In this manuscript, we apply averaging theory to investigate the maximum number of limit cycles in a type of generalized Hill differential equation.<Equation ID="Equa"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="12190_2025_2442_Article_Equa.gif" Format="GIF" Height="49" Rendition="HTML" Resolution="72" Type="Linedraw" Width="366" /> </MediaObject> <EquationSource Format="TEX">\(\mathop x\limits^{..} + \varepsilon (1 + \mathop {\mathop {\sum }\limits^m }\limits_{k = 0} \left( {\matrix{ m \cr k \cr } } \right)\mathop {\sin }\nolimits^k \theta \mathop {\cos }\nolimits^{m - k} \theta ))P(x,y) + x = 0,\)</EquationSource> </Equation>that branch out from the periodic orbits of the linear center <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12190_2025_2442_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="109" /> </InlineMediaObject> <EquationSource Format="TEX">\(\dot x = y,\dot y = - x.\)</EquationSource> </InlineEquation> Here <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12190_2025_2442_Article_IEq2.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="40" /> </InlineMediaObject> <EquationSource Format="TEX">\(\varepsilon > 0\)</EquationSource> </InlineEquation> is a small parameter, <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12190_2025_2442_Article_IEq3.gif" Format="GIF" Height="43" Rendition="HTML" Resolution="72" Type="Linedraw" Width="119" /> </InlineMediaObject> <EquationSource Format="TEX">\(\left( {\matrix{m \cr k \cr } } \right) = {{m!} \over {k!(m - k)!}},\)</EquationSource> </InlineEquation> P is a polynomial of degree <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12190_2025_2442_Article_IEq5.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">\(n,\)</EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12190_2025_2442_Article_IEq6.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">\(m\)</EquationSource> </InlineEquation> is an arbitrary non-negative integer, and <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12190_2025_2442_Article_IEq7.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="97" /> </InlineMediaObject> <EquationSource Format="TEX">\(\theta = \arctan (\frac{y}{x})\)</EquationSource> </InlineEquation>.

Limit cycles for a kind of generalized Hill differential equation

期刊论文
发表日期：

2025年3月25日

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Abdelkrim Kina,
Abdelkader Moumen,
Hamid Boulares,
Mohamed Bouye

摘要

In this manuscript, we apply averaging theory to investigate the maximum number of limit cycles in a type of generalized Hill differential equation.

\(\mathop x\limits^{..} + \varepsilon (1 + \mathop {\mathop {\sum }\limits^m }\limits_{k = 0} \left( {\matrix{ m \cr k \cr } } \right)\mathop {\sin }\nolimits^k \theta \mathop {\cos }\nolimits^{m - k} \theta ))P(x,y) + x = 0,\)

that branch out from the periodic orbits of the linear center \(\dot x = y,\dot y = - x.\) Here \(\varepsilon > 0\) is a small parameter, \(\left( {\matrix{m \cr k \cr } } \right) = {{m!} \over {k!(m - k)!}},\) P is a polynomial of degree \(n,\) and \(m\) is an arbitrary non-negative integer, and \(\theta = \arctan (\frac{y}{x})\) .