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<p>We establish sharp upper bounds for the number of limit cycles in a class of piecewise smooth vector fields composed of two subsystems separated by a straight line. One subsystem is a rigid vector field of degree <i>n</i>, while the other is either a linear center or a rigid vector field of degree <i>m</i>. By analyzing the first return map associated with a Bernoulli differential equation, we prove that at most one limit cycle exists when the linear center is located at the origin. When the center is displaced, this upper bound increases to two for any <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(n\ge 1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, and it is sharp. For systems formed by two rigid vector fields, we show that uniqueness holds when <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(n=m\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>=</mo> <mi>m</mi> </mrow> </math></EquationSource> </InlineEquation>. In the resonant case <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(m=nk\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>=</mo> <mi>n</mi> <mi>k</mi> </mrow> </math></EquationSource> </InlineEquation>, we obtain an upper bound of <i>k</i> limit cycles. We also provide a sharper bound related to an open problem on rigid smooth differential systems.</p>

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Limit Cycles of Piecewise Smooth Vector Fields Formed by Rigid Systems and Linear Centers

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年7月27日

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  • Luiz Fernando Gonçalves,
  • Rodrigo Donizete Euzébio,
  • Vitória Chaves Fernandes,
  • Jaume Llibre

摘要

We establish sharp upper bounds for the number of limit cycles in a class of piecewise smooth vector fields composed of two subsystems separated by a straight line. One subsystem is a rigid vector field of degree n, while the other is either a linear center or a rigid vector field of degree m. By analyzing the first return map associated with a Bernoulli differential equation, we prove that at most one limit cycle exists when the linear center is located at the origin. When the center is displaced, this upper bound increases to two for any \(n\ge 1\) n ≥ 1 , and it is sharp. For systems formed by two rigid vector fields, we show that uniqueness holds when \(n=m\) n = m . In the resonant case \(m=nk\) m = n k , we obtain an upper bound of k limit cycles. We also provide a sharper bound related to an open problem on rigid smooth differential systems.