Uniform upper bound for the number of limit cycles of planar piecewise linear differential systems with two zones separated by a straight line

TL;DR

This paper proves that any planar piecewise linear differential system with two zones separated by a straight line has at most 8 limit cycles, resolving a long-standing open problem in the field.

Contribution

It establishes a uniform upper bound of 8 limit cycles for such systems, combining new integral characterizations and advanced intersection theory.

Findings

Maximum of 8 limit cycles in these systems

New integral characterization of Poincaré half-maps

Extension of Khovanskii's theory for intersection analysis

Abstract

The existence of a uniform upper bound for the maximum number of limit cycles of planar piecewise linear differential systems with two zones separated by a straight line has been subject of interest of hundreds of papers. After more than 30 years of investigation since Lum-Chua's work, it has remained an open question whether this uniform upper bound exists or not. Here, we give a positive answer for this question by establishing the existence of a natural number $L^*\leq 8$ for which any planar piecewise linear differential system with two zones separated by a straight line has no more than $L^*$ limit cycles. The proof is obtained by combining a newly developed integral characterization of Poincar\'{e} half-maps for linear differential systems with an extension of Khovanski\u{\i}'s theory for investigating the number of intersection points between smooth curves and a particular kind of orbits of vector fields.