A real analytic differential system having a center at the origin of coordinates after a linear change of variables and a rescaling of the time can be written in one of the following three forms: <Equation ID="Equ32"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="12215_2025_1256_Article_Equ32.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="267" /> </MediaObject> <EquationSource Format="TEX">${\dot{x}}=-y+ X_2(x,y), \ {\dot{y}}=x+Y_2(x,y),$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mover accent="true"> <mi>x</mi> <mo>˙</mo> </mover> <mo>=</mo> <mo>-</mo> <mi>y</mi> <mo>+</mo> <msub> <mi>X</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <mspace width="4pt" /> <mover accent="true"> <mi>y</mi> <mo>˙</mo> </mover> <mo>=</mo> <mi>x</mi> <mo>+</mo> <msub> <mi>Y</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> </mrow> </math></EquationSource> </Equation>called a linear type center, <Equation ID="Equ33"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="12215_2025_1256_Article_Equ33.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="219" /> </MediaObject> <EquationSource Format="TEX">${\dot{x}}=y+ X_2(x,y), \ {\dot{y}}=Y_2(x,y)$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mover accent="true"> <mi>x</mi> <mo>˙</mo> </mover> <mo>=</mo> <mi>y</mi> <mo>+</mo> <msub> <mi>X</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <mspace width="4pt" /> <mover accent="true"> <mi>y</mi> <mo>˙</mo> </mover> <mo>=</mo> <msub> <mi>Y</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </Equation>called a nilpotent center, and <Equation ID="Equ34"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="12215_2025_1256_Article_Equ34.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="190" /> </MediaObject> <EquationSource Format="TEX">${\dot{x}}=X_2(x,y), \ {\dot{y}}=Y_2(x,y)$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mover accent="true"> <mi>x</mi> <mo>˙</mo> </mover> <mo>=</mo> <msub> <mi>X</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <mspace width="4pt" /> <mover accent="true"> <mi>y</mi> <mo>˙</mo> </mover> <mo>=</mo> <msub> <mi>Y</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </Equation>called a degenerate center, where <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12215_2025_1256_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="60" /> </InlineMediaObject> <EquationSource Format="TEX">$X_2(x,y)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>X</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12215_2025_1256_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="56" /> </InlineMediaObject> <EquationSource Format="TEX">$Y_2(x,y)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>Y</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> are real analytic functions without constant and linear terms, defined in a neighborhood of the origin.While there are many papers dedicated to study phase portraits of different classes of linear type centers, few papers studied the phase portraits of the nilpotent and degenerate centers. Here we classify the global phase portraits in the Poincaré disc of reversible nilpotent centers with cubic nonlinearities.

Reversible nilpotent centers with cubic nonlinearities

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2025年6月13日

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Leandro Bacelar,
Jaume Llibre

摘要

A real analytic differential system having a center at the origin of coordinates after a linear change of variables and a rescaling of the time can be written in one of the following three forms: ${\dot{x}}=-y+ X_2(x,y), \ {\dot{y}}=x+Y_2(x,y),$ $\dot{x} = - y + X_{2} (x, y), \dot{y} = x + Y_{2} (x, y),$ called a linear type center, ${\dot{x}}=y+ X_2(x,y), \ {\dot{y}}=Y_2(x,y)$ $\dot{x} = y + X_{2} (x, y), \dot{y} = Y_{2} (x, y)$ called a nilpotent center, and ${\dot{x}}=X_2(x,y), \ {\dot{y}}=Y_2(x,y)$ $\dot{x} = X_{2} (x, y), \dot{y} = Y_{2} (x, y)$ called a degenerate center, where $X_2(x,y)$ $X_{2} (x, y)$ and $Y_2(x,y)$ $Y_{2} (x, y)$ are real analytic functions without constant and linear terms, defined in a neighborhood of the origin.

While there are many papers dedicated to study phase portraits of different classes of linear type centers, few papers studied the phase portraits of the nilpotent and degenerate centers. Here we classify the global phase portraits in the Poincaré disc of reversible nilpotent centers with cubic nonlinearities.