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

Volume 10, Issue 7, pp 3477--3489

Publication Date: 2017-07-22

http://dx.doi.org/10.22436/jnsa.010.07.11

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.

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