrade JF: Global asymptotic behavior of a two-dimensional difference equation modelling competition. Nonlinear Anal. 2003, 52: 1765–1776. 10.1016/S0362-546X(02)00294-8MathSciNetView ArticleGoogle Scholar
- Sedaghat H: Nonlinear Difference Equations: Theory with Applications to Social Science Models. Kluwer Academic, Dordrecht; 2003.View ArticleGoogle Scholar
- Kocic, VL, Ladas, G: Global Behavior of Nonlinear Difference Equations of Higher Order with Applications. Kluwer Academic, Dordrecht (1993)View ArticleGoogle Scholar
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