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

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...

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Veröffentlicht in:Qualitative theory of dynamical systems Jg. 20; H. 3
Hauptverfasser: Nurkanović, Zehra, Nurkanović, Mehmed, Garić-Demirović, Mirela
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Cham Springer International Publishing 01.11.2021
Springer Nature B.V
Schlagworte:
Asymptotic properties
Bifurcations
Difference and Functional Equations
Difference equations
Dynamic stability
Dynamical Systems and Ergodic Theory
Mathematics
Mathematics and Statistics
Quadratic equations
Neimark–Sacker bifurcation
39A20
39A30
Stability
39A10
Difference equations
39A23
Normal form
65L20
Invariant curve
ISSN:1575-5460, 1662-3592
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Zusammenfassung: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.
Bibliographie:ObjectType-Article-1
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ISSN:1575-5460
1662-3592
DOI:10.1007/s12346-021-00515-4