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<p>We study the maximum number of crossing limit cycles in piecewise-smooth vector fields on the two-dimensional torus, where the discontinuity set is the boundary of the fundamental square. Under the assumption of a polynomial first integral of degree <i>n</i>, we apply algebraic curve-intersection methods to obtain sharp upper bounds on the number of crossing limit cycles. In the quadratic case (<InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(n=2\)</EquationSource> </InlineEquation>), we prove that the system admits at most one crossing limit cycle with two switching crossings and at most two with three mixed switching crossings, and we provide explicit parameter criteria for their existence. For general <i>n</i>, we show that the number of crossing limit cycles with two switching crossings is bounded by <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(n-1\)</EquationSource> </InlineEquation>, while those with three switching crossings are bounded by <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(n(n-1)\)</EquationSource> </InlineEquation>. Finally, we construct concrete examples attaining these bounds, showing that they are optimal.</p>

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Limit Cycles of Piecewise Hamiltonian Vector Fields on the 2-Dimensional Torus

  • 期刊论文

  • 发表日期：

    2026年6月19日

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  • Thaylon Souza de Oliveira,
  • Ricardo Miranda Martins

摘要

We study the maximum number of crossing limit cycles in piecewise-smooth vector fields on the two-dimensional torus, where the discontinuity set is the boundary of the fundamental square. Under the assumption of a polynomial first integral of degree n, we apply algebraic curve-intersection methods to obtain sharp upper bounds on the number of crossing limit cycles. In the quadratic case ( \(n=2\) ), we prove that the system admits at most one crossing limit cycle with two switching crossings and at most two with three mixed switching crossings, and we provide explicit parameter criteria for their existence. For general n, we show that the number of crossing limit cycles with two switching crossings is bounded by \(n-1\) , while those with three switching crossings are bounded by \(n(n-1)\) . Finally, we construct concrete examples attaining these bounds, showing that they are optimal.