On The Number Of Limit Cycles In A General Planar Piecewise Linear Differential Systems

In this paper,two types of planar piecewise smooth linear differential systems with two zones separated by a straight line(i.e.the discontinuous boundary)are considered.The subsystems corresponding to two zones of one type both have a virtual focus(the focus is not in the corresponding zone),and the subsystems corresponding to two zones of the other type both have a real focus(the focus is in the corresponding zone).By constructing Poincaré mapping,the number of zero points of the new function derived from the intersection of the trajectory of the linear subsystem in the phase space and the discontinuous boundary is analyzed,and the complete results of the existence and the number of crossing limit cycles(i.e.limit cycles without sliding motion)are obtained.We proved that there are at most two limit cycles in these two types of systems,and provide a complete parameter domain for the existence of one to two limit cycles.According to the main results of this paper and the existing results,we obtain a sufficient and necessary condition that a planar piecewise linear differential dynamical system with two focus subsystems has three limit cycles: there is only one real focus,that is,one subsystem has real focus and the other subsystem has virtual focus.In addition,we also obtain that the canonical form of this type of system contains at least five parameters.