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<p>In this paper, we investigate some aspects concerning the density of periodic trajectories for piecewise smooth vector fields, where uniqueness of trajectories may not exist, in contrast to the classical theory of smooth ordinary differential equations. For this purpose, we use topological conjugacy between the metric space of the global trajectories and two-sided shift maps. Additionally, we calculate the Hausdorff and Minkowski dimensions for the trajectory space, simplifying the understanding of the complexity of the set of trajectories. We prove that, under certain assumptions, the Hausdorff and Minkowski dimensions are equal, a relevant aspect in the analysis of more complex systems.</p>

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Density of Periodic Orbits and Hausdorff Dimension for Piecewise Smooth Vector Fields

  • 期刊论文

  • 发表日期：

    2026年5月7日

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  • Marco Florentino,
  • Tiago Carvalho

摘要

In this paper, we investigate some aspects concerning the density of periodic trajectories for piecewise smooth vector fields, where uniqueness of trajectories may not exist, in contrast to the classical theory of smooth ordinary differential equations. For this purpose, we use topological conjugacy between the metric space of the global trajectories and two-sided shift maps. Additionally, we calculate the Hausdorff and Minkowski dimensions for the trajectory space, simplifying the understanding of the complexity of the set of trajectories. We prove that, under certain assumptions, the Hausdorff and Minkowski dimensions are equal, a relevant aspect in the analysis of more complex systems.