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<p>We study the simultaneous bifurcation of limit cycles in planar piecewise quadratic differential systems separated by a straight line. These limit cycles arise from a degenerate Hopf bifurcation at two equilibrium points in the positive and negative half-planes, as well as from an equilibrium on the separation line. All the limit cycles are of small amplitude. This bifurcation creates a configuration of limit cycles of type <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12346_2025_1252_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="53" /> </InlineMediaObject> <EquationSource Format="TEX">\((3,5,3)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mn>3</mn> <mo>,</mo> <mn>5</mn> <mo>,</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. Additionally, in each half-plane, the maximum number of small-amplitude hyperbolic limit cycles that a quadratic vector field can have is three.</p>

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Coexistence of Analytic and Piecewise Analytic Limit Cycles in Planar Piecewise Quadratic Differential Systems

  • 期刊论文

  • 发表日期：

    2025年3月27日

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  • Leonardo P. C. da Cruz,
  • Alex C. Rezende,
  • Joan Torregrosa

摘要

We study the simultaneous bifurcation of limit cycles in planar piecewise quadratic differential systems separated by a straight line. These limit cycles arise from a degenerate Hopf bifurcation at two equilibrium points in the positive and negative half-planes, as well as from an equilibrium on the separation line. All the limit cycles are of small amplitude. This bifurcation creates a configuration of limit cycles of type \((3,5,3)\) ( 3 , 5 , 3 ) . Additionally, in each half-plane, the maximum number of small-amplitude hyperbolic limit cycles that a quadratic vector field can have is three.