Algebraic limit cycles of planar discontinuous piecewise linear differential systems with an angular switching boundary

Abstract

Known results show that, with a θ-angular switching boundary for $\theta \in (0,\pi ]$, a planar piecewise linear differential system formed by two Hamiltonian linear sub-systems has no crossing algebraic limit cycles of type I, i.e., those cycles crossing one of the two sides of the θ-angular switching boundary twice only, and at most two crossing algebraic limit cycles of type II, i.e., those cycles crossing both sides of the θ-angular switching boundary once separately. In this paper, using the Chebyshev theory and Descartes' rule to overcome difficulties in applying Gröbner basis to solve polynomial systems, we study the number of crossing algebraic limit cycles for such a piecewise linear system having a Hamiltonian sub-system and a non-Hamiltonian sub-system. We prove that the maximum number of type I is one and the lower and upper bounds of the maximum number of type II are five and seven, respectively, and show the coexistence of type I and type II, which implies that a lower bound for the maximum number of all crossing algebraic limit cycles is six.