The elementary center–focus problem is studied for a general separable planar analytic Rayleigh–Liénard system <Equation ID="Equ33"> <EquationSource Format="TEX">$\begin{aligned} \frac{\textrm{d}x}{\textrm{d}t} = y,\quad \frac{\textrm{d}y}{\textrm{d}t} = -g(x) + f(x)\psi (y)y, \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mfrac> <mrow> <mtext>d</mtext> <mi>x</mi> </mrow> <mrow> <mtext>d</mtext> <mi>t</mi> </mrow> </mfrac> <mo>=</mo> <mi>y</mi> <mo>,</mo> <mspace width="1em" /> <mfrac> <mrow> <mtext>d</mtext> <mi>y</mi> </mrow> <mrow> <mtext>d</mtext> <mi>t</mi> </mrow> </mfrac> <mo>=</mo> <mo>-</mo> <mi>g</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mi>ψ</mi> <mrow> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <mi>y</mi> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>where g(x), f(x), and <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\psi (y)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> are analytic functions. By means of the Melnikov function method, a necessary and sufficient condition is obtained for the origin to be an elementary center: <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$g'(0)>0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>g</mi> <mo>′</mo> </msup> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> and either <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\dfrac{xf\!\left( \phi ^{-1}(x)\right) }{g\!\left( \phi ^{-1}(x)\right) }$</EquationSource> <EquationSource Format="MATHML"><math> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mrow> <mi>x</mi> <mi>f</mi> <mspace width="-0.166667em" /> <mfenced close=")" open="("> <msup> <mi>ϕ</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mfenced> </mrow> <mrow> <mi>g</mi> <mspace width="-0.166667em" /> <mfenced close=")" open="("> <msup> <mi>ϕ</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mfenced> </mrow> </mfrac> </mstyle> </math></EquationSource> </InlineEquation> or <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\psi (y)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is an odd function in a neighborhood of the origin, where <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\phi (x)= \sqrt{2\!\int _0^x g(s)\, \textrm{d}s}\cdot \text {sgn}(x).$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ϕ</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msqrt> <mrow> <mn>2</mn> <mspace width="-0.166667em" /> <msubsup> <mo>∫</mo> <mn>0</mn> <mi>x</mi> </msubsup> <mi>g</mi> <mrow> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> <mspace width="0.166667em" /> <mtext>d</mtext> <mi>s</mi> </mrow> </msqrt> <mo>·</mo> <mtext>sgn</mtext> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation>

A Necessary and Sufficient Condition for General Rayleigh–Liénard System with an Elementary Center

期刊论文
发表日期：

2026年4月27日

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Xiuli Cen,
Hebai Chen,
Zhaosheng Feng,
Rui Zhang

摘要

The elementary center–focus problem is studied for a general separable planar analytic Rayleigh–Liénard system $\begin{aligned} \frac{\textrm{d}x}{\textrm{d}t} = y,\quad \frac{\textrm{d}y}{\textrm{d}t} = -g(x) + f(x)\psi (y)y, \end{aligned}$ $\begin{matrix} \frac{d x}{d t} = y, \frac{d y}{d t} = - g (x) + f (x) ψ (y) y, \end{matrix}$ where g(x), f(x), and $\psi (y)$ $ψ (y)$ are analytic functions. By means of the Melnikov function method, a necessary and sufficient condition is obtained for the origin to be an elementary center: $g'(0)>0$ $g^{'} (0) > 0$ and either $\dfrac{xf\!\left( \phi ^{-1}(x)\right) }{g\!\left( \phi ^{-1}(x)\right) }$ $\frac{x f (ϕ^{- 1}, (x))}{g (ϕ^{- 1}, (x))}$ or $\psi (y)$ $ψ (y)$ is an odd function in a neighborhood of the origin, where $\phi (x)= \sqrt{2\!\int _0^x g(s)\, \textrm{d}s}\cdot \text {sgn}(x).$ $ϕ (x) = \sqrt{2 \int_{0}^{x} g (s) d s} \cdot sgn (x) .$