Neimark-Sacker Bifurcation of a Third Order Difference Equation

Abstract

In this paper, we investigate the bifurcation of a third order rational difference equation. Firstly, we show that the equation undergoes a Neimark-Sacker bifurcation when the parameter reaches a critical value. Then, we consider the direction of the Neimark-Sacker bifurcation. Finally, we give some numerical simulations of our results.

Keywords

• Difference equation,Stability,Equilibrium,Neimark-Sacker Bifurcation,Stability

References

1. [1] E. Camouzis, Global analysis of solutions of $x_{n+1}=\frac{\beta x_n+\delta x_{n-2}}{A+Bx_n+Cx_{n-1}}$, J. Math Anal. Appl., 316 (2005), 616-627.
2. [2] Z. He, J. Qiu, Neimark-Sacker bifurcation of a third order rational difference equation, J. Differ. Equ. Appl.,19 (2013), 1513-1522.
3. [3] E. Camouzis, G. Ladas, Dynamics of Third-Order Rational Difference Equations with Open Problems and Conjectures, Chapman & Hall/CRC, New York, 2002.
4. [4] A. D. Polyanin, A. I. Chernoutsan, A. Concise, A Concise Handbook of Mathematics, Physics and Engineering Science, CRC Press, New York, 2011.
5. [5] Y. Kuznetsov, Elements of Applied Bifurcation Theory, 2nd edition, Springer, New York, 2003.