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 , ; .
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 ()
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.
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. □
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.
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).
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