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Dynamics of a system of rational third-order difference equation
Advances in Difference Equations volume 2012, Article number: 136 (2012)
In this paper, we study the dynamical behavior of positive solution for a system of a rational third-order difference equation
where , ; .
Rational difference equations that are the ratio of two polynomials are one of the most important and practical classes of nonlinear difference equations. Marwan Aloqeili  investigated the stability character, semicycle behavior of the solution of the difference equation . These difference equations appear naturally as discrete analogues and as numerical solutions of differential and delay differential equations having applications in biology, ecology, physics, etc.[2, 11]. Also, Cinar  investigated the global behavior of all positive solutions of the rational second-order difference equation
Similarly Shojaei, Saadati, and Adibi  investigated the stability and periodic character of the rational third-order difference equation
where the parameters α, β, γ, and the initial conditions , , are real numbers. Related difference equations readers can refer to the references [5–7].
Papaschinopoulos and Schinas  studied the system of two nonlinear difference equations
where p, q are positive integers.
Clark and Kulenovic [9, 10] investigated the system of rational difference equations
where , and the initial conditions and are arbitrary nonnegative numbers.
Our aim in this paper is to investigate the solutions, stability character, and asymptotic behavior of the system of difference equations
where , and the initial conditions ; .
Let , be some intervals of real number and , be continuously differentiable functions. Then for every initial conditions (), the system of difference equations
has a unique solution . A point is called an equilibrium point of (4) if , , i.e., for all .
Let , be some intervals of real numbers; interval is called invariant for system (4) if, for all ,
Definition 2.1 Assume that be a fixed point of system (4). Then
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 ()
Assume that, , is a system of difference equations andis the equilibrium point of this system, the characteristic polynomial of this system about the equilibrium pointis, with real coefficients and. Then all roots of the polynomiallie inside the open unit diskif and only if
where is the principal minor of order k of the matrix
3 Main results
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.
Theorem 3.1 Letbe positive solution of system (3), then for all,
Proof This assertion is true for . Assume that it is true for , for , we have
This completes our inductive proof. □
Corollary 3.1 If, , then by Theorem 3.1 converges exponentially to the equilibrium point.
Theorem 3.2 If
Then the equilibriumis locally asymptotically stable.
Proof We can easily obtain that the linearized system of (3) about the equilibrium is
The characteristic equation of (8) is
This shows that all the roots of the characteristic equation lie inside unit disk. So the unique equilibrium is locally asymptotically stable. □
Theorem 3.3 If
the equilibrium is locally unstable,
the positive equilibrium is locally unstable.
Proof (i) From (9), we have that all the roots of characteristic equation lie outside unit disk. So the unique equilibrium is locally unstable.
We can easily obtain that the linearized system of (3) about the equilibrium is(11)
in which , . The characteristic equation of (11) is
From (12), we have
It is clear that not all of , . Therefore, by Theorem 2.2, the positive equilibrium is locally unstable. □
Theorem 3.4 Consider system (3), and suppose that (10) holds. Then the following statements are true, for,
Proof (i) Let (), from system (3), we have
We prove by induction that
Suppose that (14) is true for . Then from (3), we have
Therefore, (14) is true. This completes the proof of (i). Similarly, we can obtain the proof of (ii). Hence, it is omitted. □
4 Conclusion and future work
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.
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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).
The authors declare that they have no competing interests.
The authors indicated in parentheses made substantial contributions to the following tasks of research: drafting the manuscript (QH Zhang, LH Yang); participating in the design of the study (JZ Liu); writing and revision of the paper (QH Zhang, LH Yang).
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Zhang, Q., Yang, L. & Liu, J. Dynamics of a system of rational third-order difference equation. Adv Differ Equ 2012, 136 (2012). https://doi.org/10.1186/1687-1847-2012-136
- difference equation
- local behavior