Global Asymptotic Stability and Naimark-Sacker Bifurcation of Certain Mix Monotone Difference Equation

Authors

Mustafa R. S. Kulenović, University of Rhode IslandFollow
S. Moranjkić
M. Nurkanović
Z. Nurkanović

Document Type

Article

Date of Original Version

11-8-2018

Department

Mathematics

Abstract

We investigate the global asymptotic stability of the following second order rational difference equation of the form x_n+1= (Bx_nx_n-1+cx²_n-1),n =0,1,..., where the parameters B, F, b, and c and initial conditions x_-1 and x₀ are positive real numbers. The map associated with this equation is always decreasing in the second variable and can be either increasing or decreasing in the first variable depending on the parametric space. In some cases, we prove that local asymptotic stability of the unique equilibrium point implies global asymptotic stability. Also, we show that considered equation exhibits the Naimark-Sacker bifurcation resulting in the existence of the locally stable periodic solution of unknown period.

Citation/Publisher Attribution

M. R. S. Kulenović, S. Moranjkić, M. Nurkanović, and Z. Nurkanović, “Global Asymptotic Stability and Naimark-Sacker Bifurcation of Certain Mix Monotone Difference Equation,” Discrete Dynamics in Nature and Society, vol. 2018, Article ID 7052935, 22 pages, 2018. https://doi.org/10.1155/2018/7052935

Available at: https://doi.org/10.1155/2018/7052935

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 License.