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Stability and Neimark–Sacker Bifurcation of Certain Mixed Monotone Rational Second-Order Difference Equation

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Abstract

This paper investigates the local and global character of the unique positive equilibrium of certain mixed monotone rational second-order difference equation with quadratic terms. The equation’s corresponding associated map is always decreasing for the second variable and can be either decreasing or increasing for the first variable depending on the corresponding parametric values. In some parametric space regions, we prove that the unique positive equilibrium point’s local asymptotic stability implies global asymptotic stability. Our main tool for studying this equation’s global dynamics is the determination of invariant interval and use so-called “m–M” theorems and semi-cycle analysis. Also, we show that the considered equation exhibits Neimark–Sacker bifurcation under certain conditions.

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Acknowledgements

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

    Zehra Nurkanović, Mehmed Nurkanović & Mirela Garić-Demirović

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  1. Zehra Nurkanović
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  2. Mehmed Nurkanović
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  3. Mirela Garić-Demirović
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Correspondence to Mehmed Nurkanović.

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Nurkanović, Z., Nurkanović, M. & Garić-Demirović, M. Stability and Neimark–Sacker Bifurcation of Certain Mixed Monotone Rational Second-Order Difference Equation. Qual. Theory Dyn. Syst. 20, 75 (2021). https://doi.org/10.1007/s12346-021-00515-4

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