Lower Bounds for the Number of Limit Cycles of Discontinuous Piecewise-Linear Systems Without Equilibrium Points but with a Nonregular Separation Line

In this paper, we investigate the crossing limit cycles of a class of discontinuous planar piecewise-linear dynamical systems without equilibrium points but with two regions and separated by a nonregular line [Formula: see text]. First, we prove that the lower bound of the maximum number of crossing limit cycles having a unique point with each of the two branches of [Formula: see text] is 3, the maximum number of limit cycles having four intersection points with [Formula: see text] are at least 2, and the maximum number of limit cycles intersecting one branch of [Formula: see text] is 1. Second, we show that the three different types of crossing limit cycles can exist simultaneously by providing concrete examples with four crossing limit cycles. But until this moment, it is an open problem to figure out if the sharp upper bound is four or more.