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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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.
AbstractList 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.
ArticleNumber 75
Author Nurkanović, Zehra
Garić-Demirović, Mirela
Nurkanović, Mehmed
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  givenname: Mirela
  surname: Garić-Demirović
  fullname: Garić-Demirović, Mirela
  organization: Department of Mathematics, University of Tuzla
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CitedBy_id crossref_primary_10_18514_MMN_2025_4546
crossref_primary_10_1016_j_matcom_2025_07_013
crossref_primary_10_1080_10236198_2024_2378824
crossref_primary_10_3390_axioms13040280
Cites_doi 10.3390/math6010010
10.1007/s00285-006-0004-3
10.1155/2013/210846
10.1155/2018/1613709
10.1155/2018/7052935
10.1155/JIA.2005.127
10.1007/s12346-015-0148-x
10.1201/9781420035384
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10.22436/jnsa.010.07.11
10.1080/10236190802054126
10.3934/dcdsb.2018062
10.1186/1687-1847-2012-153
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Issue 3
Keywords Neimark–Sacker bifurcation
39A20
39A30
Stability
39A10
Difference equations
39A23
Normal form
65L20
Invariant curve
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  start-page: 59
  issue: 1
  year: 2002
  ident: 515_CR16
  publication-title: Radovi Mat. (Sarajevo J. Math.)
– volume: 2018
  start-page: 1613709
  year: 2018
  ident: 515_CR26
  publication-title: Discrete Dyn. Nat. Soc.
  doi: 10.1155/2018/1613709
– volume: 23
  start-page: 1581
  issue: 4
  year: 2018
  ident: 515_CR25
  publication-title: Discrete Contin. Dyn. Syst. Ser. B
  doi: 10.3934/dcdsb.2018062
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Snippet This paper investigates the local and global character of the unique positive equilibrium of certain mixed monotone rational second-order difference equation...
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SubjectTerms Asymptotic properties
Bifurcations
Difference and Functional Equations
Difference equations
Dynamic stability
Dynamical Systems and Ergodic Theory
Mathematics
Mathematics and Statistics
Quadratic equations
Title Stability and Neimark–Sacker Bifurcation of Certain Mixed Monotone Rational Second-Order Difference Equation
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https://www.proquest.com/docview/2563304220
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