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<p>We consider a class of piecewise smooth Liénard systems and study the Hopf bifurcation of limit cycles from the origin as parameters vary. We introduce the cyclicity <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(N_{k,l}(m,n)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>N</mi> <mrow> <mi>k</mi> <mo>,</mo> <mi>l</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> of such systems at the origin and apply combinatorial methods to give its sharp estimation when <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(0\le k,l\le 1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo>≤</mo> <mi>k</mi> <mo>,</mo> <mi>l</mi> <mo>≤</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. Moreover, the cyclicity of such systems at the origin is also studied when some parameters are missing.</p>

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The Maximum Number of Limit Cycles in Hopf Bifurcation of a Piecewise Smooth Liénard System

  • 期刊论文

  • 发表日期：

    2026年3月31日

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  • Dekai Zhu,
  • Maoan Han

摘要

We consider a class of piecewise smooth Liénard systems and study the Hopf bifurcation of limit cycles from the origin as parameters vary. We introduce the cyclicity \(N_{k,l}(m,n)\) N k , l ( m , n ) of such systems at the origin and apply combinatorial methods to give its sharp estimation when \(0\le k,l\le 1\) 0 ≤ k , l ≤ 1 . Moreover, the cyclicity of such systems at the origin is also studied when some parameters are missing.