标题： 环状曲面上传播的分段光滑向量场的极限环 显示英文标题

标题： Limit cycles of piecewise smooth vector fields on torus with non-regular switching manifold

摘要： 我们研究在二维环面上分段光滑向量场中交叉极限环的最大数量，其中不连续集是基本正方形的边界。 在假设存在次数为n的多项式首次积分的前提下，我们应用代数曲线交点方法来获得环数的精确上界。 在二次情况（n=2）下，我们证明类型为aa或bb的环最多有一个，类型为aba或bab的环最多有两个，并给出了它们存在的显式参数条件。 对于一般的n，我们证明类型为bb的环受n-1的限制，类型为aba的环受n(n-1)的限制。 最后，我们构造了达到这些界限的具体例子，证明了它们的最优性。 显示英文摘要

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 cycle counts. In the quadratic case (n=2), we prove at most one cycle of type aa or bb and at most two of type aba or bab, and we give explicit parameter criteria for their existence. For general n, we show that cycles of type bb are bounded by n-1 and those of type aba by n(n-1). Finally, we construct concrete examples achieving these bounds, demonstrating their optimality.