Limit Cycles in the Discontinuous Planar Piecewise Linear Systems with Three Zones

In this paper, we investigate the existence of limit cycles for the discontinuous planar piecewise linear systems with three zones separated by two parallel straight lines. Based on the methods of first integral and Poincaré map, we present the maximum number of limit cycles in the normal forms of systems with boundary focus-center-boundary focus and boundary focus-center-center types, respectively. Then we show that such discontinuous piecewise linear systems can have at most three limit cycles, being two of them of four intersection points type and the third one of two intersection points type.