The generalized Liénard polynomial systems of type (m, n) are the planar differential systems of the form <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\dot{x}=y$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mover accent="true"> <mi>x</mi> <mo>˙</mo> </mover> <mo>=</mo> <mi>y</mi> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\dot{y}=-f_m(x)y-g_n(x)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mover accent="true"> <mi>y</mi> <mo>˙</mo> </mover> <mo>=</mo> <mo>-</mo> <msub> <mi>f</mi> <mi>m</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mi>y</mi> <mo>-</mo> <msub> <mi>g</mi> <mi>n</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$f_m(x)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>f</mi> <mi>m</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$g_n(x)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>g</mi> <mi>n</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> are two polynomials with real coefficients of degree m and n, respectively. We mainly present in this paper a complete characterization of all the generalized Liénard systems of type (m, n) having an irreducible invariant algebraic curve <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$F(x,y)= \sum _{j=0}^{s}a_j(x)y^{s-j}=0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>F</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msubsup> <mo>∑</mo> <mrow> <mi>j</mi> <mo>=</mo> <mn>0</mn> </mrow> <mi>s</mi> </msubsup> <msub> <mi>a</mi> <mi>j</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <msup> <mi>y</mi> <mrow> <mi>s</mi> <mo>-</mo> <mi>j</mi> </mrow> </msup> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$a_0(x)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>a</mi> <mn>0</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$a_1(x)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>a</mi> <mn>1</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\cdots $</EquationSource> <EquationSource Format="MATHML"><math> <mo>⋯</mo> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$a_s(x)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>a</mi> <mi>s</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> are polynomials of degree no more than s, <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$s\ge 4$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mo>≥</mo> <mn>4</mn> </mrow> </math></EquationSource> </InlineEquation>, and <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$a_0(x)\ne 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>a</mi> <mn>0</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>≠</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$(s-1)a_1^2(x)= 2sa_2(x)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo stretchy="false">(</mo> <mi>s</mi> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <msubsup> <mi>a</mi> <mn>1</mn> <mn>2</mn> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mn>2</mn> <mi>s</mi> <msub> <mi>a</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. Additionally, in some cases, we establish conditions on the invariant algebraic curve to ensure the integrability of the generalized Liénard systems.

Characterization of the generalized Liénard polynomial systems having special invariant algebraic curves

期刊论文
发表日期：

2025年10月27日

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Xinjie Qian,
Shaoqing Wang,
Jiazhong Yang

摘要

The generalized Liénard polynomial systems of type (m, n) are the planar differential systems of the form $\dot{x}=y$ $\dot{x} = y$ , $\dot{y}=-f_m(x)y-g_n(x)$ $\dot{y} = - f_{m} (x) y - g_{n} (x)$ , where $f_m(x)$ $f_{m} (x)$ and $g_n(x)$ $g_{n} (x)$ are two polynomials with real coefficients of degree m and n, respectively. We mainly present in this paper a complete characterization of all the generalized Liénard systems of type (m, n) having an irreducible invariant algebraic curve $F(x,y)= \sum _{j=0}^{s}a_j(x)y^{s-j}=0$ $F (x, y) = \sum_{j = 0}^{s} a_{j} (x) y^{s - j} = 0$ , where $a_0(x)$ $a_{0} (x)$ , $a_1(x)$ $a_{1} (x)$ , $\cdots $ $\dots$ , $a_s(x)$ $a_{s} (x)$ are polynomials of degree no more than s, $s\ge 4$ $s \geq 4$ , and $a_0(x)\ne 0$ $a_{0} (x) \neq 0$ , $(s-1)a_1^2(x)= 2sa_2(x)$ $(s - 1) a_{1}^{2} (x) = 2 s a_{2} (x)$ . Additionally, in some cases, we establish conditions on the invariant algebraic curve to ensure the integrability of the generalized Liénard systems.