# Naimark-Sacker bifurcation of second order rational difference equation with quadratic terms

Volume 10, Issue 7, pp 3477--3489 Publication Date: July 22, 2017       Article History
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### Authors

M. R. S. Kulenovic - Department of Mathematics, University of Rhode Island, Kingston, RI 02881, USA.
S. Moranjkic - Department of Mathematics, University of Tuzla, 75350 Tuzla, Bosnia and Herzegovina.
Z. Nurkanovic - Department of Mathematics, University of Tuzla, 75350 Tuzla, Bosnia and Herzegovina.

### Abstract

We investigate the global asymptotic stability and Naimark-Sacker bifurcation of the difference equation $x_{n+1} =\frac{F}{bx_nx_{n-1} + cx^2_{n-1} + f} , n = 0, 1, ... ,$ with non-negative parameters and nonnegative initial conditions $x_{-1}, x_0$ such that $bx_0x_{-1} + cx^2_{-1} + f > 0$. By using fixed point theorem for monotone maps we find the region of parameters where the unique equilibrium is globally asymptotically stable.

### Keywords

• Attractivity
• bifurcation
• difference equation
• invariant
• Naimark-Sacker bifurcation
• periodic solution.

•  39A10
•  39A28
•  39A30

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