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

TL;DR

This paper investigates the maximum number of crossing limit cycles in piecewise-smooth vector fields on a torus with a non-regular switching manifold, providing sharp bounds and explicit examples for different polynomial degrees.

Contribution

It establishes sharp upper bounds on crossing limit cycles for polynomial first integrals of degree n and constructs examples that achieve these bounds, demonstrating their optimality.

Findings

Maximum of one cycle of type aa or bb in quadratic case

Maximum of two cycles of type aba or bab in quadratic case

Bounds of n-1 for type bb and n(n-1) for type aba cycles for general n

Abstract

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.