Limit cycles in planar continuous piecewise linear systems

In this paper an asymmetric planar continuous piecewise linear differential system with three zones x˙=y−F(x),y˙=−g(x) is considered. The aim of this paper gives a completely study of limit cycles when this system satisfies such conditions and the uniqueness equilibrium does not lie in the central region. When (x−x0)g(x)>0 for ∀x ≠ x0 and y=F(x) is a Z-shaped curve, it owns at most two limit cycles, which exist between a linear Hopf bifurcation surface and a double limit cycle bifurcation surface. Moreover, we prove the conjectures proposed by Ponce et al. [27]. When the uniqueness equilibrium lies in the central region, this system has exactly one limit cycles by others. Finally, some numerical examples are demonstrated.