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<p>In the present paper, we investigate the maximum number of limit cycles bifurcating from the period annulus of a cubic isochronous center when it is perturbed by piecewise polynomials of arbitrary degree <i>n</i> with the switching line <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(x=0\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>x</mi> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. By analyzing the number of zeros of the first order Melnikov function with a new Chebyshev criterion in [<CitationRef CitationID="CR3">3</CitationRef>], we show that the sharp upper bound on the number of limit cycles for the perturbed systems is <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(2n+1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> for <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(n\ge 1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>.</p>

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Bifurcation of Limit Cycles through Perturbing a Cubic Isochronous Center in Piecewise Polynomial Differential Systems

  • 期刊论文

  • 发表日期：

    2025年11月13日

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  • Xiuli Cen,
  • Shangming Chen,
  • Zhe Zhang

摘要

In the present paper, we investigate the maximum number of limit cycles bifurcating from the period annulus of a cubic isochronous center when it is perturbed by piecewise polynomials of arbitrary degree n with the switching line \(x=0\) x = 0 . By analyzing the number of zeros of the first order Melnikov function with a new Chebyshev criterion in [3], we show that the sharp upper bound on the number of limit cycles for the perturbed systems is \(2n+1\) 2 n + 1 for \(n\ge 1\) n ≥ 1 .