Bounding the number of limit cycles in piecewise linear differential systems: methodology and worked examples

Abstract

In 2012, Huan and Yang introduced the first piecewise linear differential system with two zones separated by a straight line having at least three limit cycles, serving as a counterexample to the Han-Zhang conjecture that said that such systems have no more than two limit cycles. Over the past decade, extensive research has been conducted to explore periodic solutions in piecewise linear differential systems. However, the question of whether the Huan-Yang example indeed has exactly three limit cycles has remained unresolved, primarily due to the lack of techniques for bounding the number of limit cycles in these systems. Based on the authors' recent results, this paper presents a methodology for bounding the number of limit cycles in piecewise linear systems. This methodology conclusively establishes that the Huan-Yang example has exactly three limit cycles. Our methodology has a broader applicability and constitutes a powerful tool for analyzing and bounding the number of limit cycles in any explicit example of piecewise linear differential systems. We extend our analysis to several recent examples of significance in the literature and show that they also exhibit exactly three limit cycles. Finally, we present an algebraic criterion for bounding the number of crossing limit cycles in the focus-focus case.