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<p>In this paper, we deal with a discrete-time dynamical system given in the form <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40065_2025_538_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="151" /> </InlineMediaObject> <EquationSource Format="TEX">\(x_{n+1}=f(x_{n-1},x_n,a)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>x</mi> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mi>n</mi> </msub> <mo>,</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, where <i>a</i> is a real parameter and <i>f</i> is assumed smooth enough. More specifically, we will focus on giving sufficient conditions to ensure the system undergoes a bifurcation of the fold, transcritical, pitchfork, and flip types. In our analysis, all the bifurcations are of codimension one and related to a unique eigenvalue, either one or minus one.</p>

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On sufficient conditions of fold and flip bifurcations in second-order difference equations

  • 期刊论文

  • 发表日期：

    2025年7月14日

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  • Jose S. Cánovas,
  • María Muñoz-Guillermo

摘要

In this paper, we deal with a discrete-time dynamical system given in the form \(x_{n+1}=f(x_{n-1},x_n,a)\) x n + 1 = f ( x n - 1 , x n , a ) , where a is a real parameter and f is assumed smooth enough. More specifically, we will focus on giving sufficient conditions to ensure the system undergoes a bifurcation of the fold, transcritical, pitchfork, and flip types. In our analysis, all the bifurcations are of codimension one and related to a unique eigenvalue, either one or minus one.