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<p>In this paper, we study the maximum number of limit cycles bifurcating from the period annulus of the planar linear center <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">\(\dot{y}=-x\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mover accent="true"> <mi>y</mi> <mo>˙</mo> </mover> <mo>=</mo> <mo>-</mo> <mi>x</mi> </mrow> </math></EquationSource> </InlineEquation> with the switching curves <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(y=\pm x^2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>y</mi> <mo>=</mo> <mo>±</mo> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </math></EquationSource> </InlineEquation>, under arbitrary polynomial perturbations of degree <i>n</i> in <i>x</i> and <i>y</i> (<InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(n\in \mathbb {N}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>∈</mo> <mi mathvariant="double-struck">N</mi> </mrow> </math></EquationSource> </InlineEquation>). By using the first order Melnikov function, we establish both a lower bound and an upper bound for the maximum number of limit cycles.</p>

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On the Number of Limit Cycles Bifurcating from a Class of Planar Piecewise Linear Systems with Four Zones

  • 期刊论文

  • 发表日期：

    2026年2月18日

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  • Ranran Jia,
  • Liqin Zhao

摘要

In this paper, we study the maximum number of limit cycles bifurcating from the period annulus of the planar linear center \(\dot{x}=y\) x ˙ = y , \(\dot{y}=-x\) y ˙ = - x with the switching curves \(y=\pm x^2\) y = ± x 2 , under arbitrary polynomial perturbations of degree n in x and y ( \(n\in \mathbb {N}\) n ∈ N ). By using the first order Melnikov function, we establish both a lower bound and an upper bound for the maximum number of limit cycles.