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<p>In the paper we generalize some classic results on limit cycles of Liénard system</p><p><Equation ID="Equa"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="10114_2025_3420_Article_Equa.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="219" /> </MediaObject> <EquationSource Format="TEX">\(\dot{x}=\phi(y)-F(x),\quad\dot{y}=-g(x)\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mover> <mi>x</mi> <mo>˙</mo> </mover> </mrow> <mo>=</mo> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="1em" /> <mrow> <mover> <mi>y</mi> <mo>˙</mo> </mover> </mrow> <mo>=</mo> <mo>−</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </math></EquationSource> </Equation></p><p>having a unique equilibrium to that of the system with several equilibria. As applications, we strictly prove the number of limit cycles and obtain the distribution of limit cycles for three classes of Liénard systems, in which we correct a mistake in the literature.</p>

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Limit Cycles of Liénard Systems with Several Equilibria

  • 期刊论文

  • 发表日期：

    2025年4月15日

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  • Hebai Chen,
  • Yilei Tang,
  • Dongmei Xiao

摘要

In the paper we generalize some classic results on limit cycles of Liénard system

\(\dot{x}=\phi(y)-F(x),\quad\dot{y}=-g(x)\) x ˙ = ϕ ( y ) − F ( x ) , y ˙ = − g ( x )

having a unique equilibrium to that of the system with several equilibria. As applications, we strictly prove the number of limit cycles and obtain the distribution of limit cycles for three classes of Liénard systems, in which we correct a mistake in the literature.