A proof of Euzébio–Pazim–Ponce’s conjectures for a degenerate planar piecewise linear differential system with three zones

Abstract

In this paper we study bifurcations and dynamics in a planar piecewise linear differential system with three zones ẋ=F(x)−y,ẏ=g(x)−α. When the system is degenerate in the central zone, i.e., g′(x)=0 in the central zone, and F(x) is a flute linear function, Euzébio, Pazim and Ponce in Euzébio et al. (2016) proposed three conjectures on limit cycles. The aim of this paper is to prove Euzébio–Pazim–Ponce’s conjectures so that the number and the bifurcation of limit cycles of the degenerate planar piecewise linear differential system with three zones, i.e., under the same assumptions as in Euzébio et al. (2016), are studied completely. Finally, the bifurcation diagrams and the phase portraits of this planar piecewise linear differential system are given completely, including scabbard bifurcation, grazing bifurcation and double limit cycle bifurcation.