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

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* ≤ 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 Poincaré half-maps for linear differential systems with an extension of Khovanskiĭ's theory for investigating the number of intersection points between smooth curves and a particular kind of orbits of vector fields.