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<p>Piecewise differential systems have received significant attention over the last decades because of their relevance in numerous applications. A key topic in the qualitative study of these systems is the existence of limit cycles, that is, periodic trajectories that are isolated from other nearby periodic trajectories. In this paper, we investigate a class of planar piecewise differential systems defined on <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathbb {R}^2\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> and separated by the analytic curve given by the graphic of the function <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(y=e^x\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>y</mi> <mo>=</mo> <msup> <mi>e</mi> <mi>x</mi> </msup> </mrow> </math></EquationSource> </InlineEquation>. On each side of this curve, the dynamics is governed by an arbitrary linear Hamiltonian system. The considered systems are continuous–discontinuous in the sense that the first coordinates of the corresponding vector fields agree on the switching curve, whereas the second coordinates do not necessarily coincide there. Our main result establishes that systems within this family may possess at most three limit cycles.</p>

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On the Existence of Limit Cycles of Continuous–Discontinuous Piecewise Linear Hamiltonian Systems in the Plane with Exponential Switching Curve

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  • 发表日期：

    2026年7月19日

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  • N. Chachapoyas,
  • J. Llibre,
  • I. S. Meza-Sarmiento,
  • J. Vidarte

摘要

Piecewise differential systems have received significant attention over the last decades because of their relevance in numerous applications. A key topic in the qualitative study of these systems is the existence of limit cycles, that is, periodic trajectories that are isolated from other nearby periodic trajectories. In this paper, we investigate a class of planar piecewise differential systems defined on \(\mathbb {R}^2\) R 2 and separated by the analytic curve given by the graphic of the function \(y=e^x\) y = e x . On each side of this curve, the dynamics is governed by an arbitrary linear Hamiltonian system. The considered systems are continuous–discontinuous in the sense that the first coordinates of the corresponding vector fields agree on the switching curve, whereas the second coordinates do not necessarily coincide there. Our main result establishes that systems within this family may possess at most three limit cycles.