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Global Dynamics and Bifurcations of Certain Second Order Rational Difference Equation with Quadratic Terms

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Abstract

We investigate global dynamics of the equation

$$\begin{aligned} x_{n+1}=\frac{x_{n-1}}{ax_{n}^{2}+ex_{n-1}+f},\quad n=0,1,2,\ldots , \end{aligned}$$

where the parameters a, e and f are nonnegative numbers with condition \(a+e+f>0\) and the initial conditions \(x_{-1},x_{0}\) are arbitrary nonnegative numbers such that \(x_{-1}+x_{0}>0\). The global dynamics of this equation consists of three bifurcations, two exchange of stability bifurcations and one global period doubling bifurcation.

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Acknowledgments

The authors are grateful to the anonymous referee for a number of helpful and constructive suggestions which improve the presentation of results.

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Authors and Affiliations

  1. Department of Mathematics, University of Tuzla, 75000, Tuzla, Bosnia and Herzegovina

    Sabina Jašarević Hrustić & M. Nurkanović

  2. Department of Mathematics, University of Rhode Island, Kingston, RI, 02881-0816, USA

    M. R. S. Kulenović

Authors
  1. Sabina Jašarević Hrustić
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  2. M. R. S. Kulenović
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  3. M. Nurkanović
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Correspondence to M. R. S. Kulenović.

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Hrustić, S.J., Kulenović, M.R.S. & Nurkanović, M. Global Dynamics and Bifurcations of Certain Second Order Rational Difference Equation with Quadratic Terms. Qual. Theory Dyn. Syst. 15, 283–307 (2016). https://doi.org/10.1007/s12346-015-0148-x

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  • DOI: https://doi.org/10.1007/s12346-015-0148-x

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