In this paper, we give a complete algebraic characterization of the bifurcation of critical periods and non-degenerate isochronous center for the Cherkas polynomial differential equation <Equation ID="Equ28"> <EquationSource Format="TEX">$\begin{aligned} \ddot{x}+h(x)\dot{x}^2+f(x)\dot{x}+g(x)=0, \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mover accent="true"> <mi>x</mi> <mo>¨</mo> </mover> <mo>+</mo> <mi>h</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <msup> <mover accent="true"> <mi>x</mi> <mo>˙</mo> </mover> <mn>2</mn> </msup> <mo>+</mo> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mover accent="true"> <mi>x</mi> <mo>˙</mo> </mover> <mo>+</mo> <mi>g</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>with f, g and h of degree <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\le 4$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>≤</mo> <mn>4</mn> </mrow> </math></EquationSource> </InlineEquation>. The main tools used in our proof are the classical results of period constant, crossed resultant and numerical calculation. By computing period constants, we give weak center of exact order and parameter condition for isochronous center. Moreover, we prove that the system has 6 critical periods.

Critical Period Bifurcation and Isochronous Center for a Sextic Cherkas System

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2026年4月20日

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Yusen Wu,
Wenyi Wang,
Yanbin Shang

摘要

In this paper, we give a complete algebraic characterization of the bifurcation of critical periods and non-degenerate isochronous center for the Cherkas polynomial differential equation $\begin{aligned} \ddot{x}+h(x)\dot{x}^2+f(x)\dot{x}+g(x)=0, \end{aligned}$ $\begin{matrix} \ddot{x} + h (x) {\dot{x}}^{2} + f (x) \dot{x} + g (x) = 0, \end{matrix}$ with f, g and h of degree $\le 4$ $\leq 4$ . The main tools used in our proof are the classical results of period constant, crossed resultant and numerical calculation. By computing period constants, we give weak center of exact order and parameter condition for isochronous center. Moreover, we prove that the system has 6 critical periods.