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<p>In piecewise-smooth differential systems, tangent points are the singularities on the switching manifold where the vector field exhibits degeneracy along the normal direction of the switching manifold. In this paper, we focus on investigating bifurcations of tangent points and loops connecting them of any degeneracy. The functional parameter method used for low degeneracy is invalid for high degeneracy because the recurrences of orbits near tangent points are unclear under perturbations, which makes defining an appropriate Poincaré return map very difficult. To overcome this difficulty, we not only provide a functional function method for clarifying the recurrences but also combine this method and Poincaré return map, and hence establish the relationships between the degeneracy and the numbers of crossing limit cycles and sliding loops bifurcating from a critical loop.</p>

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Bifurcations of Tangent Points and Critical Loops in Piecewise-Smooth Systems

  • 期刊论文

  • 发表日期：

    2025年10月7日

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  • Zhihao Fang,
  • Xingwu Chen

摘要

In piecewise-smooth differential systems, tangent points are the singularities on the switching manifold where the vector field exhibits degeneracy along the normal direction of the switching manifold. In this paper, we focus on investigating bifurcations of tangent points and loops connecting them of any degeneracy. The functional parameter method used for low degeneracy is invalid for high degeneracy because the recurrences of orbits near tangent points are unclear under perturbations, which makes defining an appropriate Poincaré return map very difficult. To overcome this difficulty, we not only provide a functional function method for clarifying the recurrences but also combine this method and Poincaré return map, and hence establish the relationships between the degeneracy and the numbers of crossing limit cycles and sliding loops bifurcating from a critical loop.